Sec 2 G1 Maths Tutor Bukit Timah

Reader note: All examples are original fictional learning charts, not actual Bukit Timah demographics, authentic survey results or copied examination items. A chart’s categories, key, units and sample size are essential to correct interpretation. Current tuition place, fees and exact course fit must be confirmed directly.

Bring a misread pie chart or pictogram worksheet to eduKate Singapore and ask how the tutor will test the learner’s understanding of the whole and the chart key.

Official syllabus and accuracy note

Bring a misread pie chart or pictogram worksheet to eduKate Singapore and ask how the tutor will test the learner’s understanding of the whole and the chart key.

Official syllabus and accuracy note

Bring a misread pie chart or pictogram worksheet to eduKate Singapore and ask how the tutor will test the learner’s understanding of the whole and the chart key.

Day one: change 90°,120°,180° into fractions of a circle.
Day two: find category counts using stated fictional survey totals.
Day three: reverse a pie sector to recover total sample size.
Day four: create a pictogram with a consistent symbol key and a valid half symbol.
Day five: explain a sample-size limitation or category labelling issue in one sentence.

Current G1 curriculum versus the SEC year

MOE’s 2020 G1 Mathematics content introduces tables, bar graphs, pictograms, line graphs and pie charts in the earlier lower-secondary stage and further data representations in Secondary Two. The official 2027 K110 G1 SEC syllabus includes these and other forms of Data Handling and Analysis. A pupil in Secondary Two in 2026 should follow the school’s current topic sequence and later the official examination specification for the pupil’s own graduating year.

Frequently asked questions

Does 90° mean 90% in a pie chart?

No. It is ninety of the full 360 degrees, or one quarter = 25%.

How do you find total observations from one sector?

Divide its category count by the fraction angle/360°.

Is a half icon half a person?

No. It is half of the key’s represented count, which should yield a valid count or be labelled approximate.

Can pie charts and histograms replace each other?

No. Pie charts show shares of one whole; histograms show distributions across numerical intervals.

Can equal pie sectors represent different respondent counts?

Yes, if the pies have different sample sizes.

What should a tutor check?

Can the pupil move independently among sector angles, fractions, percentages and absolute counts, and use a pictogram key accurately?

Official syllabus and accuracy note

Bring a misread pie chart or pictogram worksheet to eduKate Singapore and ask how the tutor will test the learner’s understanding of the whole and the chart key.

Day one: change 90°,120°,180° into fractions of a circle.
Day two: find category counts using stated fictional survey totals.
Day three: reverse a pie sector to recover total sample size.
Day four: create a pictogram with a consistent symbol key and a valid half symbol.
Day five: explain a sample-size limitation or category labelling issue in one sentence.

Current G1 curriculum versus the SEC year

Frequently asked questions

Does 90° mean 90% in a pie chart?

No. It is ninety of the full 360 degrees, or one quarter = 25%.

How do you find total observations from one sector?

Divide its category count by the fraction angle/360°.

Is a half icon half a person?

No. It is half of the key’s represented count, which should yield a valid count or be labelled approximate.

Can pie charts and histograms replace each other?

No. Pie charts show shares of one whole; histograms show distributions across numerical intervals.

Can equal pie sectors represent different respondent counts?

Yes, if the pies have different sample sizes.

What should a tutor check?

Can the pupil move independently among sector angles, fractions, percentages and absolute counts, and use a pictogram key accurately?

Official syllabus and accuracy note

Bring a misread pie chart or pictogram worksheet to eduKate Singapore and ask how the tutor will test the learner’s understanding of the whole and the chart key.

10 minutes: retrieve one-whole fractions and percentage equivalents.
20 minutes: draw pie sectors from simple fractions and counts.
25 minutes: individually solve sector-to-frequency and reverse-total questions.
15 minutes: count pictogram full and half-symbol equivalents.
10 minutes: identify one conclusion a chart does not justify.
10 minutes: correct the first mistake and retest with new data.

Five manageable home practice sessions

Day one: change 90°,120°,180° into fractions of a circle.
Day two: find category counts using stated fictional survey totals.
Day three: reverse a pie sector to recover total sample size.
Day four: create a pictogram with a consistent symbol key and a valid half symbol.
Day five: explain a sample-size limitation or category labelling issue in one sentence.

Current G1 curriculum versus the SEC year

Frequently asked questions

Does 90° mean 90% in a pie chart?

No. It is ninety of the full 360 degrees, or one quarter = 25%.

How do you find total observations from one sector?

Divide its category count by the fraction angle/360°.

Is a half icon half a person?

No. It is half of the key’s represented count, which should yield a valid count or be labelled approximate.

Can pie charts and histograms replace each other?

No. Pie charts show shares of one whole; histograms show distributions across numerical intervals.

Can equal pie sectors represent different respondent counts?

Yes, if the pies have different sample sizes.

What should a tutor check?

Can the pupil move independently among sector angles, fractions, percentages and absolute counts, and use a pictogram key accurately?

Official syllabus and accuracy note

Bring a misread pie chart or pictogram worksheet to eduKate Singapore and ask how the tutor will test the learner’s understanding of the whole and the chart key.

10 minutes: retrieve one-whole fractions and percentage equivalents.
20 minutes: draw pie sectors from simple fractions and counts.
25 minutes: individually solve sector-to-frequency and reverse-total questions.
15 minutes: count pictogram full and half-symbol equivalents.
10 minutes: identify one conclusion a chart does not justify.
10 minutes: correct the first mistake and retest with new data.

Five manageable home practice sessions

Day one: change 90°,120°,180° into fractions of a circle.
Day two: find category counts using stated fictional survey totals.
Day three: reverse a pie sector to recover total sample size.
Day four: create a pictogram with a consistent symbol key and a valid half symbol.
Day five: explain a sample-size limitation or category labelling issue in one sentence.

Current G1 curriculum versus the SEC year

Frequently asked questions

Does 90° mean 90% in a pie chart?

No. It is ninety of the full 360 degrees, or one quarter = 25%.

How do you find total observations from one sector?

Divide its category count by the fraction angle/360°.

Is a half icon half a person?

No. It is half of the key’s represented count, which should yield a valid count or be labelled approximate.

Can pie charts and histograms replace each other?

No. Pie charts show shares of one whole; histograms show distributions across numerical intervals.

Can equal pie sectors represent different respondent counts?

Yes, if the pies have different sample sizes.

What should a tutor check?

Can the pupil move independently among sector angles, fractions, percentages and absolute counts, and use a pictogram key accurately?

Official syllabus and accuracy note

Bring a misread pie chart or pictogram worksheet to eduKate Singapore and ask how the tutor will test the learner’s understanding of the whole and the chart key.

One-minute diagnostic
How a sector represents a fraction
How to reverse a pie calculation
Reading the pictogram key
What charts do not prove
Original chart clinics
First mistakes to repair
Three-pupil tuition model
Parent revision plan
FAQs
Official sources

A one-minute chart diagnostic

Ask the pupil: what percentage is 90° of a circle, how many of forty responses are in that sector, and how many objects do three icons represent if the key says each icon equals five? Answers are 25%, ten responses and fifteen objects. A wrong first answer suggests confusion between degrees and percent; a wrong second suggests a missing whole; a wrong third suggests the key was ignored. They need different teaching responses.

The 360-degree circle is a mathematical whole

One complete pie is 360°. A sector of 72° is 72/360=1/5=20% of the sample. If the chart describes sixty responses, this is twelve responses. Conversely, if twelve of sixty responses belong to a category, its angle is twelve divided by sixty times 360°, or 72°. The two calculations should reverse each other. A student who knows both directions has a better chance of checking unfamiliar questions.

Two identical sectors can represent different counts

A half-circle sector in a sample of twenty people represents ten; in a sample of one hundred it represents fifty. The sector share is 50% in both. A reader who claims the count must be equal is ignoring the size of each sample. State the sample size beside any absolute frequency claim and never infer wider population behaviour without sampling evidence.

How to recover the sample total

If a category occupies 60° and contains ten responses, the sector fraction is 60/360=1/6. Divide ten by one sixth to get total sixty. Multiplying ten by one sixth gives a smaller part and therefore cannot produce the whole from the supplied category. This is the same part–whole logic students use in reverse percentages and ratios.

An icon key is a unit conversion

If one picture symbol means twelve books, two full icons and a half-icon represent 12+12+6=30 books. The drawing uses two-and-a-half symbols, but the frequency is thirty. Reading a pictogram is therefore a multiplication with units rather than just visual counting. For indivisible people, select a key that allows whole-person counts for any partial symbols; a half of five people is not an exact person count.

What a chart does not tell you

A pie sector labelled Other may contain several separate categories whose individual sizes are unknown. A pictogram with an unclear key cannot support reliable counts. A fictional poll of ten students describes the ten respondents, not everyone in Singapore. Knowing the mathematical calculation is only one part of good data literacy: the conclusion must remain within what was actually measured.

124 original pie-chart and pictogram clinics

All counts and category labels below are hypothetical learning examples. They include converting angle to count, reversing the calculation, building angles from frequencies, pictogram keys, partial icons and chart interpretation. Use a few at a time. Cover the worked answer first and state the full sample size or key aloud before any arithmetic.

1. Pie chart 1: sector to category frequency

Question: A fictional pie chart shows 16 observations. A category occupies 180 degrees of the full circle. Find how many observations belong to it.

Work it out: Share is 180/360=1/2. Multiply by total 16: 8 observations. This also represents 50% of the whole.

Common first error: Treating 180 degrees as 180% or as 180 individual observations confuses angle, proportion and frequency. A sector does not have an independent count without the total.

Independent check: The reverse calculation (8/16)×360°=180°. The category count is no more than the full sample 16.

Transfer: If the same share applied to 32 responses, the count would double to 16, but the angle would remain 180°.

2. Pie chart 1: recover the complete sample

Question: An imaginary category has 8 responses and occupies 180° of a pie chart representing every recorded response. Find the whole sample size.

Work it out: The category is 1/2 of its whole. Divide the count 8 by 1/2 to recover 16 responses.

Common first error: Multiplying the category count by 1/2 instead would yield less than the category and cannot represent the total population being counted.

Independent check: 8/16=1/2, and (8/16)×360°=180°, matching the supplied angle.

Transfer: Ask what happens to the total if the same category frequency instead occupies a smaller sector angle. The total must be larger if the recorded count stays fixed.

3. Pie chart 1: frequency to angle

Question: In a fictional survey of 16, category A has 8 observations. Work out the sector angle for A.

Work it out: Sector angle=(part/whole)×360° = (8/16)×360° = 180°. The fraction is 1/2 and share approximately 50%.

Common first error: Reversing total and count gives a fraction greater than one, which could produce an impossible sector larger than a full 360° circle.

Independent check: The rest of the pie occupies 180° and corresponds to 8 other observations. Together they reconstruct the whole.

Transfer: If another category has the same frequency, it must receive the same sector angle within this pie, even if its label is different.

4. Pie chart 2: sector to category frequency

Question: A fictional pie chart shows 27 observations. A category occupies 120 degrees of the full circle. Find how many observations belong to it.

Work it out: Share is 120/360=1/3. Multiply by total 27: 9 observations. This also represents 33.333% of the whole.

Common first error: Treating 120 degrees as 120% or as 120 individual observations confuses angle, proportion and frequency. A sector does not have an independent count without the total.

Independent check: The reverse calculation (9/27)×360°=120°. The category count is no more than the full sample 27.

Transfer: If the same share applied to 54 responses, the count would double to 18, but the angle would remain 120°.

5. Pie chart 2: recover the complete sample

Question: An imaginary category has 9 responses and occupies 120° of a pie chart representing every recorded response. Find the whole sample size.

Work it out: The category is 1/3 of its whole. Divide the count 9 by 1/3 to recover 27 responses.

Common first error: Multiplying the category count by 1/3 instead would yield less than the category and cannot represent the total population being counted.

Independent check: 9/27=1/3, and (9/27)×360°=120°, matching the supplied angle.

Transfer: Ask what happens to the total if the same category frequency instead occupies a smaller sector angle. The total must be larger if the recorded count stays fixed.

6. Pie chart 2: frequency to angle

Question: In a fictional survey of 27, category A has 9 observations. Work out the sector angle for A.

Work it out: Sector angle=(part/whole)×360° = (9/27)×360° = 120°. The fraction is 1/3 and share approximately 33.333%.

Common first error: Reversing total and count gives a fraction greater than one, which could produce an impossible sector larger than a full 360° circle.

Independent check: The rest of the pie occupies 240° and corresponds to 18 other observations. Together they reconstruct the whole.

Transfer: If another category has the same frequency, it must receive the same sector angle within this pie, even if its label is different.

7. Pie chart 3: sector to category frequency

Question: A fictional pie chart shows 40 observations. A category occupies 90 degrees of the full circle. Find how many observations belong to it.

Work it out: Share is 90/360=1/4. Multiply by total 40: 10 observations. This also represents 25% of the whole.

Common first error: Treating 90 degrees as 90% or as 90 individual observations confuses angle, proportion and frequency. A sector does not have an independent count without the total.

Independent check: The reverse calculation (10/40)×360°=90°. The category count is no more than the full sample 40.

Transfer: If the same share applied to 80 responses, the count would double to 20, but the angle would remain 90°.

8. Pie chart 3: recover the complete sample

Question: An imaginary category has 10 responses and occupies 90° of a pie chart representing every recorded response. Find the whole sample size.

Work it out: The category is 1/4 of its whole. Divide the count 10 by 1/4 to recover 40 responses.

Common first error: Multiplying the category count by 1/4 instead would yield less than the category and cannot represent the total population being counted.

Independent check: 10/40=1/4, and (10/40)×360°=90°, matching the supplied angle.

Transfer: Ask what happens to the total if the same category frequency instead occupies a smaller sector angle. The total must be larger if the recorded count stays fixed.

9. Pie chart 3: frequency to angle

Question: In a fictional survey of 40, category A has 10 observations. Work out the sector angle for A.

Work it out: Sector angle=(part/whole)×360° = (10/40)×360° = 90°. The fraction is 1/4 and share approximately 25%.

Common first error: Reversing total and count gives a fraction greater than one, which could produce an impossible sector larger than a full 360° circle.

Independent check: The rest of the pie occupies 270° and corresponds to 30 other observations. Together they reconstruct the whole.

Transfer: If another category has the same frequency, it must receive the same sector angle within this pie, even if its label is different.

10. Pie chart 4: sector to category frequency

Question: A fictional pie chart shows 55 observations. A category occupies 72 degrees of the full circle. Find how many observations belong to it.

Work it out: Share is 72/360=1/5. Multiply by total 55: 11 observations. This also represents 20% of the whole.

Common first error: Treating 72 degrees as 72% or as 72 individual observations confuses angle, proportion and frequency. A sector does not have an independent count without the total.

Independent check: The reverse calculation (11/55)×360°=72°. The category count is no more than the full sample 55.

Transfer: If the same share applied to 110 responses, the count would double to 22, but the angle would remain 72°.

11. Pie chart 4: recover the complete sample

Question: An imaginary category has 11 responses and occupies 72° of a pie chart representing every recorded response. Find the whole sample size.

Work it out: The category is 1/5 of its whole. Divide the count 11 by 1/5 to recover 55 responses.

Common first error: Multiplying the category count by 1/5 instead would yield less than the category and cannot represent the total population being counted.

Independent check: 11/55=1/5, and (11/55)×360°=72°, matching the supplied angle.

Transfer: Ask what happens to the total if the same category frequency instead occupies a smaller sector angle. The total must be larger if the recorded count stays fixed.

12. Pie chart 4: frequency to angle

Question: In a fictional survey of 55, category A has 11 observations. Work out the sector angle for A.

Work it out: Sector angle=(part/whole)×360° = (11/55)×360° = 72°. The fraction is 1/5 and share approximately 20%.

Common first error: Reversing total and count gives a fraction greater than one, which could produce an impossible sector larger than a full 360° circle.

Independent check: The rest of the pie occupies 288° and corresponds to 44 other observations. Together they reconstruct the whole.

Transfer: If another category has the same frequency, it must receive the same sector angle within this pie, even if its label is different.

13. Pie chart 5: sector to category frequency

Question: A fictional pie chart shows 72 observations. A category occupies 60 degrees of the full circle. Find how many observations belong to it.

Work it out: Share is 60/360=1/6. Multiply by total 72: 12 observations. This also represents 16.667% of the whole.

Common first error: Treating 60 degrees as 60% or as 60 individual observations confuses angle, proportion and frequency. A sector does not have an independent count without the total.

Independent check: The reverse calculation (12/72)×360°=60°. The category count is no more than the full sample 72.

Transfer: If the same share applied to 144 responses, the count would double to 24, but the angle would remain 60°.

14. Pie chart 5: recover the complete sample

Question: An imaginary category has 12 responses and occupies 60° of a pie chart representing every recorded response. Find the whole sample size.

Work it out: The category is 1/6 of its whole. Divide the count 12 by 1/6 to recover 72 responses.

Common first error: Multiplying the category count by 1/6 instead would yield less than the category and cannot represent the total population being counted.

Independent check: 12/72=1/6, and (12/72)×360°=60°, matching the supplied angle.

Transfer: Ask what happens to the total if the same category frequency instead occupies a smaller sector angle. The total must be larger if the recorded count stays fixed.

15. Pie chart 5: frequency to angle

Question: In a fictional survey of 72, category A has 12 observations. Work out the sector angle for A.

Work it out: Sector angle=(part/whole)×360° = (12/72)×360° = 60°. The fraction is 1/6 and share approximately 16.667%.

Common first error: Reversing total and count gives a fraction greater than one, which could produce an impossible sector larger than a full 360° circle.

Independent check: The rest of the pie occupies 300° and corresponds to 60 other observations. Together they reconstruct the whole.

Transfer: If another category has the same frequency, it must receive the same sector angle within this pie, even if its label is different.

16. Pie chart 6: sector to category frequency

Question: A fictional pie chart shows 65 observations. A category occupies 144 degrees of the full circle. Find how many observations belong to it.

Work it out: Share is 144/360=2/5. Multiply by total 65: 26 observations. This also represents 40% of the whole.

Common first error: Treating 144 degrees as 144% or as 144 individual observations confuses angle, proportion and frequency. A sector does not have an independent count without the total.

Independent check: The reverse calculation (26/65)×360°=144°. The category count is no more than the full sample 65.

Transfer: If the same share applied to 130 responses, the count would double to 52, but the angle would remain 144°.

17. Pie chart 6: recover the complete sample

Question: An imaginary category has 26 responses and occupies 144° of a pie chart representing every recorded response. Find the whole sample size.

Work it out: The category is 2/5 of its whole. Divide the count 26 by 2/5 to recover 65 responses.

Common first error: Multiplying the category count by 2/5 instead would yield less than the category and cannot represent the total population being counted.

Independent check: 26/65=2/5, and (26/65)×360°=144°, matching the supplied angle.

Transfer: Ask what happens to the total if the same category frequency instead occupies a smaller sector angle. The total must be larger if the recorded count stays fixed.

18. Pie chart 6: frequency to angle

Question: In a fictional survey of 65, category A has 26 observations. Work out the sector angle for A.

Work it out: Sector angle=(part/whole)×360° = (26/65)×360° = 144°. The fraction is 2/5 and share approximately 40%.

Common first error: Reversing total and count gives a fraction greater than one, which could produce an impossible sector larger than a full 360° circle.

Independent check: The rest of the pie occupies 216° and corresponds to 39 other observations. Together they reconstruct the whole.

Transfer: If another category has the same frequency, it must receive the same sector angle within this pie, even if its label is different.

19. Pie chart 7: sector to category frequency

Question: A fictional pie chart shows 70 observations. A category occupies 216 degrees of the full circle. Find how many observations belong to it.

Work it out: Share is 216/360=3/5. Multiply by total 70: 42 observations. This also represents 60% of the whole.

Common first error: Treating 216 degrees as 216% or as 216 individual observations confuses angle, proportion and frequency. A sector does not have an independent count without the total.

Independent check: The reverse calculation (42/70)×360°=216°. The category count is no more than the full sample 70.

Transfer: If the same share applied to 140 responses, the count would double to 84, but the angle would remain 216°.

20. Pie chart 7: recover the complete sample

Question: An imaginary category has 42 responses and occupies 216° of a pie chart representing every recorded response. Find the whole sample size.

Work it out: The category is 3/5 of its whole. Divide the count 42 by 3/5 to recover 70 responses.

Common first error: Multiplying the category count by 3/5 instead would yield less than the category and cannot represent the total population being counted.

Independent check: 42/70=3/5, and (42/70)×360°=216°, matching the supplied angle.

Transfer: Ask what happens to the total if the same category frequency instead occupies a smaller sector angle. The total must be larger if the recorded count stays fixed.

21. Pie chart 7: frequency to angle

Question: In a fictional survey of 70, category A has 42 observations. Work out the sector angle for A.

Work it out: Sector angle=(part/whole)×360° = (42/70)×360° = 216°. The fraction is 3/5 and share approximately 60%.

Common first error: Reversing total and count gives a fraction greater than one, which could produce an impossible sector larger than a full 360° circle.

Independent check: The rest of the pie occupies 144° and corresponds to 28 other observations. Together they reconstruct the whole.

Transfer: If another category has the same frequency, it must receive the same sector angle within this pie, even if its label is different.

22. Pie chart 8: sector to category frequency

Question: A fictional pie chart shows 45 observations. A category occupies 240 degrees of the full circle. Find how many observations belong to it.

Work it out: Share is 240/360=2/3. Multiply by total 45: 30 observations. This also represents 66.667% of the whole.

Common first error: Treating 240 degrees as 240% or as 240 individual observations confuses angle, proportion and frequency. A sector does not have an independent count without the total.

Independent check: The reverse calculation (30/45)×360°=240°. The category count is no more than the full sample 45.

Transfer: If the same share applied to 90 responses, the count would double to 60, but the angle would remain 240°.

23. Pie chart 8: recover the complete sample

Question: An imaginary category has 30 responses and occupies 240° of a pie chart representing every recorded response. Find the whole sample size.

Work it out: The category is 2/3 of its whole. Divide the count 30 by 2/3 to recover 45 responses.

Common first error: Multiplying the category count by 2/3 instead would yield less than the category and cannot represent the total population being counted.

Independent check: 30/45=2/3, and (30/45)×360°=240°, matching the supplied angle.

Transfer: Ask what happens to the total if the same category frequency instead occupies a smaller sector angle. The total must be larger if the recorded count stays fixed.

24. Pie chart 8: frequency to angle

Question: In a fictional survey of 45, category A has 30 observations. Work out the sector angle for A.

Work it out: Sector angle=(part/whole)×360° = (30/45)×360° = 240°. The fraction is 2/3 and share approximately 66.667%.

Common first error: Reversing total and count gives a fraction greater than one, which could produce an impossible sector larger than a full 360° circle.

Independent check: The rest of the pie occupies 120° and corresponds to 15 other observations. Together they reconstruct the whole.

Transfer: If another category has the same frequency, it must receive the same sector angle within this pie, even if its label is different.

25. Pie chart 9: sector to category frequency

Question: A fictional pie chart shows 64 observations. A category occupies 270 degrees of the full circle. Find how many observations belong to it.

Work it out: Share is 270/360=3/4. Multiply by total 64: 48 observations. This also represents 75% of the whole.

Common first error: Treating 270 degrees as 270% or as 270 individual observations confuses angle, proportion and frequency. A sector does not have an independent count without the total.

Independent check: The reverse calculation (48/64)×360°=270°. The category count is no more than the full sample 64.

Transfer: If the same share applied to 128 responses, the count would double to 96, but the angle would remain 270°.

26. Pie chart 9: recover the complete sample

Question: An imaginary category has 48 responses and occupies 270° of a pie chart representing every recorded response. Find the whole sample size.

Work it out: The category is 3/4 of its whole. Divide the count 48 by 3/4 to recover 64 responses.

Common first error: Multiplying the category count by 3/4 instead would yield less than the category and cannot represent the total population being counted.

Independent check: 48/64=3/4, and (48/64)×360°=270°, matching the supplied angle.

Transfer: Ask what happens to the total if the same category frequency instead occupies a smaller sector angle. The total must be larger if the recorded count stays fixed.

27. Pie chart 9: frequency to angle

Question: In a fictional survey of 64, category A has 48 observations. Work out the sector angle for A.

Work it out: Sector angle=(part/whole)×360° = (48/64)×360° = 270°. The fraction is 3/4 and share approximately 75%.

Common first error: Reversing total and count gives a fraction greater than one, which could produce an impossible sector larger than a full 360° circle.

Independent check: The rest of the pie occupies 90° and corresponds to 16 other observations. Together they reconstruct the whole.

Transfer: If another category has the same frequency, it must receive the same sector angle within this pie, even if its label is different.

28. Pie chart 10: sector to category frequency

Question: A fictional pie chart shows 136 observations. A category occupies 45 degrees of the full circle. Find how many observations belong to it.

Work it out: Share is 45/360=1/8. Multiply by total 136: 17 observations. This also represents 12.5% of the whole.

Common first error: Treating 45 degrees as 45% or as 45 individual observations confuses angle, proportion and frequency. A sector does not have an independent count without the total.

Independent check: The reverse calculation (17/136)×360°=45°. The category count is no more than the full sample 136.

Transfer: If the same share applied to 272 responses, the count would double to 34, but the angle would remain 45°.

29. Pie chart 10: recover the complete sample

Question: An imaginary category has 17 responses and occupies 45° of a pie chart representing every recorded response. Find the whole sample size.

Work it out: The category is 1/8 of its whole. Divide the count 17 by 1/8 to recover 136 responses.

Common first error: Multiplying the category count by 1/8 instead would yield less than the category and cannot represent the total population being counted.

Independent check: 17/136=1/8, and (17/136)×360°=45°, matching the supplied angle.

Transfer: Ask what happens to the total if the same category frequency instead occupies a smaller sector angle. The total must be larger if the recorded count stays fixed.

30. Pie chart 10: frequency to angle

Question: In a fictional survey of 136, category A has 17 observations. Work out the sector angle for A.

Work it out: Sector angle=(part/whole)×360° = (17/136)×360° = 45°. The fraction is 1/8 and share approximately 12.5%.

Common first error: Reversing total and count gives a fraction greater than one, which could produce an impossible sector larger than a full 360° circle.

Independent check: The rest of the pie occupies 315° and corresponds to 119 other observations. Together they reconstruct the whole.

Transfer: If another category has the same frequency, it must receive the same sector angle within this pie, even if its label is different.

31. Pie chart 11: sector to category frequency

Question: A fictional pie chart shows 144 observations. A category occupies 135 degrees of the full circle. Find how many observations belong to it.

Work it out: Share is 135/360=3/8. Multiply by total 144: 54 observations. This also represents 37.5% of the whole.

Common first error: Treating 135 degrees as 135% or as 135 individual observations confuses angle, proportion and frequency. A sector does not have an independent count without the total.

Independent check: The reverse calculation (54/144)×360°=135°. The category count is no more than the full sample 144.

Transfer: If the same share applied to 288 responses, the count would double to 108, but the angle would remain 135°.

32. Pie chart 11: recover the complete sample

Question: An imaginary category has 54 responses and occupies 135° of a pie chart representing every recorded response. Find the whole sample size.

Work it out: The category is 3/8 of its whole. Divide the count 54 by 3/8 to recover 144 responses.

Common first error: Multiplying the category count by 3/8 instead would yield less than the category and cannot represent the total population being counted.

Independent check: 54/144=3/8, and (54/144)×360°=135°, matching the supplied angle.

Transfer: Ask what happens to the total if the same category frequency instead occupies a smaller sector angle. The total must be larger if the recorded count stays fixed.

33. Pie chart 11: frequency to angle

Question: In a fictional survey of 144, category A has 54 observations. Work out the sector angle for A.

Work it out: Sector angle=(part/whole)×360° = (54/144)×360° = 135°. The fraction is 3/8 and share approximately 37.5%.

Common first error: Reversing total and count gives a fraction greater than one, which could produce an impossible sector larger than a full 360° circle.

Independent check: The rest of the pie occupies 225° and corresponds to 90 other observations. Together they reconstruct the whole.

Transfer: If another category has the same frequency, it must receive the same sector angle within this pie, even if its label is different.

34. Pie chart 12: sector to category frequency

Question: A fictional pie chart shows 228 observations. A category occupies 150 degrees of the full circle. Find how many observations belong to it.

Work it out: Share is 150/360=5/12. Multiply by total 228: 95 observations. This also represents 41.667% of the whole.

Common first error: Treating 150 degrees as 150% or as 150 individual observations confuses angle, proportion and frequency. A sector does not have an independent count without the total.

Independent check: The reverse calculation (95/228)×360°=150°. The category count is no more than the full sample 228.

Transfer: If the same share applied to 456 responses, the count would double to 190, but the angle would remain 150°.

35. Pie chart 12: recover the complete sample

Question: An imaginary category has 95 responses and occupies 150° of a pie chart representing every recorded response. Find the whole sample size.

Work it out: The category is 5/12 of its whole. Divide the count 95 by 5/12 to recover 228 responses.

Common first error: Multiplying the category count by 5/12 instead would yield less than the category and cannot represent the total population being counted.

Independent check: 95/228=5/12, and (95/228)×360°=150°, matching the supplied angle.

Transfer: Ask what happens to the total if the same category frequency instead occupies a smaller sector angle. The total must be larger if the recorded count stays fixed.

36. Pie chart 12: frequency to angle

Question: In a fictional survey of 228, category A has 95 observations. Work out the sector angle for A.

Work it out: Sector angle=(part/whole)×360° = (95/228)×360° = 150°. The fraction is 5/12 and share approximately 41.667%.

Common first error: Reversing total and count gives a fraction greater than one, which could produce an impossible sector larger than a full 360° circle.

Independent check: The rest of the pie occupies 210° and corresponds to 133 other observations. Together they reconstruct the whole.

Transfer: If another category has the same frequency, it must receive the same sector angle within this pie, even if its label is different.

37. Pie chart 13: sector to category frequency

Question: A fictional pie chart shows 240 observations. A category occupies 210 degrees of the full circle. Find how many observations belong to it.

Work it out: Share is 210/360=7/12. Multiply by total 240: 140 observations. This also represents 58.333% of the whole.

Common first error: Treating 210 degrees as 210% or as 210 individual observations confuses angle, proportion and frequency. A sector does not have an independent count without the total.

Independent check: The reverse calculation (140/240)×360°=210°. The category count is no more than the full sample 240.

Transfer: If the same share applied to 480 responses, the count would double to 280, but the angle would remain 210°.

38. Pie chart 13: recover the complete sample

Question: An imaginary category has 140 responses and occupies 210° of a pie chart representing every recorded response. Find the whole sample size.

Work it out: The category is 7/12 of its whole. Divide the count 140 by 7/12 to recover 240 responses.

Common first error: Multiplying the category count by 7/12 instead would yield less than the category and cannot represent the total population being counted.

Independent check: 140/240=7/12, and (140/240)×360°=210°, matching the supplied angle.

Transfer: Ask what happens to the total if the same category frequency instead occupies a smaller sector angle. The total must be larger if the recorded count stays fixed.

39. Pie chart 13: frequency to angle

Question: In a fictional survey of 240, category A has 140 observations. Work out the sector angle for A.

Work it out: Sector angle=(part/whole)×360° = (140/240)×360° = 210°. The fraction is 7/12 and share approximately 58.333%.

Common first error: Reversing total and count gives a fraction greater than one, which could produce an impossible sector larger than a full 360° circle.

Independent check: The rest of the pie occupies 150° and corresponds to 100 other observations. Together they reconstruct the whole.

Transfer: If another category has the same frequency, it must receive the same sector angle within this pie, even if its label is different.

40. Pie chart 14: sector to category frequency

Question: A fictional pie chart shows 80 observations. A category occupies 36 degrees of the full circle. Find how many observations belong to it.

Work it out: Share is 36/360=1/10. Multiply by total 80: 8 observations. This also represents 10% of the whole.

Common first error: Treating 36 degrees as 36% or as 36 individual observations confuses angle, proportion and frequency. A sector does not have an independent count without the total.

Independent check: The reverse calculation (8/80)×360°=36°. The category count is no more than the full sample 80.

Transfer: If the same share applied to 160 responses, the count would double to 16, but the angle would remain 36°.

41. Pie chart 14: recover the complete sample

Question: An imaginary category has 8 responses and occupies 36° of a pie chart representing every recorded response. Find the whole sample size.

Work it out: The category is 1/10 of its whole. Divide the count 8 by 1/10 to recover 80 responses.

Common first error: Multiplying the category count by 1/10 instead would yield less than the category and cannot represent the total population being counted.

Independent check: 8/80=1/10, and (8/80)×360°=36°, matching the supplied angle.

Transfer: Ask what happens to the total if the same category frequency instead occupies a smaller sector angle. The total must be larger if the recorded count stays fixed.

42. Pie chart 14: frequency to angle

Question: In a fictional survey of 80, category A has 8 observations. Work out the sector angle for A.

Work it out: Sector angle=(part/whole)×360° = (8/80)×360° = 36°. The fraction is 1/10 and share approximately 10%.

Common first error: Reversing total and count gives a fraction greater than one, which could produce an impossible sector larger than a full 360° circle.

Independent check: The rest of the pie occupies 324° and corresponds to 72 other observations. Together they reconstruct the whole.

Transfer: If another category has the same frequency, it must receive the same sector angle within this pie, even if its label is different.

43. Pie chart 15: sector to category frequency

Question: A fictional pie chart shows 90 observations. A category occupies 108 degrees of the full circle. Find how many observations belong to it.

Work it out: Share is 108/360=3/10. Multiply by total 90: 27 observations. This also represents 30% of the whole.

Common first error: Treating 108 degrees as 108% or as 108 individual observations confuses angle, proportion and frequency. A sector does not have an independent count without the total.

Independent check: The reverse calculation (27/90)×360°=108°. The category count is no more than the full sample 90.

Transfer: If the same share applied to 180 responses, the count would double to 54, but the angle would remain 108°.

44. Pie chart 15: recover the complete sample

Question: An imaginary category has 27 responses and occupies 108° of a pie chart representing every recorded response. Find the whole sample size.

Work it out: The category is 3/10 of its whole. Divide the count 27 by 3/10 to recover 90 responses.

Common first error: Multiplying the category count by 3/10 instead would yield less than the category and cannot represent the total population being counted.

Independent check: 27/90=3/10, and (27/90)×360°=108°, matching the supplied angle.

Transfer: Ask what happens to the total if the same category frequency instead occupies a smaller sector angle. The total must be larger if the recorded count stays fixed.

45. Pie chart 15: frequency to angle

Question: In a fictional survey of 90, category A has 27 observations. Work out the sector angle for A.

Work it out: Sector angle=(part/whole)×360° = (27/90)×360° = 108°. The fraction is 3/10 and share approximately 30%.

Common first error: Reversing total and count gives a fraction greater than one, which could produce an impossible sector larger than a full 360° circle.

Independent check: The rest of the pie occupies 252° and corresponds to 63 other observations. Together they reconstruct the whole.

Transfer: If another category has the same frequency, it must receive the same sector angle within this pie, even if its label is different.

46. Pie chart 16: sector to category frequency

Question: A fictional pie chart shows 90 observations. A category occupies 80 degrees of the full circle. Find how many observations belong to it.

Work it out: Share is 80/360=2/9. Multiply by total 90: 20 observations. This also represents 22.222% of the whole.

Common first error: Treating 80 degrees as 80% or as 80 individual observations confuses angle, proportion and frequency. A sector does not have an independent count without the total.

Independent check: The reverse calculation (20/90)×360°=80°. The category count is no more than the full sample 90.

Transfer: If the same share applied to 180 responses, the count would double to 40, but the angle would remain 80°.

47. Pie chart 16: recover the complete sample

Question: An imaginary category has 20 responses and occupies 80° of a pie chart representing every recorded response. Find the whole sample size.

Work it out: The category is 2/9 of its whole. Divide the count 20 by 2/9 to recover 90 responses.

Common first error: Multiplying the category count by 2/9 instead would yield less than the category and cannot represent the total population being counted.

Independent check: 20/90=2/9, and (20/90)×360°=80°, matching the supplied angle.

Transfer: Ask what happens to the total if the same category frequency instead occupies a smaller sector angle. The total must be larger if the recorded count stays fixed.

48. Pie chart 16: frequency to angle

Question: In a fictional survey of 90, category A has 20 observations. Work out the sector angle for A.

Work it out: Sector angle=(part/whole)×360° = (20/90)×360° = 80°. The fraction is 2/9 and share approximately 22.222%.

Common first error: Reversing total and count gives a fraction greater than one, which could produce an impossible sector larger than a full 360° circle.

Independent check: The rest of the pie occupies 280° and corresponds to 70 other observations. Together they reconstruct the whole.

Transfer: If another category has the same frequency, it must receive the same sector angle within this pie, even if its label is different.

49. Pie chart 17: sector to category frequency

Question: A fictional pie chart shows 99 observations. A category occupies 160 degrees of the full circle. Find how many observations belong to it.

Work it out: Share is 160/360=4/9. Multiply by total 99: 44 observations. This also represents 44.444% of the whole.

Common first error: Treating 160 degrees as 160% or as 160 individual observations confuses angle, proportion and frequency. A sector does not have an independent count without the total.

Independent check: The reverse calculation (44/99)×360°=160°. The category count is no more than the full sample 99.

Transfer: If the same share applied to 198 responses, the count would double to 88, but the angle would remain 160°.

50. Pie chart 17: recover the complete sample

Question: An imaginary category has 44 responses and occupies 160° of a pie chart representing every recorded response. Find the whole sample size.

Work it out: The category is 4/9 of its whole. Divide the count 44 by 4/9 to recover 99 responses.

Common first error: Multiplying the category count by 4/9 instead would yield less than the category and cannot represent the total population being counted.

Independent check: 44/99=4/9, and (44/99)×360°=160°, matching the supplied angle.

Transfer: Ask what happens to the total if the same category frequency instead occupies a smaller sector angle. The total must be larger if the recorded count stays fixed.

51. Pie chart 17: frequency to angle

Question: In a fictional survey of 99, category A has 44 observations. Work out the sector angle for A.

Work it out: Sector angle=(part/whole)×360° = (44/99)×360° = 160°. The fraction is 4/9 and share approximately 44.444%.

Common first error: Reversing total and count gives a fraction greater than one, which could produce an impossible sector larger than a full 360° circle.

Independent check: The rest of the pie occupies 200° and corresponds to 55 other observations. Together they reconstruct the whole.

Transfer: If another category has the same frequency, it must receive the same sector angle within this pie, even if its label is different.

52. Pie chart 18: sector to category frequency

Question: A fictional pie chart shows 108 observations. A category occupies 200 degrees of the full circle. Find how many observations belong to it.

Work it out: Share is 200/360=5/9. Multiply by total 108: 60 observations. This also represents 55.556% of the whole.

Common first error: Treating 200 degrees as 200% or as 200 individual observations confuses angle, proportion and frequency. A sector does not have an independent count without the total.

Independent check: The reverse calculation (60/108)×360°=200°. The category count is no more than the full sample 108.

Transfer: If the same share applied to 216 responses, the count would double to 120, but the angle would remain 200°.

53. Pie chart 18: recover the complete sample

Question: An imaginary category has 60 responses and occupies 200° of a pie chart representing every recorded response. Find the whole sample size.

Work it out: The category is 5/9 of its whole. Divide the count 60 by 5/9 to recover 108 responses.

Common first error: Multiplying the category count by 5/9 instead would yield less than the category and cannot represent the total population being counted.

Independent check: 60/108=5/9, and (60/108)×360°=200°, matching the supplied angle.

Transfer: Ask what happens to the total if the same category frequency instead occupies a smaller sector angle. The total must be larger if the recorded count stays fixed.

54. Pie chart 18: frequency to angle

Question: In a fictional survey of 108, category A has 60 observations. Work out the sector angle for A.

Work it out: Sector angle=(part/whole)×360° = (60/108)×360° = 200°. The fraction is 5/9 and share approximately 55.556%.

Common first error: Reversing total and count gives a fraction greater than one, which could produce an impossible sector larger than a full 360° circle.

Independent check: The rest of the pie occupies 160° and corresponds to 48 other observations. Together they reconstruct the whole.

Transfer: If another category has the same frequency, it must receive the same sector angle within this pie, even if its label is different.

55. Pie chart 19: sector to category frequency

Question: A fictional pie chart shows 78 observations. A category occupies 300 degrees of the full circle. Find how many observations belong to it.

Work it out: Share is 300/360=5/6. Multiply by total 78: 65 observations. This also represents 83.333% of the whole.

Common first error: Treating 300 degrees as 300% or as 300 individual observations confuses angle, proportion and frequency. A sector does not have an independent count without the total.

Independent check: The reverse calculation (65/78)×360°=300°. The category count is no more than the full sample 78.

Transfer: If the same share applied to 156 responses, the count would double to 130, but the angle would remain 300°.

56. Pie chart 19: recover the complete sample

Question: An imaginary category has 65 responses and occupies 300° of a pie chart representing every recorded response. Find the whole sample size.

Work it out: The category is 5/6 of its whole. Divide the count 65 by 5/6 to recover 78 responses.

Common first error: Multiplying the category count by 5/6 instead would yield less than the category and cannot represent the total population being counted.

Independent check: 65/78=5/6, and (65/78)×360°=300°, matching the supplied angle.

Transfer: Ask what happens to the total if the same category frequency instead occupies a smaller sector angle. The total must be larger if the recorded count stays fixed.

57. Pie chart 19: frequency to angle

Question: In a fictional survey of 78, category A has 65 observations. Work out the sector angle for A.

Work it out: Sector angle=(part/whole)×360° = (65/78)×360° = 300°. The fraction is 5/6 and share approximately 83.333%.

Common first error: Reversing total and count gives a fraction greater than one, which could produce an impossible sector larger than a full 360° circle.

Independent check: The rest of the pie occupies 60° and corresponds to 13 other observations. Together they reconstruct the whole.

Transfer: If another category has the same frequency, it must receive the same sector angle within this pie, even if its label is different.

58. Pie chart 20: sector to category frequency

Question: A fictional pie chart shows 112 observations. A category occupies 315 degrees of the full circle. Find how many observations belong to it.

Work it out: Share is 315/360=7/8. Multiply by total 112: 98 observations. This also represents 87.5% of the whole.

Common first error: Treating 315 degrees as 315% or as 315 individual observations confuses angle, proportion and frequency. A sector does not have an independent count without the total.

Independent check: The reverse calculation (98/112)×360°=315°. The category count is no more than the full sample 112.

Transfer: If the same share applied to 224 responses, the count would double to 196, but the angle would remain 315°.

59. Pie chart 20: recover the complete sample

Question: An imaginary category has 98 responses and occupies 315° of a pie chart representing every recorded response. Find the whole sample size.

Work it out: The category is 7/8 of its whole. Divide the count 98 by 7/8 to recover 112 responses.

Common first error: Multiplying the category count by 7/8 instead would yield less than the category and cannot represent the total population being counted.

Independent check: 98/112=7/8, and (98/112)×360°=315°, matching the supplied angle.

Transfer: Ask what happens to the total if the same category frequency instead occupies a smaller sector angle. The total must be larger if the recorded count stays fixed.

60. Pie chart 20: frequency to angle

Question: In a fictional survey of 112, category A has 98 observations. Work out the sector angle for A.

Work it out: Sector angle=(part/whole)×360° = (98/112)×360° = 315°. The fraction is 7/8 and share approximately 87.5%.

Common first error: Reversing total and count gives a fraction greater than one, which could produce an impossible sector larger than a full 360° circle.

Independent check: The rest of the pie occupies 45° and corresponds to 14 other observations. Together they reconstruct the whole.

Transfer: If another category has the same frequency, it must receive the same sector angle within this pie, even if its label is different.

61. Pie chart 21: sector to category frequency

Question: A fictional pie chart shows 180 observations. A category occupies 30 degrees of the full circle. Find how many observations belong to it.

Work it out: Share is 30/360=1/12. Multiply by total 180: 15 observations. This also represents 8.333% of the whole.

Common first error: Treating 30 degrees as 30% or as 30 individual observations confuses angle, proportion and frequency. A sector does not have an independent count without the total.

Independent check: The reverse calculation (15/180)×360°=30°. The category count is no more than the full sample 180.

Transfer: If the same share applied to 360 responses, the count would double to 30, but the angle would remain 30°.

62. Pie chart 21: recover the complete sample

Question: An imaginary category has 15 responses and occupies 30° of a pie chart representing every recorded response. Find the whole sample size.

Work it out: The category is 1/12 of its whole. Divide the count 15 by 1/12 to recover 180 responses.

Common first error: Multiplying the category count by 1/12 instead would yield less than the category and cannot represent the total population being counted.

Independent check: 15/180=1/12, and (15/180)×360°=30°, matching the supplied angle.

Transfer: Ask what happens to the total if the same category frequency instead occupies a smaller sector angle. The total must be larger if the recorded count stays fixed.

63. Pie chart 21: frequency to angle

Question: In a fictional survey of 180, category A has 15 observations. Work out the sector angle for A.

Work it out: Sector angle=(part/whole)×360° = (15/180)×360° = 30°. The fraction is 1/12 and share approximately 8.333%.

Common first error: Reversing total and count gives a fraction greater than one, which could produce an impossible sector larger than a full 360° circle.

Independent check: The rest of the pie occupies 330° and corresponds to 165 other observations. Together they reconstruct the whole.

Transfer: If another category has the same frequency, it must receive the same sector angle within this pie, even if its label is different.

64. Pie chart 22: sector to category frequency

Question: A fictional pie chart shows 192 observations. A category occupies 330 degrees of the full circle. Find how many observations belong to it.

Work it out: Share is 330/360=11/12. Multiply by total 192: 176 observations. This also represents 91.667% of the whole.

Common first error: Treating 330 degrees as 330% or as 330 individual observations confuses angle, proportion and frequency. A sector does not have an independent count without the total.

Independent check: The reverse calculation (176/192)×360°=330°. The category count is no more than the full sample 192.

Transfer: If the same share applied to 384 responses, the count would double to 352, but the angle would remain 330°.

65. Pie chart 22: recover the complete sample

Question: An imaginary category has 176 responses and occupies 330° of a pie chart representing every recorded response. Find the whole sample size.

Work it out: The category is 11/12 of its whole. Divide the count 176 by 11/12 to recover 192 responses.

Common first error: Multiplying the category count by 11/12 instead would yield less than the category and cannot represent the total population being counted.

Independent check: 176/192=11/12, and (176/192)×360°=330°, matching the supplied angle.

Transfer: Ask what happens to the total if the same category frequency instead occupies a smaller sector angle. The total must be larger if the recorded count stays fixed.

66. Pie chart 22: frequency to angle

Question: In a fictional survey of 192, category A has 176 observations. Work out the sector angle for A.

Work it out: Sector angle=(part/whole)×360° = (176/192)×360° = 330°. The fraction is 11/12 and share approximately 91.667%.

Common first error: Reversing total and count gives a fraction greater than one, which could produce an impossible sector larger than a full 360° circle.

Independent check: The rest of the pie occupies 30° and corresponds to 16 other observations. Together they reconstruct the whole.

Transfer: If another category has the same frequency, it must receive the same sector angle within this pie, even if its label is different.

67. Pie chart 23: sector to category frequency

Question: A fictional pie chart shows 153 observations. A category occupies 40 degrees of the full circle. Find how many observations belong to it.

Work it out: Share is 40/360=1/9. Multiply by total 153: 17 observations. This also represents 11.111% of the whole.

Common first error: Treating 40 degrees as 40% or as 40 individual observations confuses angle, proportion and frequency. A sector does not have an independent count without the total.

Independent check: The reverse calculation (17/153)×360°=40°. The category count is no more than the full sample 153.

Transfer: If the same share applied to 306 responses, the count would double to 34, but the angle would remain 40°.

68. Pie chart 23: recover the complete sample

Question: An imaginary category has 17 responses and occupies 40° of a pie chart representing every recorded response. Find the whole sample size.

Work it out: The category is 1/9 of its whole. Divide the count 17 by 1/9 to recover 153 responses.

Common first error: Multiplying the category count by 1/9 instead would yield less than the category and cannot represent the total population being counted.

Independent check: 17/153=1/9, and (17/153)×360°=40°, matching the supplied angle.

Transfer: Ask what happens to the total if the same category frequency instead occupies a smaller sector angle. The total must be larger if the recorded count stays fixed.

69. Pie chart 23: frequency to angle

Question: In a fictional survey of 153, category A has 17 observations. Work out the sector angle for A.

Work it out: Sector angle=(part/whole)×360° = (17/153)×360° = 40°. The fraction is 1/9 and share approximately 11.111%.

Common first error: Reversing total and count gives a fraction greater than one, which could produce an impossible sector larger than a full 360° circle.

Independent check: The rest of the pie occupies 320° and corresponds to 136 other observations. Together they reconstruct the whole.

Transfer: If another category has the same frequency, it must receive the same sector angle within this pie, even if its label is different.

70. Pie chart 24: sector to category frequency

Question: A fictional pie chart shows 162 observations. A category occupies 280 degrees of the full circle. Find how many observations belong to it.

Work it out: Share is 280/360=7/9. Multiply by total 162: 126 observations. This also represents 77.778% of the whole.

Common first error: Treating 280 degrees as 280% or as 280 individual observations confuses angle, proportion and frequency. A sector does not have an independent count without the total.

Independent check: The reverse calculation (126/162)×360°=280°. The category count is no more than the full sample 162.

Transfer: If the same share applied to 324 responses, the count would double to 252, but the angle would remain 280°.

71. Pie chart 24: recover the complete sample

Question: An imaginary category has 126 responses and occupies 280° of a pie chart representing every recorded response. Find the whole sample size.

Work it out: The category is 7/9 of its whole. Divide the count 126 by 7/9 to recover 162 responses.

Common first error: Multiplying the category count by 7/9 instead would yield less than the category and cannot represent the total population being counted.

Independent check: 126/162=7/9, and (126/162)×360°=280°, matching the supplied angle.

Transfer: Ask what happens to the total if the same category frequency instead occupies a smaller sector angle. The total must be larger if the recorded count stays fixed.

72. Pie chart 24: frequency to angle

Question: In a fictional survey of 162, category A has 126 observations. Work out the sector angle for A.

Work it out: Sector angle=(part/whole)×360° = (126/162)×360° = 280°. The fraction is 7/9 and share approximately 77.778%.

Common first error: Reversing total and count gives a fraction greater than one, which could produce an impossible sector larger than a full 360° circle.

Independent check: The rest of the pie occupies 80° and corresponds to 36 other observations. Together they reconstruct the whole.

Transfer: If another category has the same frequency, it must receive the same sector angle within this pie, even if its label is different.

73. Pictogram 1: use the key to count

Question: A hypothetical pictogram key says each full symbol stands for 4 items. One category shows 2 full symbols and a half symbol. Find its represented frequency.

Work it out: A full symbol represents 4, so 2 symbols count 8. The half-symbol adds 2. The total is 10 items.

Common first error: Counting symbols rather than the key-scaled items gives only 2.5, not the represented frequency. A half symbol is half the key’s value, not automatically half an item.

Independent check: The icon-equivalent count is 2.5. Multiplying by 4 returns 10, the reported frequency.

Transfer: A second category has 1 full icons and represents 4. Compare the two by counts, not visual decoration. Their absolute difference is 6 items.

74. Pictogram 2: use the key to count

Question: A hypothetical pictogram key says each full symbol stands for 6 items. One category shows 5 full symbols. Find its represented frequency.

Work it out: A full symbol represents 6, so 5 symbols count 30. There is no additional fraction of an icon. The total is 30 items.

Common first error: Counting symbols rather than the key-scaled items gives only 5, not the represented frequency. A half symbol is half the key’s value, not automatically half an item.

Independent check: The icon-equivalent count is 5. Multiplying by 6 returns 30, the reported frequency.

Transfer: A second category has 6 full icons and represents 36. Compare the two by counts, not visual decoration. Their absolute difference is 6 items.

75. Pictogram 3: use the key to count

Question: A hypothetical pictogram key says each full symbol stands for 8 items. One category shows 2 full symbols. Find its represented frequency.

Work it out: A full symbol represents 8, so 2 symbols count 16. There is no additional fraction of an icon. The total is 16 items.

Common first error: Counting symbols rather than the key-scaled items gives only 2, not the represented frequency. A half symbol is half the key’s value, not automatically half an item.

Independent check: The icon-equivalent count is 2. Multiplying by 8 returns 16, the reported frequency.

Transfer: A second category has 4 full icons and represents 32. Compare the two by counts, not visual decoration. Their absolute difference is 16 items.

76. Pictogram 4: use the key to count

Question: A hypothetical pictogram key says each full symbol stands for 10 items. One category shows 5 full symbols and a half symbol. Find its represented frequency.

Work it out: A full symbol represents 10, so 5 symbols count 50. The half-symbol adds 5. The total is 55 items.

Common first error: Counting symbols rather than the key-scaled items gives only 5.5, not the represented frequency. A half symbol is half the key’s value, not automatically half an item.

Independent check: The icon-equivalent count is 5.5. Multiplying by 10 returns 55, the reported frequency.

Transfer: A second category has 2 full icons and represents 20. Compare the two by counts, not visual decoration. Their absolute difference is 35 items.

77. Pictogram 5: use the key to count

Question: A hypothetical pictogram key says each full symbol stands for 12 items. One category shows 2 full symbols. Find its represented frequency.

Work it out: A full symbol represents 12, so 2 symbols count 24. There is no additional fraction of an icon. The total is 24 items.

Common first error: Counting symbols rather than the key-scaled items gives only 2, not the represented frequency. A half symbol is half the key’s value, not automatically half an item.

Independent check: The icon-equivalent count is 2. Multiplying by 12 returns 24, the reported frequency.

Transfer: A second category has 7 full icons and represents 84. Compare the two by counts, not visual decoration. Their absolute difference is 60 items.

78. Pictogram 6: use the key to count

Question: A hypothetical pictogram key says each full symbol stands for 14 items. One category shows 5 full symbols. Find its represented frequency.

Work it out: A full symbol represents 14, so 5 symbols count 70. There is no additional fraction of an icon. The total is 70 items.

Common first error: Counting symbols rather than the key-scaled items gives only 5, not the represented frequency. A half symbol is half the key’s value, not automatically half an item.

Independent check: The icon-equivalent count is 5. Multiplying by 14 returns 70, the reported frequency.

Transfer: A second category has 5 full icons and represents 70. Compare the two by counts, not visual decoration. Their absolute difference is 0 items.

79. Pictogram 7: use the key to count

Question: A hypothetical pictogram key says each full symbol stands for 16 items. One category shows 2 full symbols and a half symbol. Find its represented frequency.

Work it out: A full symbol represents 16, so 2 symbols count 32. The half-symbol adds 8. The total is 40 items.

Common first error: Counting symbols rather than the key-scaled items gives only 2.5, not the represented frequency. A half symbol is half the key’s value, not automatically half an item.

Independent check: The icon-equivalent count is 2.5. Multiplying by 16 returns 40, the reported frequency.

Transfer: A second category has 3 full icons and represents 48. Compare the two by counts, not visual decoration. Their absolute difference is 8 items.

80. Pictogram 8: use the key to count

Question: A hypothetical pictogram key says each full symbol stands for 18 items. One category shows 5 full symbols. Find its represented frequency.

Work it out: A full symbol represents 18, so 5 symbols count 90. There is no additional fraction of an icon. The total is 90 items.

Common first error: Counting symbols rather than the key-scaled items gives only 5, not the represented frequency. A half symbol is half the key’s value, not automatically half an item.

Independent check: The icon-equivalent count is 5. Multiplying by 18 returns 90, the reported frequency.

Transfer: A second category has 1 full icons and represents 18. Compare the two by counts, not visual decoration. Their absolute difference is 72 items.

81. Pictogram 9: use the key to count

Question: A hypothetical pictogram key says each full symbol stands for 20 items. One category shows 2 full symbols. Find its represented frequency.

Work it out: A full symbol represents 20, so 2 symbols count 40. There is no additional fraction of an icon. The total is 40 items.

Common first error: Counting symbols rather than the key-scaled items gives only 2, not the represented frequency. A half symbol is half the key’s value, not automatically half an item.

Independent check: The icon-equivalent count is 2. Multiplying by 20 returns 40, the reported frequency.

Transfer: A second category has 6 full icons and represents 120. Compare the two by counts, not visual decoration. Their absolute difference is 80 items.

82. Pictogram 10: use the key to count

Question: A hypothetical pictogram key says each full symbol stands for 22 items. One category shows 5 full symbols and a half symbol. Find its represented frequency.

Work it out: A full symbol represents 22, so 5 symbols count 110. The half-symbol adds 11. The total is 121 items.

Common first error: Counting symbols rather than the key-scaled items gives only 5.5, not the represented frequency. A half symbol is half the key’s value, not automatically half an item.

Independent check: The icon-equivalent count is 5.5. Multiplying by 22 returns 121, the reported frequency.

Transfer: A second category has 4 full icons and represents 88. Compare the two by counts, not visual decoration. Their absolute difference is 33 items.

83. Pictogram 11: use the key to count

Question: A hypothetical pictogram key says each full symbol stands for 24 items. One category shows 2 full symbols. Find its represented frequency.

Work it out: A full symbol represents 24, so 2 symbols count 48. There is no additional fraction of an icon. The total is 48 items.

Common first error: Counting symbols rather than the key-scaled items gives only 2, not the represented frequency. A half symbol is half the key’s value, not automatically half an item.

Independent check: The icon-equivalent count is 2. Multiplying by 24 returns 48, the reported frequency.

Transfer: A second category has 2 full icons and represents 48. Compare the two by counts, not visual decoration. Their absolute difference is 0 items.

84. Pictogram 12: use the key to count

Question: A hypothetical pictogram key says each full symbol stands for 4 items. One category shows 5 full symbols. Find its represented frequency.

Work it out: A full symbol represents 4, so 5 symbols count 20. There is no additional fraction of an icon. The total is 20 items.

Common first error: Counting symbols rather than the key-scaled items gives only 5, not the represented frequency. A half symbol is half the key’s value, not automatically half an item.

Independent check: The icon-equivalent count is 5. Multiplying by 4 returns 20, the reported frequency.

Transfer: A second category has 7 full icons and represents 28. Compare the two by counts, not visual decoration. Their absolute difference is 8 items.

85. Pictogram 13: use the key to count

Question: A hypothetical pictogram key says each full symbol stands for 6 items. One category shows 2 full symbols and a half symbol. Find its represented frequency.

Work it out: A full symbol represents 6, so 2 symbols count 12. The half-symbol adds 3. The total is 15 items.

Common first error: Counting symbols rather than the key-scaled items gives only 2.5, not the represented frequency. A half symbol is half the key’s value, not automatically half an item.

Independent check: The icon-equivalent count is 2.5. Multiplying by 6 returns 15, the reported frequency.

Transfer: A second category has 5 full icons and represents 30. Compare the two by counts, not visual decoration. Their absolute difference is 15 items.

86. Pictogram 14: use the key to count

Question: A hypothetical pictogram key says each full symbol stands for 8 items. One category shows 5 full symbols. Find its represented frequency.

Work it out: A full symbol represents 8, so 5 symbols count 40. There is no additional fraction of an icon. The total is 40 items.

Common first error: Counting symbols rather than the key-scaled items gives only 5, not the represented frequency. A half symbol is half the key’s value, not automatically half an item.

Independent check: The icon-equivalent count is 5. Multiplying by 8 returns 40, the reported frequency.

Transfer: A second category has 3 full icons and represents 24. Compare the two by counts, not visual decoration. Their absolute difference is 16 items.

87. Pictogram 15: use the key to count

Question: A hypothetical pictogram key says each full symbol stands for 10 items. One category shows 2 full symbols. Find its represented frequency.

Work it out: A full symbol represents 10, so 2 symbols count 20. There is no additional fraction of an icon. The total is 20 items.

Common first error: Counting symbols rather than the key-scaled items gives only 2, not the represented frequency. A half symbol is half the key’s value, not automatically half an item.

Independent check: The icon-equivalent count is 2. Multiplying by 10 returns 20, the reported frequency.

Transfer: A second category has 1 full icons and represents 10. Compare the two by counts, not visual decoration. Their absolute difference is 10 items.

88. Pictogram 16: use the key to count

Question: A hypothetical pictogram key says each full symbol stands for 12 items. One category shows 5 full symbols and a half symbol. Find its represented frequency.

Work it out: A full symbol represents 12, so 5 symbols count 60. The half-symbol adds 6. The total is 66 items.

Common first error: Counting symbols rather than the key-scaled items gives only 5.5, not the represented frequency. A half symbol is half the key’s value, not automatically half an item.

Independent check: The icon-equivalent count is 5.5. Multiplying by 12 returns 66, the reported frequency.

Transfer: A second category has 6 full icons and represents 72. Compare the two by counts, not visual decoration. Their absolute difference is 6 items.

89. Pictogram 17: use the key to count

Question: A hypothetical pictogram key says each full symbol stands for 14 items. One category shows 2 full symbols. Find its represented frequency.

Work it out: A full symbol represents 14, so 2 symbols count 28. There is no additional fraction of an icon. The total is 28 items.

Common first error: Counting symbols rather than the key-scaled items gives only 2, not the represented frequency. A half symbol is half the key’s value, not automatically half an item.

Independent check: The icon-equivalent count is 2. Multiplying by 14 returns 28, the reported frequency.

Transfer: A second category has 4 full icons and represents 56. Compare the two by counts, not visual decoration. Their absolute difference is 28 items.

90. Pictogram 18: use the key to count

Question: A hypothetical pictogram key says each full symbol stands for 16 items. One category shows 5 full symbols. Find its represented frequency.

Work it out: A full symbol represents 16, so 5 symbols count 80. There is no additional fraction of an icon. The total is 80 items.

Common first error: Counting symbols rather than the key-scaled items gives only 5, not the represented frequency. A half symbol is half the key’s value, not automatically half an item.

Independent check: The icon-equivalent count is 5. Multiplying by 16 returns 80, the reported frequency.

Transfer: A second category has 2 full icons and represents 32. Compare the two by counts, not visual decoration. Their absolute difference is 48 items.

91. Pictogram 19: use the key to count

Question: A hypothetical pictogram key says each full symbol stands for 18 items. One category shows 2 full symbols and a half symbol. Find its represented frequency.

Work it out: A full symbol represents 18, so 2 symbols count 36. The half-symbol adds 9. The total is 45 items.

Common first error: Counting symbols rather than the key-scaled items gives only 2.5, not the represented frequency. A half symbol is half the key’s value, not automatically half an item.

Independent check: The icon-equivalent count is 2.5. Multiplying by 18 returns 45, the reported frequency.

Transfer: A second category has 7 full icons and represents 126. Compare the two by counts, not visual decoration. Their absolute difference is 81 items.

92. Pictogram 20: use the key to count

Question: A hypothetical pictogram key says each full symbol stands for 20 items. One category shows 5 full symbols. Find its represented frequency.

Work it out: A full symbol represents 20, so 5 symbols count 100. There is no additional fraction of an icon. The total is 100 items.

Common first error: Counting symbols rather than the key-scaled items gives only 5, not the represented frequency. A half symbol is half the key’s value, not automatically half an item.

Independent check: The icon-equivalent count is 5. Multiplying by 20 returns 100, the reported frequency.

Transfer: A second category has 5 full icons and represents 100. Compare the two by counts, not visual decoration. Their absolute difference is 0 items.

93. Pictogram 21: use the key to count

Question: A hypothetical pictogram key says each full symbol stands for 22 items. One category shows 2 full symbols. Find its represented frequency.

Work it out: A full symbol represents 22, so 2 symbols count 44. There is no additional fraction of an icon. The total is 44 items.

Common first error: Counting symbols rather than the key-scaled items gives only 2, not the represented frequency. A half symbol is half the key’s value, not automatically half an item.

Independent check: The icon-equivalent count is 2. Multiplying by 22 returns 44, the reported frequency.

Transfer: A second category has 3 full icons and represents 66. Compare the two by counts, not visual decoration. Their absolute difference is 22 items.

94. Pictogram 22: use the key to count

Question: A hypothetical pictogram key says each full symbol stands for 24 items. One category shows 5 full symbols and a half symbol. Find its represented frequency.

Work it out: A full symbol represents 24, so 5 symbols count 120. The half-symbol adds 12. The total is 132 items.

Common first error: Counting symbols rather than the key-scaled items gives only 5.5, not the represented frequency. A half symbol is half the key’s value, not automatically half an item.

Independent check: The icon-equivalent count is 5.5. Multiplying by 24 returns 132, the reported frequency.

Transfer: A second category has 1 full icons and represents 24. Compare the two by counts, not visual decoration. Their absolute difference is 108 items.

95. Pictogram 23: use the key to count

Question: A hypothetical pictogram key says each full symbol stands for 4 items. One category shows 2 full symbols. Find its represented frequency.

Work it out: A full symbol represents 4, so 2 symbols count 8. There is no additional fraction of an icon. The total is 8 items.

Common first error: Counting symbols rather than the key-scaled items gives only 2, not the represented frequency. A half symbol is half the key’s value, not automatically half an item.

Independent check: The icon-equivalent count is 2. Multiplying by 4 returns 8, the reported frequency.

Transfer: A second category has 6 full icons and represents 24. Compare the two by counts, not visual decoration. Their absolute difference is 16 items.

96. Pictogram 24: use the key to count

Question: A hypothetical pictogram key says each full symbol stands for 6 items. One category shows 5 full symbols. Find its represented frequency.

Work it out: A full symbol represents 6, so 5 symbols count 30. There is no additional fraction of an icon. The total is 30 items.

Common first error: Counting symbols rather than the key-scaled items gives only 5, not the represented frequency. A half symbol is half the key’s value, not automatically half an item.

Independent check: The icon-equivalent count is 5. Multiplying by 6 returns 30, the reported frequency.

Transfer: A second category has 4 full icons and represents 24. Compare the two by counts, not visual decoration. Their absolute difference is 6 items.

97. Pictogram 25: use the key to count

Question: A hypothetical pictogram key says each full symbol stands for 8 items. One category shows 2 full symbols and a half symbol. Find its represented frequency.

Work it out: A full symbol represents 8, so 2 symbols count 16. The half-symbol adds 4. The total is 20 items.

Common first error: Counting symbols rather than the key-scaled items gives only 2.5, not the represented frequency. A half symbol is half the key’s value, not automatically half an item.

Independent check: The icon-equivalent count is 2.5. Multiplying by 8 returns 20, the reported frequency.

Transfer: A second category has 2 full icons and represents 16. Compare the two by counts, not visual decoration. Their absolute difference is 4 items.

98. Pictogram 26: use the key to count

Question: A hypothetical pictogram key says each full symbol stands for 10 items. One category shows 5 full symbols. Find its represented frequency.

Work it out: A full symbol represents 10, so 5 symbols count 50. There is no additional fraction of an icon. The total is 50 items.

Common first error: Counting symbols rather than the key-scaled items gives only 5, not the represented frequency. A half symbol is half the key’s value, not automatically half an item.

Independent check: The icon-equivalent count is 5. Multiplying by 10 returns 50, the reported frequency.

Transfer: A second category has 7 full icons and represents 70. Compare the two by counts, not visual decoration. Their absolute difference is 20 items.

99. A full pie chart is 360 degrees

Question: What angle corresponds to 100% of a complete pie?

Work it out: 360°. A full circle represents the one whole partition.

Common first error: Using 180°, the angle along a straight line, instead of a full turn.

Independent check: Four sectors of ninety degrees each fill the full circle.

Transfer: For a new chart, decide whether the task asks for angle, percentage, absolute frequency or total. Identify what one whole means and what key is being used. A valid calculation also needs a supported interpretation; never invent details about people or dates not recorded in the data.

100. Half a circle is not 180 percent

Question: A sector spans 180°. What percentage is it?

Work it out: 180/360=1/2=50%.

Common first error: Treating the degree number 180 as the percentage number.

Independent check: Two equal 180° halves fill a full 360°.

101. A right-angle sector

Question: A sector is 90°. Find its share of the whole.

Work it out: 90/360=1/4=25%.

Common first error: Reporting 90% because 90 appears in the picture.

Independent check: Four such sectors fill the whole.

102. A 72-degree sector

Question: Find the percentage represented by a 72° slice.

Work it out: 72/360=1/5=20%.

Common first error: Reporting 72%, which exceeds the actual share.

Independent check: Five equal slices of 72° cover 360°.

103. Equal categories

Question: An exhaustive pie has three categories of equal frequency. What angle for each?

Work it out: 360/3=120° for each.

Common first error: Using thirty-three degrees after mixing percentage with angle units.

Independent check: 120×3=360°.

104. Pie sector missing angle

Question: Three sectors are 90°,108°,72°. What is the fourth in a complete four-category pie?

Work it out: Total known is 270°, so missing angle is 90°.

Common first error: Subtracting from 180°, not 360°.

Independent check: All four angles add to 360°.

105. Category frequency sum

Question: A survey total is 60, with category counts 10,15,20 and unknown. Find the unknown.

Work it out: 60−10−15−20=15 observations.

Common first error: Assuming angles without first finding the missing count.

Independent check: The four counts sum to sixty.

106. Pie categories exceed one whole

Question: Shares are 45%,35%,30% in three supposedly disjoint exhaustive categories. Is the pie valid?

Work it out: No. They total 110%, which cannot describe one partition.

Common first error: Trusting a nice-looking circle rather than arithmetic.

Independent check: Recheck denominators and whether categories overlap.

107. Pictogram with zero symbols

Question: A row explicitly labelled B has zero symbols under key one icon=5. What is its frequency?

Work it out: Zero items in this represented dataset.

Common first error: Assuming an entirely omitted category also necessarily means zero.

Independent check: Distinguish a labelled empty row from missing data.

108. Half an icon for six people

Question: A key says one full icon means six people. What does a half icon mean?

Work it out: Three people.

Common first error: Calling it half a person because one half-symbol is visible.

Independent check: 3+3=6, reconstructing a full icon.

109. A questionable key for discrete people

Question: One icon represents five people, but the chart shows a half icon. Does this represent an exact whole-person count?

Work it out: No, not without some alternate annotation: half of five is 2.5 people, which is not a valid exact count of individuals.

Common first error: Using fractions of persons without considering the object being counted.

Independent check: Choose a compatible key or state that the pictogram is approximate.

110. Pie and histogram are different

Question: A display partitions a whole sample into favourite-subject sectors. Is it a histogram?

Work it out: No. It is a categorical pie chart. A histogram groups numerical measurements into intervals.

Common first error: Calling any visual about frequency a histogram.

Independent check: Read the kind of horizontal numerical or categorical variable.

111. One chart may have a smaller sample

Question: Two pies have 50% sectors, one over 40 people and the other over 100. Are category counts equal?

Work it out: No. Counts are 20 and 50, respectively.

Common first error: Equating equal percentages with equal absolute frequencies.

Independent check: Multiply each proportion by its own total.

112. Sample does not mean all residents

Question: Six of ten fictional survey respondents choose an option. Does this establish a 60% preference among all Singapore residents?

Work it out: No. It shows 60% of the ten respondents only; population generalisation needs appropriate sampling.

Common first error: Making unsupported broader claims from a small convenience sample.

Independent check: State the sample and the observation period.

113. Pie ‘Other’ hides detail

Question: A sector labelled Other is 30%. Can you name its most common subcategory?

Work it out: No. The pie shows only the combined share of Other, not its internal breakdown.

Common first error: Guessing a category from its overall sector size.

Independent check: Request the raw subgroup frequency table.

114. A pie sector cannot exceed one whole

Question: A category is labelled 140% of the same complete pie. Is that possible?

Work it out: No. In a nonoverlapping partition, one category cannot exceed the 100% whole.

Common first error: Confusing a percentage increase over 100% with a share of a fixed sample.

Independent check: Its angle would be 504°, exceeding 360°.

115. Quarter of a pictogram key

Question: If one icon is eight items, what does a quarter icon stand for?

Work it out: Two items.

Common first error: Calling it 0.25 item regardless of key value.

Independent check: Four quarters of two items reconstruct eight.

116. Pictogram rows require one scale

Question: Could two rows with different icon sizes be compared accurately with no legend?

Work it out: Not reliably. A shared key and consistent representation are required.

Common first error: Comparing visual area instead of encoded values.

Independent check: Read the legend before drawing conclusions.

117. A percentage can be used to draw a pie

Question: If 12.5% of a class is in category A, what is its angle?

Work it out: 0.125×360°=45°.

Common first error: Using 12.5° as a sector angle.

Independent check: 45/360=0.125.

118. Frequency from five percent

Question: If five percent of a fictional eighty-response sample belongs to A, how many responses are A?

Work it out: 0.05×80=4, corresponding to 18°.

Common first error: Reporting five people directly from a five percent label.

Independent check: Four of eighty is one twentieth, or five percent.

119. Pictogram with 1.5 full equivalents

Question: One full icon=12 books. What do 1.5 icons represent?

Work it out: 1.5×12=18 books.

Common first error: Reading 1.5 icons as 1.5 books.

Independent check: Eighteen divided by twelve gives 1.5 icon equivalents.

120. No total, no absolute count

Question: A category is 25% of a pie. Can its exact frequency be found without the sample total?

Work it out: No. One quarter of an unspecified total has no determined absolute frequency.

Common first error: Reporting twenty-five observations from a twenty-five percent share.

Independent check: For totals forty and eighty, the category counts would be ten and twenty.

121. Chart date is necessary for ‘today’

Question: A pie chart has no publication date or collection period. Can it tell us what happened today?

Work it out: No. The chart’s time window is not specified.

Common first error: Treating an old or undated chart as current evidence.

Independent check: Find source date and measurement period.

122. Two close angles

Question: Sector A is 71° and B is 72°. Which has more frequency in the same pie?

Work it out: B, by a small difference corresponding to one degree of the same total.

Common first error: Reading similar-looking wedges as exactly equal.

Independent check: Compare printed angles, not line thickness.

123. A key may encode tens

Question: A pictogram key is one symbol=20 objects. A row has four symbols. Find count.

Work it out: 4×20=80 objects.

Common first error: Treating four visible icons as four objects.

Independent check: 80÷20=4 icon equivalents.

124. The whole must be exclusive and exhaustive

Question: Why should all pie categories add to one whole?

Work it out: They form a partition: each observed item appears in exactly one displayed category, and all are included.

Common first error: Double-counting overlapping categories or omitting unlabeled groups.

Independent check: Sum counts to sample total and angles to 360°.

Five common G1 chart mistakes

Sector angle is read as a percentage

The learner sees ninety degrees and writes ninety percent. Ask them to draw four quarters of a circle: each quarter is ninety degrees but only twenty-five percent of a whole. Compare 90/360 and 90/100 explicitly. The pupil can then use 180° as a new check and should get fifty percent.

Pictogram key is ignored

A row of four icons does not necessarily mean four objects. With key one icon=ten, it means forty. Ask the pupil to label one symbol with ‘×10’ and extend the mapping to the row. If the key changes to five, the represented frequency must change despite the same count of drawn icons.

A category is mistaken for the total

The question says twelve observations are one fifth of the sample, but the learner multiplies twelve by one fifth. That computes a smaller quantity, not the whole. Divide twelve by one fifth and check the total sixty. Reverse-direction questions are especially useful for exposing a missing part–whole distinction.

Categories do not sum to one whole

If supposedly mutually exclusive exhaustive shares total 110%, the chart is inconsistent. A circle cannot represent more than its own 100%. Check category definitions, original counts and arithmetic. A neat drawing is not proof the data were classified properly.

A chart is treated as a universal survey

Six of ten fictitious respondents choosing one option does not establish a sixty-percent preference across Singapore. The chart records those respondents only. Ask the pupil to name the measured group, sample size and collection period. This is mathematical interpretation, not simply politeness about uncertain information.

How a small-group tutor can teach charts

The immutable eduKateSG Clementi three-pupil Maths reference describes 90-minute sessions near Sixth Avenue MRT. A Secondary 2 G1 teaching adaptation could have one learner convert angle and percent, another recover sample totals and a third decode pictogram keys and unsupported statements. The format does not establish current group openings, exact fees or guaranteed outcomes.

10 minutes: retrieve one-whole fractions and percentage equivalents.
20 minutes: draw pie sectors from simple fractions and counts.
25 minutes: individually solve sector-to-frequency and reverse-total questions.
15 minutes: count pictogram full and half-symbol equivalents.
10 minutes: identify one conclusion a chart does not justify.
10 minutes: correct the first mistake and retest with new data.

Five manageable home practice sessions

Day one: change 90°,120°,180° into fractions of a circle.
Day two: find category counts using stated fictional survey totals.
Day three: reverse a pie sector to recover total sample size.
Day four: create a pictogram with a consistent symbol key and a valid half symbol.
Day five: explain a sample-size limitation or category labelling issue in one sentence.

Current G1 curriculum versus the SEC year

Frequently asked questions

Does 90° mean 90% in a pie chart?

No. It is ninety of the full 360 degrees, or one quarter = 25%.

How do you find total observations from one sector?

Divide its category count by the fraction angle/360°.

Is a half icon half a person?

No. It is half of the key’s represented count, which should yield a valid count or be labelled approximate.

Can pie charts and histograms replace each other?

No. Pie charts show shares of one whole; histograms show distributions across numerical intervals.

Can equal pie sectors represent different respondent counts?

Yes, if the pies have different sample sizes.

What should a tutor check?

Can the pupil move independently among sector angles, fractions, percentages and absolute counts, and use a pictogram key accurately?

Official syllabus and accuracy note

Bring a misread pie chart or pictogram worksheet to eduKate Singapore and ask how the tutor will test the learner’s understanding of the whole and the chart key.

One-minute diagnostic
How a sector represents a fraction
How to reverse a pie calculation
Reading the pictogram key
What charts do not prove
Original chart clinics
First mistakes to repair
Three-pupil tuition model
Parent revision plan
FAQs
Official sources

A one-minute chart diagnostic

The 360-degree circle is a mathematical whole

Two identical sectors can represent different counts

How to recover the sample total

An icon key is a unit conversion

What a chart does not tell you

124 original pie-chart and pictogram clinics

1. Pie chart 1: sector to category frequency

Question: A fictional pie chart shows 16 observations. A category occupies 180 degrees of the full circle. Find how many observations belong to it.

Work it out: Share is 180/360=1/2. Multiply by total 16: 8 observations. This also represents 50% of the whole.

Common first error: Treating 180 degrees as 180% or as 180 individual observations confuses angle, proportion and frequency. A sector does not have an independent count without the total.

Independent check: The reverse calculation (8/16)×360°=180°. The category count is no more than the full sample 16.

Transfer: If the same share applied to 32 responses, the count would double to 16, but the angle would remain 180°.

2. Pie chart 1: recover the complete sample

Question: An imaginary category has 8 responses and occupies 180° of a pie chart representing every recorded response. Find the whole sample size.

Work it out: The category is 1/2 of its whole. Divide the count 8 by 1/2 to recover 16 responses.

Common first error: Multiplying the category count by 1/2 instead would yield less than the category and cannot represent the total population being counted.

Independent check: 8/16=1/2, and (8/16)×360°=180°, matching the supplied angle.

Transfer: Ask what happens to the total if the same category frequency instead occupies a smaller sector angle. The total must be larger if the recorded count stays fixed.

3. Pie chart 1: frequency to angle

Question: In a fictional survey of 16, category A has 8 observations. Work out the sector angle for A.

Work it out: Sector angle=(part/whole)×360° = (8/16)×360° = 180°. The fraction is 1/2 and share approximately 50%.

Common first error: Reversing total and count gives a fraction greater than one, which could produce an impossible sector larger than a full 360° circle.

Independent check: The rest of the pie occupies 180° and corresponds to 8 other observations. Together they reconstruct the whole.

Transfer: If another category has the same frequency, it must receive the same sector angle within this pie, even if its label is different.

4. Pie chart 2: sector to category frequency

Question: A fictional pie chart shows 27 observations. A category occupies 120 degrees of the full circle. Find how many observations belong to it.

Work it out: Share is 120/360=1/3. Multiply by total 27: 9 observations. This also represents 33.333% of the whole.

Common first error: Treating 120 degrees as 120% or as 120 individual observations confuses angle, proportion and frequency. A sector does not have an independent count without the total.

Independent check: The reverse calculation (9/27)×360°=120°. The category count is no more than the full sample 27.

Transfer: If the same share applied to 54 responses, the count would double to 18, but the angle would remain 120°.

5. Pie chart 2: recover the complete sample

Question: An imaginary category has 9 responses and occupies 120° of a pie chart representing every recorded response. Find the whole sample size.

Work it out: The category is 1/3 of its whole. Divide the count 9 by 1/3 to recover 27 responses.

Common first error: Multiplying the category count by 1/3 instead would yield less than the category and cannot represent the total population being counted.

Independent check: 9/27=1/3, and (9/27)×360°=120°, matching the supplied angle.

Transfer: Ask what happens to the total if the same category frequency instead occupies a smaller sector angle. The total must be larger if the recorded count stays fixed.

6. Pie chart 2: frequency to angle

Question: In a fictional survey of 27, category A has 9 observations. Work out the sector angle for A.

Work it out: Sector angle=(part/whole)×360° = (9/27)×360° = 120°. The fraction is 1/3 and share approximately 33.333%.

Common first error: Reversing total and count gives a fraction greater than one, which could produce an impossible sector larger than a full 360° circle.

Independent check: The rest of the pie occupies 240° and corresponds to 18 other observations. Together they reconstruct the whole.

Transfer: If another category has the same frequency, it must receive the same sector angle within this pie, even if its label is different.

7. Pie chart 3: sector to category frequency

Question: A fictional pie chart shows 40 observations. A category occupies 90 degrees of the full circle. Find how many observations belong to it.

Work it out: Share is 90/360=1/4. Multiply by total 40: 10 observations. This also represents 25% of the whole.

Common first error: Treating 90 degrees as 90% or as 90 individual observations confuses angle, proportion and frequency. A sector does not have an independent count without the total.

Independent check: The reverse calculation (10/40)×360°=90°. The category count is no more than the full sample 40.

Transfer: If the same share applied to 80 responses, the count would double to 20, but the angle would remain 90°.

8. Pie chart 3: recover the complete sample

Question: An imaginary category has 10 responses and occupies 90° of a pie chart representing every recorded response. Find the whole sample size.

Work it out: The category is 1/4 of its whole. Divide the count 10 by 1/4 to recover 40 responses.

Common first error: Multiplying the category count by 1/4 instead would yield less than the category and cannot represent the total population being counted.

Independent check: 10/40=1/4, and (10/40)×360°=90°, matching the supplied angle.

Transfer: Ask what happens to the total if the same category frequency instead occupies a smaller sector angle. The total must be larger if the recorded count stays fixed.

9. Pie chart 3: frequency to angle

Question: In a fictional survey of 40, category A has 10 observations. Work out the sector angle for A.

Work it out: Sector angle=(part/whole)×360° = (10/40)×360° = 90°. The fraction is 1/4 and share approximately 25%.

Common first error: Reversing total and count gives a fraction greater than one, which could produce an impossible sector larger than a full 360° circle.

Independent check: The rest of the pie occupies 270° and corresponds to 30 other observations. Together they reconstruct the whole.

Transfer: If another category has the same frequency, it must receive the same sector angle within this pie, even if its label is different.

10. Pie chart 4: sector to category frequency

Question: A fictional pie chart shows 55 observations. A category occupies 72 degrees of the full circle. Find how many observations belong to it.

Work it out: Share is 72/360=1/5. Multiply by total 55: 11 observations. This also represents 20% of the whole.

Common first error: Treating 72 degrees as 72% or as 72 individual observations confuses angle, proportion and frequency. A sector does not have an independent count without the total.

Independent check: The reverse calculation (11/55)×360°=72°. The category count is no more than the full sample 55.

Transfer: If the same share applied to 110 responses, the count would double to 22, but the angle would remain 72°.

11. Pie chart 4: recover the complete sample

Question: An imaginary category has 11 responses and occupies 72° of a pie chart representing every recorded response. Find the whole sample size.

Work it out: The category is 1/5 of its whole. Divide the count 11 by 1/5 to recover 55 responses.

Common first error: Multiplying the category count by 1/5 instead would yield less than the category and cannot represent the total population being counted.

Independent check: 11/55=1/5, and (11/55)×360°=72°, matching the supplied angle.

Transfer: Ask what happens to the total if the same category frequency instead occupies a smaller sector angle. The total must be larger if the recorded count stays fixed.

12. Pie chart 4: frequency to angle

Question: In a fictional survey of 55, category A has 11 observations. Work out the sector angle for A.

Work it out: Sector angle=(part/whole)×360° = (11/55)×360° = 72°. The fraction is 1/5 and share approximately 20%.

Common first error: Reversing total and count gives a fraction greater than one, which could produce an impossible sector larger than a full 360° circle.

Independent check: The rest of the pie occupies 288° and corresponds to 44 other observations. Together they reconstruct the whole.

Transfer: If another category has the same frequency, it must receive the same sector angle within this pie, even if its label is different.

13. Pie chart 5: sector to category frequency

Question: A fictional pie chart shows 72 observations. A category occupies 60 degrees of the full circle. Find how many observations belong to it.

Work it out: Share is 60/360=1/6. Multiply by total 72: 12 observations. This also represents 16.667% of the whole.

Common first error: Treating 60 degrees as 60% or as 60 individual observations confuses angle, proportion and frequency. A sector does not have an independent count without the total.

Independent check: The reverse calculation (12/72)×360°=60°. The category count is no more than the full sample 72.

Transfer: If the same share applied to 144 responses, the count would double to 24, but the angle would remain 60°.

14. Pie chart 5: recover the complete sample

Question: An imaginary category has 12 responses and occupies 60° of a pie chart representing every recorded response. Find the whole sample size.

Work it out: The category is 1/6 of its whole. Divide the count 12 by 1/6 to recover 72 responses.

Common first error: Multiplying the category count by 1/6 instead would yield less than the category and cannot represent the total population being counted.

Independent check: 12/72=1/6, and (12/72)×360°=60°, matching the supplied angle.

Transfer: Ask what happens to the total if the same category frequency instead occupies a smaller sector angle. The total must be larger if the recorded count stays fixed.

15. Pie chart 5: frequency to angle

Question: In a fictional survey of 72, category A has 12 observations. Work out the sector angle for A.

Work it out: Sector angle=(part/whole)×360° = (12/72)×360° = 60°. The fraction is 1/6 and share approximately 16.667%.

Common first error: Reversing total and count gives a fraction greater than one, which could produce an impossible sector larger than a full 360° circle.

Independent check: The rest of the pie occupies 300° and corresponds to 60 other observations. Together they reconstruct the whole.

Transfer: If another category has the same frequency, it must receive the same sector angle within this pie, even if its label is different.

16. Pie chart 6: sector to category frequency

Question: A fictional pie chart shows 65 observations. A category occupies 144 degrees of the full circle. Find how many observations belong to it.

Work it out: Share is 144/360=2/5. Multiply by total 65: 26 observations. This also represents 40% of the whole.

Common first error: Treating 144 degrees as 144% or as 144 individual observations confuses angle, proportion and frequency. A sector does not have an independent count without the total.

Independent check: The reverse calculation (26/65)×360°=144°. The category count is no more than the full sample 65.

Transfer: If the same share applied to 130 responses, the count would double to 52, but the angle would remain 144°.

17. Pie chart 6: recover the complete sample

Question: An imaginary category has 26 responses and occupies 144° of a pie chart representing every recorded response. Find the whole sample size.

Work it out: The category is 2/5 of its whole. Divide the count 26 by 2/5 to recover 65 responses.

Common first error: Multiplying the category count by 2/5 instead would yield less than the category and cannot represent the total population being counted.

Independent check: 26/65=2/5, and (26/65)×360°=144°, matching the supplied angle.

Transfer: Ask what happens to the total if the same category frequency instead occupies a smaller sector angle. The total must be larger if the recorded count stays fixed.

18. Pie chart 6: frequency to angle

Question: In a fictional survey of 65, category A has 26 observations. Work out the sector angle for A.

Work it out: Sector angle=(part/whole)×360° = (26/65)×360° = 144°. The fraction is 2/5 and share approximately 40%.

Common first error: Reversing total and count gives a fraction greater than one, which could produce an impossible sector larger than a full 360° circle.

Independent check: The rest of the pie occupies 216° and corresponds to 39 other observations. Together they reconstruct the whole.

Transfer: If another category has the same frequency, it must receive the same sector angle within this pie, even if its label is different.

19. Pie chart 7: sector to category frequency

Question: A fictional pie chart shows 70 observations. A category occupies 216 degrees of the full circle. Find how many observations belong to it.

Work it out: Share is 216/360=3/5. Multiply by total 70: 42 observations. This also represents 60% of the whole.

Common first error: Treating 216 degrees as 216% or as 216 individual observations confuses angle, proportion and frequency. A sector does not have an independent count without the total.

Independent check: The reverse calculation (42/70)×360°=216°. The category count is no more than the full sample 70.

Transfer: If the same share applied to 140 responses, the count would double to 84, but the angle would remain 216°.

20. Pie chart 7: recover the complete sample

Question: An imaginary category has 42 responses and occupies 216° of a pie chart representing every recorded response. Find the whole sample size.

Work it out: The category is 3/5 of its whole. Divide the count 42 by 3/5 to recover 70 responses.

Common first error: Multiplying the category count by 3/5 instead would yield less than the category and cannot represent the total population being counted.

Independent check: 42/70=3/5, and (42/70)×360°=216°, matching the supplied angle.

Transfer: Ask what happens to the total if the same category frequency instead occupies a smaller sector angle. The total must be larger if the recorded count stays fixed.

21. Pie chart 7: frequency to angle

Question: In a fictional survey of 70, category A has 42 observations. Work out the sector angle for A.

Work it out: Sector angle=(part/whole)×360° = (42/70)×360° = 216°. The fraction is 3/5 and share approximately 60%.

Common first error: Reversing total and count gives a fraction greater than one, which could produce an impossible sector larger than a full 360° circle.

Independent check: The rest of the pie occupies 144° and corresponds to 28 other observations. Together they reconstruct the whole.

Transfer: If another category has the same frequency, it must receive the same sector angle within this pie, even if its label is different.

22. Pie chart 8: sector to category frequency

Question: A fictional pie chart shows 45 observations. A category occupies 240 degrees of the full circle. Find how many observations belong to it.

Work it out: Share is 240/360=2/3. Multiply by total 45: 30 observations. This also represents 66.667% of the whole.

Common first error: Treating 240 degrees as 240% or as 240 individual observations confuses angle, proportion and frequency. A sector does not have an independent count without the total.

Independent check: The reverse calculation (30/45)×360°=240°. The category count is no more than the full sample 45.

Transfer: If the same share applied to 90 responses, the count would double to 60, but the angle would remain 240°.

23. Pie chart 8: recover the complete sample

Question: An imaginary category has 30 responses and occupies 240° of a pie chart representing every recorded response. Find the whole sample size.

Work it out: The category is 2/3 of its whole. Divide the count 30 by 2/3 to recover 45 responses.

Common first error: Multiplying the category count by 2/3 instead would yield less than the category and cannot represent the total population being counted.

Independent check: 30/45=2/3, and (30/45)×360°=240°, matching the supplied angle.

Transfer: Ask what happens to the total if the same category frequency instead occupies a smaller sector angle. The total must be larger if the recorded count stays fixed.

24. Pie chart 8: frequency to angle

Question: In a fictional survey of 45, category A has 30 observations. Work out the sector angle for A.

Work it out: Sector angle=(part/whole)×360° = (30/45)×360° = 240°. The fraction is 2/3 and share approximately 66.667%.

Common first error: Reversing total and count gives a fraction greater than one, which could produce an impossible sector larger than a full 360° circle.

Independent check: The rest of the pie occupies 120° and corresponds to 15 other observations. Together they reconstruct the whole.

Transfer: If another category has the same frequency, it must receive the same sector angle within this pie, even if its label is different.

25. Pie chart 9: sector to category frequency

Question: A fictional pie chart shows 64 observations. A category occupies 270 degrees of the full circle. Find how many observations belong to it.

Work it out: Share is 270/360=3/4. Multiply by total 64: 48 observations. This also represents 75% of the whole.

Common first error: Treating 270 degrees as 270% or as 270 individual observations confuses angle, proportion and frequency. A sector does not have an independent count without the total.

Independent check: The reverse calculation (48/64)×360°=270°. The category count is no more than the full sample 64.

Transfer: If the same share applied to 128 responses, the count would double to 96, but the angle would remain 270°.

26. Pie chart 9: recover the complete sample

Question: An imaginary category has 48 responses and occupies 270° of a pie chart representing every recorded response. Find the whole sample size.

Work it out: The category is 3/4 of its whole. Divide the count 48 by 3/4 to recover 64 responses.

Common first error: Multiplying the category count by 3/4 instead would yield less than the category and cannot represent the total population being counted.

Independent check: 48/64=3/4, and (48/64)×360°=270°, matching the supplied angle.

Transfer: Ask what happens to the total if the same category frequency instead occupies a smaller sector angle. The total must be larger if the recorded count stays fixed.

27. Pie chart 9: frequency to angle

Question: In a fictional survey of 64, category A has 48 observations. Work out the sector angle for A.

Work it out: Sector angle=(part/whole)×360° = (48/64)×360° = 270°. The fraction is 3/4 and share approximately 75%.

Common first error: Reversing total and count gives a fraction greater than one, which could produce an impossible sector larger than a full 360° circle.

Independent check: The rest of the pie occupies 90° and corresponds to 16 other observations. Together they reconstruct the whole.

Transfer: If another category has the same frequency, it must receive the same sector angle within this pie, even if its label is different.

28. Pie chart 10: sector to category frequency

Question: A fictional pie chart shows 136 observations. A category occupies 45 degrees of the full circle. Find how many observations belong to it.

Work it out: Share is 45/360=1/8. Multiply by total 136: 17 observations. This also represents 12.5% of the whole.

Common first error: Treating 45 degrees as 45% or as 45 individual observations confuses angle, proportion and frequency. A sector does not have an independent count without the total.

Independent check: The reverse calculation (17/136)×360°=45°. The category count is no more than the full sample 136.

Transfer: If the same share applied to 272 responses, the count would double to 34, but the angle would remain 45°.

29. Pie chart 10: recover the complete sample

Question: An imaginary category has 17 responses and occupies 45° of a pie chart representing every recorded response. Find the whole sample size.

Work it out: The category is 1/8 of its whole. Divide the count 17 by 1/8 to recover 136 responses.

Common first error: Multiplying the category count by 1/8 instead would yield less than the category and cannot represent the total population being counted.

Independent check: 17/136=1/8, and (17/136)×360°=45°, matching the supplied angle.

Transfer: Ask what happens to the total if the same category frequency instead occupies a smaller sector angle. The total must be larger if the recorded count stays fixed.

30. Pie chart 10: frequency to angle

Question: In a fictional survey of 136, category A has 17 observations. Work out the sector angle for A.

Work it out: Sector angle=(part/whole)×360° = (17/136)×360° = 45°. The fraction is 1/8 and share approximately 12.5%.

Common first error: Reversing total and count gives a fraction greater than one, which could produce an impossible sector larger than a full 360° circle.

Independent check: The rest of the pie occupies 315° and corresponds to 119 other observations. Together they reconstruct the whole.

Transfer: If another category has the same frequency, it must receive the same sector angle within this pie, even if its label is different.

31. Pie chart 11: sector to category frequency

Question: A fictional pie chart shows 144 observations. A category occupies 135 degrees of the full circle. Find how many observations belong to it.

Work it out: Share is 135/360=3/8. Multiply by total 144: 54 observations. This also represents 37.5% of the whole.

Common first error: Treating 135 degrees as 135% or as 135 individual observations confuses angle, proportion and frequency. A sector does not have an independent count without the total.

Independent check: The reverse calculation (54/144)×360°=135°. The category count is no more than the full sample 144.

Transfer: If the same share applied to 288 responses, the count would double to 108, but the angle would remain 135°.

32. Pie chart 11: recover the complete sample

Question: An imaginary category has 54 responses and occupies 135° of a pie chart representing every recorded response. Find the whole sample size.

Work it out: The category is 3/8 of its whole. Divide the count 54 by 3/8 to recover 144 responses.

Common first error: Multiplying the category count by 3/8 instead would yield less than the category and cannot represent the total population being counted.

Independent check: 54/144=3/8, and (54/144)×360°=135°, matching the supplied angle.

Transfer: Ask what happens to the total if the same category frequency instead occupies a smaller sector angle. The total must be larger if the recorded count stays fixed.

33. Pie chart 11: frequency to angle

Question: In a fictional survey of 144, category A has 54 observations. Work out the sector angle for A.

Work it out: Sector angle=(part/whole)×360° = (54/144)×360° = 135°. The fraction is 3/8 and share approximately 37.5%.

Common first error: Reversing total and count gives a fraction greater than one, which could produce an impossible sector larger than a full 360° circle.

Independent check: The rest of the pie occupies 225° and corresponds to 90 other observations. Together they reconstruct the whole.

Transfer: If another category has the same frequency, it must receive the same sector angle within this pie, even if its label is different.

34. Pie chart 12: sector to category frequency

Question: A fictional pie chart shows 228 observations. A category occupies 150 degrees of the full circle. Find how many observations belong to it.

Work it out: Share is 150/360=5/12. Multiply by total 228: 95 observations. This also represents 41.667% of the whole.

Common first error: Treating 150 degrees as 150% or as 150 individual observations confuses angle, proportion and frequency. A sector does not have an independent count without the total.

Independent check: The reverse calculation (95/228)×360°=150°. The category count is no more than the full sample 228.

Transfer: If the same share applied to 456 responses, the count would double to 190, but the angle would remain 150°.

35. Pie chart 12: recover the complete sample

Question: An imaginary category has 95 responses and occupies 150° of a pie chart representing every recorded response. Find the whole sample size.

Work it out: The category is 5/12 of its whole. Divide the count 95 by 5/12 to recover 228 responses.

Common first error: Multiplying the category count by 5/12 instead would yield less than the category and cannot represent the total population being counted.

Independent check: 95/228=5/12, and (95/228)×360°=150°, matching the supplied angle.

Transfer: Ask what happens to the total if the same category frequency instead occupies a smaller sector angle. The total must be larger if the recorded count stays fixed.

36. Pie chart 12: frequency to angle

Question: In a fictional survey of 228, category A has 95 observations. Work out the sector angle for A.

Work it out: Sector angle=(part/whole)×360° = (95/228)×360° = 150°. The fraction is 5/12 and share approximately 41.667%.

Common first error: Reversing total and count gives a fraction greater than one, which could produce an impossible sector larger than a full 360° circle.

Independent check: The rest of the pie occupies 210° and corresponds to 133 other observations. Together they reconstruct the whole.

Transfer: If another category has the same frequency, it must receive the same sector angle within this pie, even if its label is different.

37. Pie chart 13: sector to category frequency

Question: A fictional pie chart shows 240 observations. A category occupies 210 degrees of the full circle. Find how many observations belong to it.

Work it out: Share is 210/360=7/12. Multiply by total 240: 140 observations. This also represents 58.333% of the whole.

Common first error: Treating 210 degrees as 210% or as 210 individual observations confuses angle, proportion and frequency. A sector does not have an independent count without the total.

Independent check: The reverse calculation (140/240)×360°=210°. The category count is no more than the full sample 240.

Transfer: If the same share applied to 480 responses, the count would double to 280, but the angle would remain 210°.

38. Pie chart 13: recover the complete sample

Question: An imaginary category has 140 responses and occupies 210° of a pie chart representing every recorded response. Find the whole sample size.

Work it out: The category is 7/12 of its whole. Divide the count 140 by 7/12 to recover 240 responses.

Common first error: Multiplying the category count by 7/12 instead would yield less than the category and cannot represent the total population being counted.

Independent check: 140/240=7/12, and (140/240)×360°=210°, matching the supplied angle.

Transfer: Ask what happens to the total if the same category frequency instead occupies a smaller sector angle. The total must be larger if the recorded count stays fixed.

39. Pie chart 13: frequency to angle

Question: In a fictional survey of 240, category A has 140 observations. Work out the sector angle for A.

Work it out: Sector angle=(part/whole)×360° = (140/240)×360° = 210°. The fraction is 7/12 and share approximately 58.333%.

Common first error: Reversing total and count gives a fraction greater than one, which could produce an impossible sector larger than a full 360° circle.

Independent check: The rest of the pie occupies 150° and corresponds to 100 other observations. Together they reconstruct the whole.

Transfer: If another category has the same frequency, it must receive the same sector angle within this pie, even if its label is different.

40. Pie chart 14: sector to category frequency

Question: A fictional pie chart shows 80 observations. A category occupies 36 degrees of the full circle. Find how many observations belong to it.

Work it out: Share is 36/360=1/10. Multiply by total 80: 8 observations. This also represents 10% of the whole.

Common first error: Treating 36 degrees as 36% or as 36 individual observations confuses angle, proportion and frequency. A sector does not have an independent count without the total.

Independent check: The reverse calculation (8/80)×360°=36°. The category count is no more than the full sample 80.

Transfer: If the same share applied to 160 responses, the count would double to 16, but the angle would remain 36°.

41. Pie chart 14: recover the complete sample

Question: An imaginary category has 8 responses and occupies 36° of a pie chart representing every recorded response. Find the whole sample size.

Work it out: The category is 1/10 of its whole. Divide the count 8 by 1/10 to recover 80 responses.

Common first error: Multiplying the category count by 1/10 instead would yield less than the category and cannot represent the total population being counted.

Independent check: 8/80=1/10, and (8/80)×360°=36°, matching the supplied angle.

Transfer: Ask what happens to the total if the same category frequency instead occupies a smaller sector angle. The total must be larger if the recorded count stays fixed.

42. Pie chart 14: frequency to angle

Question: In a fictional survey of 80, category A has 8 observations. Work out the sector angle for A.

Work it out: Sector angle=(part/whole)×360° = (8/80)×360° = 36°. The fraction is 1/10 and share approximately 10%.

Common first error: Reversing total and count gives a fraction greater than one, which could produce an impossible sector larger than a full 360° circle.

Independent check: The rest of the pie occupies 324° and corresponds to 72 other observations. Together they reconstruct the whole.

Transfer: If another category has the same frequency, it must receive the same sector angle within this pie, even if its label is different.

43. Pie chart 15: sector to category frequency

Question: A fictional pie chart shows 90 observations. A category occupies 108 degrees of the full circle. Find how many observations belong to it.

Work it out: Share is 108/360=3/10. Multiply by total 90: 27 observations. This also represents 30% of the whole.

Common first error: Treating 108 degrees as 108% or as 108 individual observations confuses angle, proportion and frequency. A sector does not have an independent count without the total.

Independent check: The reverse calculation (27/90)×360°=108°. The category count is no more than the full sample 90.

Transfer: If the same share applied to 180 responses, the count would double to 54, but the angle would remain 108°.

44. Pie chart 15: recover the complete sample

Question: An imaginary category has 27 responses and occupies 108° of a pie chart representing every recorded response. Find the whole sample size.

Work it out: The category is 3/10 of its whole. Divide the count 27 by 3/10 to recover 90 responses.

Common first error: Multiplying the category count by 3/10 instead would yield less than the category and cannot represent the total population being counted.

Independent check: 27/90=3/10, and (27/90)×360°=108°, matching the supplied angle.

Transfer: Ask what happens to the total if the same category frequency instead occupies a smaller sector angle. The total must be larger if the recorded count stays fixed.

45. Pie chart 15: frequency to angle

Question: In a fictional survey of 90, category A has 27 observations. Work out the sector angle for A.

Work it out: Sector angle=(part/whole)×360° = (27/90)×360° = 108°. The fraction is 3/10 and share approximately 30%.

Common first error: Reversing total and count gives a fraction greater than one, which could produce an impossible sector larger than a full 360° circle.

Independent check: The rest of the pie occupies 252° and corresponds to 63 other observations. Together they reconstruct the whole.

Transfer: If another category has the same frequency, it must receive the same sector angle within this pie, even if its label is different.

46. Pie chart 16: sector to category frequency

Question: A fictional pie chart shows 90 observations. A category occupies 80 degrees of the full circle. Find how many observations belong to it.

Work it out: Share is 80/360=2/9. Multiply by total 90: 20 observations. This also represents 22.222% of the whole.

Common first error: Treating 80 degrees as 80% or as 80 individual observations confuses angle, proportion and frequency. A sector does not have an independent count without the total.

Independent check: The reverse calculation (20/90)×360°=80°. The category count is no more than the full sample 90.

Transfer: If the same share applied to 180 responses, the count would double to 40, but the angle would remain 80°.

47. Pie chart 16: recover the complete sample

Question: An imaginary category has 20 responses and occupies 80° of a pie chart representing every recorded response. Find the whole sample size.

Work it out: The category is 2/9 of its whole. Divide the count 20 by 2/9 to recover 90 responses.

Common first error: Multiplying the category count by 2/9 instead would yield less than the category and cannot represent the total population being counted.

Independent check: 20/90=2/9, and (20/90)×360°=80°, matching the supplied angle.

Transfer: Ask what happens to the total if the same category frequency instead occupies a smaller sector angle. The total must be larger if the recorded count stays fixed.

48. Pie chart 16: frequency to angle

Question: In a fictional survey of 90, category A has 20 observations. Work out the sector angle for A.

Work it out: Sector angle=(part/whole)×360° = (20/90)×360° = 80°. The fraction is 2/9 and share approximately 22.222%.

Common first error: Reversing total and count gives a fraction greater than one, which could produce an impossible sector larger than a full 360° circle.

Independent check: The rest of the pie occupies 280° and corresponds to 70 other observations. Together they reconstruct the whole.

Transfer: If another category has the same frequency, it must receive the same sector angle within this pie, even if its label is different.

49. Pie chart 17: sector to category frequency

Question: A fictional pie chart shows 99 observations. A category occupies 160 degrees of the full circle. Find how many observations belong to it.

Work it out: Share is 160/360=4/9. Multiply by total 99: 44 observations. This also represents 44.444% of the whole.

Common first error: Treating 160 degrees as 160% or as 160 individual observations confuses angle, proportion and frequency. A sector does not have an independent count without the total.

Independent check: The reverse calculation (44/99)×360°=160°. The category count is no more than the full sample 99.

Transfer: If the same share applied to 198 responses, the count would double to 88, but the angle would remain 160°.

50. Pie chart 17: recover the complete sample

Question: An imaginary category has 44 responses and occupies 160° of a pie chart representing every recorded response. Find the whole sample size.

Work it out: The category is 4/9 of its whole. Divide the count 44 by 4/9 to recover 99 responses.

Common first error: Multiplying the category count by 4/9 instead would yield less than the category and cannot represent the total population being counted.

Independent check: 44/99=4/9, and (44/99)×360°=160°, matching the supplied angle.

Transfer: Ask what happens to the total if the same category frequency instead occupies a smaller sector angle. The total must be larger if the recorded count stays fixed.

51. Pie chart 17: frequency to angle

Question: In a fictional survey of 99, category A has 44 observations. Work out the sector angle for A.

Work it out: Sector angle=(part/whole)×360° = (44/99)×360° = 160°. The fraction is 4/9 and share approximately 44.444%.

Common first error: Reversing total and count gives a fraction greater than one, which could produce an impossible sector larger than a full 360° circle.

Independent check: The rest of the pie occupies 200° and corresponds to 55 other observations. Together they reconstruct the whole.

Transfer: If another category has the same frequency, it must receive the same sector angle within this pie, even if its label is different.

52. Pie chart 18: sector to category frequency

Question: A fictional pie chart shows 108 observations. A category occupies 200 degrees of the full circle. Find how many observations belong to it.

Work it out: Share is 200/360=5/9. Multiply by total 108: 60 observations. This also represents 55.556% of the whole.

Common first error: Treating 200 degrees as 200% or as 200 individual observations confuses angle, proportion and frequency. A sector does not have an independent count without the total.

Independent check: The reverse calculation (60/108)×360°=200°. The category count is no more than the full sample 108.

Transfer: If the same share applied to 216 responses, the count would double to 120, but the angle would remain 200°.

53. Pie chart 18: recover the complete sample

Question: An imaginary category has 60 responses and occupies 200° of a pie chart representing every recorded response. Find the whole sample size.

Work it out: The category is 5/9 of its whole. Divide the count 60 by 5/9 to recover 108 responses.

Common first error: Multiplying the category count by 5/9 instead would yield less than the category and cannot represent the total population being counted.

Independent check: 60/108=5/9, and (60/108)×360°=200°, matching the supplied angle.

Transfer: Ask what happens to the total if the same category frequency instead occupies a smaller sector angle. The total must be larger if the recorded count stays fixed.

54. Pie chart 18: frequency to angle

Question: In a fictional survey of 108, category A has 60 observations. Work out the sector angle for A.

Work it out: Sector angle=(part/whole)×360° = (60/108)×360° = 200°. The fraction is 5/9 and share approximately 55.556%.

Common first error: Reversing total and count gives a fraction greater than one, which could produce an impossible sector larger than a full 360° circle.

Independent check: The rest of the pie occupies 160° and corresponds to 48 other observations. Together they reconstruct the whole.

Transfer: If another category has the same frequency, it must receive the same sector angle within this pie, even if its label is different.

55. Pie chart 19: sector to category frequency

Question: A fictional pie chart shows 78 observations. A category occupies 300 degrees of the full circle. Find how many observations belong to it.

Work it out: Share is 300/360=5/6. Multiply by total 78: 65 observations. This also represents 83.333% of the whole.

Common first error: Treating 300 degrees as 300% or as 300 individual observations confuses angle, proportion and frequency. A sector does not have an independent count without the total.

Independent check: The reverse calculation (65/78)×360°=300°. The category count is no more than the full sample 78.

Transfer: If the same share applied to 156 responses, the count would double to 130, but the angle would remain 300°.

56. Pie chart 19: recover the complete sample

Question: An imaginary category has 65 responses and occupies 300° of a pie chart representing every recorded response. Find the whole sample size.

Work it out: The category is 5/6 of its whole. Divide the count 65 by 5/6 to recover 78 responses.

Common first error: Multiplying the category count by 5/6 instead would yield less than the category and cannot represent the total population being counted.

Independent check: 65/78=5/6, and (65/78)×360°=300°, matching the supplied angle.

Transfer: Ask what happens to the total if the same category frequency instead occupies a smaller sector angle. The total must be larger if the recorded count stays fixed.

57. Pie chart 19: frequency to angle

Question: In a fictional survey of 78, category A has 65 observations. Work out the sector angle for A.

Work it out: Sector angle=(part/whole)×360° = (65/78)×360° = 300°. The fraction is 5/6 and share approximately 83.333%.

Common first error: Reversing total and count gives a fraction greater than one, which could produce an impossible sector larger than a full 360° circle.

Independent check: The rest of the pie occupies 60° and corresponds to 13 other observations. Together they reconstruct the whole.

Transfer: If another category has the same frequency, it must receive the same sector angle within this pie, even if its label is different.

58. Pie chart 20: sector to category frequency

Question: A fictional pie chart shows 112 observations. A category occupies 315 degrees of the full circle. Find how many observations belong to it.

Work it out: Share is 315/360=7/8. Multiply by total 112: 98 observations. This also represents 87.5% of the whole.

Common first error: Treating 315 degrees as 315% or as 315 individual observations confuses angle, proportion and frequency. A sector does not have an independent count without the total.

Independent check: The reverse calculation (98/112)×360°=315°. The category count is no more than the full sample 112.

Transfer: If the same share applied to 224 responses, the count would double to 196, but the angle would remain 315°.

59. Pie chart 20: recover the complete sample

Question: An imaginary category has 98 responses and occupies 315° of a pie chart representing every recorded response. Find the whole sample size.

Work it out: The category is 7/8 of its whole. Divide the count 98 by 7/8 to recover 112 responses.

Common first error: Multiplying the category count by 7/8 instead would yield less than the category and cannot represent the total population being counted.

Independent check: 98/112=7/8, and (98/112)×360°=315°, matching the supplied angle.

Transfer: Ask what happens to the total if the same category frequency instead occupies a smaller sector angle. The total must be larger if the recorded count stays fixed.

60. Pie chart 20: frequency to angle

Question: In a fictional survey of 112, category A has 98 observations. Work out the sector angle for A.

Work it out: Sector angle=(part/whole)×360° = (98/112)×360° = 315°. The fraction is 7/8 and share approximately 87.5%.

Common first error: Reversing total and count gives a fraction greater than one, which could produce an impossible sector larger than a full 360° circle.

Independent check: The rest of the pie occupies 45° and corresponds to 14 other observations. Together they reconstruct the whole.

Transfer: If another category has the same frequency, it must receive the same sector angle within this pie, even if its label is different.

61. Pie chart 21: sector to category frequency

Question: A fictional pie chart shows 180 observations. A category occupies 30 degrees of the full circle. Find how many observations belong to it.

Work it out: Share is 30/360=1/12. Multiply by total 180: 15 observations. This also represents 8.333% of the whole.

Common first error: Treating 30 degrees as 30% or as 30 individual observations confuses angle, proportion and frequency. A sector does not have an independent count without the total.

Independent check: The reverse calculation (15/180)×360°=30°. The category count is no more than the full sample 180.

Transfer: If the same share applied to 360 responses, the count would double to 30, but the angle would remain 30°.

62. Pie chart 21: recover the complete sample

Question: An imaginary category has 15 responses and occupies 30° of a pie chart representing every recorded response. Find the whole sample size.

Work it out: The category is 1/12 of its whole. Divide the count 15 by 1/12 to recover 180 responses.

Common first error: Multiplying the category count by 1/12 instead would yield less than the category and cannot represent the total population being counted.

Independent check: 15/180=1/12, and (15/180)×360°=30°, matching the supplied angle.

Transfer: Ask what happens to the total if the same category frequency instead occupies a smaller sector angle. The total must be larger if the recorded count stays fixed.

63. Pie chart 21: frequency to angle

Question: In a fictional survey of 180, category A has 15 observations. Work out the sector angle for A.

Work it out: Sector angle=(part/whole)×360° = (15/180)×360° = 30°. The fraction is 1/12 and share approximately 8.333%.

Common first error: Reversing total and count gives a fraction greater than one, which could produce an impossible sector larger than a full 360° circle.

Independent check: The rest of the pie occupies 330° and corresponds to 165 other observations. Together they reconstruct the whole.

Transfer: If another category has the same frequency, it must receive the same sector angle within this pie, even if its label is different.

64. Pie chart 22: sector to category frequency

Question: A fictional pie chart shows 192 observations. A category occupies 330 degrees of the full circle. Find how many observations belong to it.

Work it out: Share is 330/360=11/12. Multiply by total 192: 176 observations. This also represents 91.667% of the whole.

Common first error: Treating 330 degrees as 330% or as 330 individual observations confuses angle, proportion and frequency. A sector does not have an independent count without the total.

Independent check: The reverse calculation (176/192)×360°=330°. The category count is no more than the full sample 192.

Transfer: If the same share applied to 384 responses, the count would double to 352, but the angle would remain 330°.

65. Pie chart 22: recover the complete sample

Question: An imaginary category has 176 responses and occupies 330° of a pie chart representing every recorded response. Find the whole sample size.

Work it out: The category is 11/12 of its whole. Divide the count 176 by 11/12 to recover 192 responses.

Common first error: Multiplying the category count by 11/12 instead would yield less than the category and cannot represent the total population being counted.

Independent check: 176/192=11/12, and (176/192)×360°=330°, matching the supplied angle.

Transfer: Ask what happens to the total if the same category frequency instead occupies a smaller sector angle. The total must be larger if the recorded count stays fixed.

66. Pie chart 22: frequency to angle

Question: In a fictional survey of 192, category A has 176 observations. Work out the sector angle for A.

Work it out: Sector angle=(part/whole)×360° = (176/192)×360° = 330°. The fraction is 11/12 and share approximately 91.667%.

Common first error: Reversing total and count gives a fraction greater than one, which could produce an impossible sector larger than a full 360° circle.

Independent check: The rest of the pie occupies 30° and corresponds to 16 other observations. Together they reconstruct the whole.

Transfer: If another category has the same frequency, it must receive the same sector angle within this pie, even if its label is different.

67. Pie chart 23: sector to category frequency

Question: A fictional pie chart shows 153 observations. A category occupies 40 degrees of the full circle. Find how many observations belong to it.

Work it out: Share is 40/360=1/9. Multiply by total 153: 17 observations. This also represents 11.111% of the whole.

Common first error: Treating 40 degrees as 40% or as 40 individual observations confuses angle, proportion and frequency. A sector does not have an independent count without the total.

Independent check: The reverse calculation (17/153)×360°=40°. The category count is no more than the full sample 153.

Transfer: If the same share applied to 306 responses, the count would double to 34, but the angle would remain 40°.

68. Pie chart 23: recover the complete sample

Question: An imaginary category has 17 responses and occupies 40° of a pie chart representing every recorded response. Find the whole sample size.

Work it out: The category is 1/9 of its whole. Divide the count 17 by 1/9 to recover 153 responses.

Common first error: Multiplying the category count by 1/9 instead would yield less than the category and cannot represent the total population being counted.

Independent check: 17/153=1/9, and (17/153)×360°=40°, matching the supplied angle.

Transfer: Ask what happens to the total if the same category frequency instead occupies a smaller sector angle. The total must be larger if the recorded count stays fixed.

69. Pie chart 23: frequency to angle

Question: In a fictional survey of 153, category A has 17 observations. Work out the sector angle for A.

Work it out: Sector angle=(part/whole)×360° = (17/153)×360° = 40°. The fraction is 1/9 and share approximately 11.111%.

Common first error: Reversing total and count gives a fraction greater than one, which could produce an impossible sector larger than a full 360° circle.

Independent check: The rest of the pie occupies 320° and corresponds to 136 other observations. Together they reconstruct the whole.

Transfer: If another category has the same frequency, it must receive the same sector angle within this pie, even if its label is different.

70. Pie chart 24: sector to category frequency

Question: A fictional pie chart shows 162 observations. A category occupies 280 degrees of the full circle. Find how many observations belong to it.

Work it out: Share is 280/360=7/9. Multiply by total 162: 126 observations. This also represents 77.778% of the whole.

Common first error: Treating 280 degrees as 280% or as 280 individual observations confuses angle, proportion and frequency. A sector does not have an independent count without the total.

Independent check: The reverse calculation (126/162)×360°=280°. The category count is no more than the full sample 162.

Transfer: If the same share applied to 324 responses, the count would double to 252, but the angle would remain 280°.

71. Pie chart 24: recover the complete sample

Question: An imaginary category has 126 responses and occupies 280° of a pie chart representing every recorded response. Find the whole sample size.

Work it out: The category is 7/9 of its whole. Divide the count 126 by 7/9 to recover 162 responses.

Common first error: Multiplying the category count by 7/9 instead would yield less than the category and cannot represent the total population being counted.

Independent check: 126/162=7/9, and (126/162)×360°=280°, matching the supplied angle.

Transfer: Ask what happens to the total if the same category frequency instead occupies a smaller sector angle. The total must be larger if the recorded count stays fixed.

72. Pie chart 24: frequency to angle

Question: In a fictional survey of 162, category A has 126 observations. Work out the sector angle for A.

Work it out: Sector angle=(part/whole)×360° = (126/162)×360° = 280°. The fraction is 7/9 and share approximately 77.778%.

Common first error: Reversing total and count gives a fraction greater than one, which could produce an impossible sector larger than a full 360° circle.

Independent check: The rest of the pie occupies 80° and corresponds to 36 other observations. Together they reconstruct the whole.

Transfer: If another category has the same frequency, it must receive the same sector angle within this pie, even if its label is different.

73. Pictogram 1: use the key to count

Question: A hypothetical pictogram key says each full symbol stands for 4 items. One category shows 2 full symbols and a half symbol. Find its represented frequency.

Work it out: A full symbol represents 4, so 2 symbols count 8. The half-symbol adds 2. The total is 10 items.

Common first error: Counting symbols rather than the key-scaled items gives only 2.5, not the represented frequency. A half symbol is half the key’s value, not automatically half an item.

Independent check: The icon-equivalent count is 2.5. Multiplying by 4 returns 10, the reported frequency.

Transfer: A second category has 1 full icons and represents 4. Compare the two by counts, not visual decoration. Their absolute difference is 6 items.

74. Pictogram 2: use the key to count

Question: A hypothetical pictogram key says each full symbol stands for 6 items. One category shows 5 full symbols. Find its represented frequency.

Work it out: A full symbol represents 6, so 5 symbols count 30. There is no additional fraction of an icon. The total is 30 items.

Common first error: Counting symbols rather than the key-scaled items gives only 5, not the represented frequency. A half symbol is half the key’s value, not automatically half an item.

Independent check: The icon-equivalent count is 5. Multiplying by 6 returns 30, the reported frequency.

Transfer: A second category has 6 full icons and represents 36. Compare the two by counts, not visual decoration. Their absolute difference is 6 items.

75. Pictogram 3: use the key to count

Question: A hypothetical pictogram key says each full symbol stands for 8 items. One category shows 2 full symbols. Find its represented frequency.

Work it out: A full symbol represents 8, so 2 symbols count 16. There is no additional fraction of an icon. The total is 16 items.

Common first error: Counting symbols rather than the key-scaled items gives only 2, not the represented frequency. A half symbol is half the key’s value, not automatically half an item.

Independent check: The icon-equivalent count is 2. Multiplying by 8 returns 16, the reported frequency.

Transfer: A second category has 4 full icons and represents 32. Compare the two by counts, not visual decoration. Their absolute difference is 16 items.

76. Pictogram 4: use the key to count

Question: A hypothetical pictogram key says each full symbol stands for 10 items. One category shows 5 full symbols and a half symbol. Find its represented frequency.

Work it out: A full symbol represents 10, so 5 symbols count 50. The half-symbol adds 5. The total is 55 items.

Common first error: Counting symbols rather than the key-scaled items gives only 5.5, not the represented frequency. A half symbol is half the key’s value, not automatically half an item.

Independent check: The icon-equivalent count is 5.5. Multiplying by 10 returns 55, the reported frequency.

Transfer: A second category has 2 full icons and represents 20. Compare the two by counts, not visual decoration. Their absolute difference is 35 items.

77. Pictogram 5: use the key to count

Question: A hypothetical pictogram key says each full symbol stands for 12 items. One category shows 2 full symbols. Find its represented frequency.

Work it out: A full symbol represents 12, so 2 symbols count 24. There is no additional fraction of an icon. The total is 24 items.

Common first error: Counting symbols rather than the key-scaled items gives only 2, not the represented frequency. A half symbol is half the key’s value, not automatically half an item.

Independent check: The icon-equivalent count is 2. Multiplying by 12 returns 24, the reported frequency.

Transfer: A second category has 7 full icons and represents 84. Compare the two by counts, not visual decoration. Their absolute difference is 60 items.

78. Pictogram 6: use the key to count

Question: A hypothetical pictogram key says each full symbol stands for 14 items. One category shows 5 full symbols. Find its represented frequency.

Work it out: A full symbol represents 14, so 5 symbols count 70. There is no additional fraction of an icon. The total is 70 items.

Common first error: Counting symbols rather than the key-scaled items gives only 5, not the represented frequency. A half symbol is half the key’s value, not automatically half an item.

Independent check: The icon-equivalent count is 5. Multiplying by 14 returns 70, the reported frequency.

Transfer: A second category has 5 full icons and represents 70. Compare the two by counts, not visual decoration. Their absolute difference is 0 items.

79. Pictogram 7: use the key to count

Question: A hypothetical pictogram key says each full symbol stands for 16 items. One category shows 2 full symbols and a half symbol. Find its represented frequency.

Work it out: A full symbol represents 16, so 2 symbols count 32. The half-symbol adds 8. The total is 40 items.

Common first error: Counting symbols rather than the key-scaled items gives only 2.5, not the represented frequency. A half symbol is half the key’s value, not automatically half an item.

Independent check: The icon-equivalent count is 2.5. Multiplying by 16 returns 40, the reported frequency.

Transfer: A second category has 3 full icons and represents 48. Compare the two by counts, not visual decoration. Their absolute difference is 8 items.

80. Pictogram 8: use the key to count

Question: A hypothetical pictogram key says each full symbol stands for 18 items. One category shows 5 full symbols. Find its represented frequency.

Work it out: A full symbol represents 18, so 5 symbols count 90. There is no additional fraction of an icon. The total is 90 items.

Common first error: Counting symbols rather than the key-scaled items gives only 5, not the represented frequency. A half symbol is half the key’s value, not automatically half an item.

Independent check: The icon-equivalent count is 5. Multiplying by 18 returns 90, the reported frequency.

Transfer: A second category has 1 full icons and represents 18. Compare the two by counts, not visual decoration. Their absolute difference is 72 items.

81. Pictogram 9: use the key to count

Question: A hypothetical pictogram key says each full symbol stands for 20 items. One category shows 2 full symbols. Find its represented frequency.

Work it out: A full symbol represents 20, so 2 symbols count 40. There is no additional fraction of an icon. The total is 40 items.

Common first error: Counting symbols rather than the key-scaled items gives only 2, not the represented frequency. A half symbol is half the key’s value, not automatically half an item.

Independent check: The icon-equivalent count is 2. Multiplying by 20 returns 40, the reported frequency.

Transfer: A second category has 6 full icons and represents 120. Compare the two by counts, not visual decoration. Their absolute difference is 80 items.

82. Pictogram 10: use the key to count

Question: A hypothetical pictogram key says each full symbol stands for 22 items. One category shows 5 full symbols and a half symbol. Find its represented frequency.

Work it out: A full symbol represents 22, so 5 symbols count 110. The half-symbol adds 11. The total is 121 items.

Common first error: Counting symbols rather than the key-scaled items gives only 5.5, not the represented frequency. A half symbol is half the key’s value, not automatically half an item.

Independent check: The icon-equivalent count is 5.5. Multiplying by 22 returns 121, the reported frequency.

Transfer: A second category has 4 full icons and represents 88. Compare the two by counts, not visual decoration. Their absolute difference is 33 items.

83. Pictogram 11: use the key to count

Question: A hypothetical pictogram key says each full symbol stands for 24 items. One category shows 2 full symbols. Find its represented frequency.

Work it out: A full symbol represents 24, so 2 symbols count 48. There is no additional fraction of an icon. The total is 48 items.

Common first error: Counting symbols rather than the key-scaled items gives only 2, not the represented frequency. A half symbol is half the key’s value, not automatically half an item.

Independent check: The icon-equivalent count is 2. Multiplying by 24 returns 48, the reported frequency.

Transfer: A second category has 2 full icons and represents 48. Compare the two by counts, not visual decoration. Their absolute difference is 0 items.

84. Pictogram 12: use the key to count

Question: A hypothetical pictogram key says each full symbol stands for 4 items. One category shows 5 full symbols. Find its represented frequency.

Work it out: A full symbol represents 4, so 5 symbols count 20. There is no additional fraction of an icon. The total is 20 items.

Common first error: Counting symbols rather than the key-scaled items gives only 5, not the represented frequency. A half symbol is half the key’s value, not automatically half an item.

Independent check: The icon-equivalent count is 5. Multiplying by 4 returns 20, the reported frequency.

Transfer: A second category has 7 full icons and represents 28. Compare the two by counts, not visual decoration. Their absolute difference is 8 items.

85. Pictogram 13: use the key to count

Question: A hypothetical pictogram key says each full symbol stands for 6 items. One category shows 2 full symbols and a half symbol. Find its represented frequency.

Work it out: A full symbol represents 6, so 2 symbols count 12. The half-symbol adds 3. The total is 15 items.

Common first error: Counting symbols rather than the key-scaled items gives only 2.5, not the represented frequency. A half symbol is half the key’s value, not automatically half an item.

Independent check: The icon-equivalent count is 2.5. Multiplying by 6 returns 15, the reported frequency.

Transfer: A second category has 5 full icons and represents 30. Compare the two by counts, not visual decoration. Their absolute difference is 15 items.

86. Pictogram 14: use the key to count

Question: A hypothetical pictogram key says each full symbol stands for 8 items. One category shows 5 full symbols. Find its represented frequency.

Work it out: A full symbol represents 8, so 5 symbols count 40. There is no additional fraction of an icon. The total is 40 items.

Common first error: Counting symbols rather than the key-scaled items gives only 5, not the represented frequency. A half symbol is half the key’s value, not automatically half an item.

Independent check: The icon-equivalent count is 5. Multiplying by 8 returns 40, the reported frequency.

Transfer: A second category has 3 full icons and represents 24. Compare the two by counts, not visual decoration. Their absolute difference is 16 items.

87. Pictogram 15: use the key to count

Question: A hypothetical pictogram key says each full symbol stands for 10 items. One category shows 2 full symbols. Find its represented frequency.

Work it out: A full symbol represents 10, so 2 symbols count 20. There is no additional fraction of an icon. The total is 20 items.

Common first error: Counting symbols rather than the key-scaled items gives only 2, not the represented frequency. A half symbol is half the key’s value, not automatically half an item.

Independent check: The icon-equivalent count is 2. Multiplying by 10 returns 20, the reported frequency.

Transfer: A second category has 1 full icons and represents 10. Compare the two by counts, not visual decoration. Their absolute difference is 10 items.

88. Pictogram 16: use the key to count

Question: A hypothetical pictogram key says each full symbol stands for 12 items. One category shows 5 full symbols and a half symbol. Find its represented frequency.

Work it out: A full symbol represents 12, so 5 symbols count 60. The half-symbol adds 6. The total is 66 items.

Common first error: Counting symbols rather than the key-scaled items gives only 5.5, not the represented frequency. A half symbol is half the key’s value, not automatically half an item.

Independent check: The icon-equivalent count is 5.5. Multiplying by 12 returns 66, the reported frequency.

Transfer: A second category has 6 full icons and represents 72. Compare the two by counts, not visual decoration. Their absolute difference is 6 items.

89. Pictogram 17: use the key to count

Question: A hypothetical pictogram key says each full symbol stands for 14 items. One category shows 2 full symbols. Find its represented frequency.

Work it out: A full symbol represents 14, so 2 symbols count 28. There is no additional fraction of an icon. The total is 28 items.

Common first error: Counting symbols rather than the key-scaled items gives only 2, not the represented frequency. A half symbol is half the key’s value, not automatically half an item.

Independent check: The icon-equivalent count is 2. Multiplying by 14 returns 28, the reported frequency.

Transfer: A second category has 4 full icons and represents 56. Compare the two by counts, not visual decoration. Their absolute difference is 28 items.

90. Pictogram 18: use the key to count

Question: A hypothetical pictogram key says each full symbol stands for 16 items. One category shows 5 full symbols. Find its represented frequency.

Work it out: A full symbol represents 16, so 5 symbols count 80. There is no additional fraction of an icon. The total is 80 items.

Common first error: Counting symbols rather than the key-scaled items gives only 5, not the represented frequency. A half symbol is half the key’s value, not automatically half an item.

Independent check: The icon-equivalent count is 5. Multiplying by 16 returns 80, the reported frequency.

Transfer: A second category has 2 full icons and represents 32. Compare the two by counts, not visual decoration. Their absolute difference is 48 items.

91. Pictogram 19: use the key to count

Question: A hypothetical pictogram key says each full symbol stands for 18 items. One category shows 2 full symbols and a half symbol. Find its represented frequency.

Work it out: A full symbol represents 18, so 2 symbols count 36. The half-symbol adds 9. The total is 45 items.

Common first error: Counting symbols rather than the key-scaled items gives only 2.5, not the represented frequency. A half symbol is half the key’s value, not automatically half an item.

Independent check: The icon-equivalent count is 2.5. Multiplying by 18 returns 45, the reported frequency.

Transfer: A second category has 7 full icons and represents 126. Compare the two by counts, not visual decoration. Their absolute difference is 81 items.

92. Pictogram 20: use the key to count

Question: A hypothetical pictogram key says each full symbol stands for 20 items. One category shows 5 full symbols. Find its represented frequency.

Work it out: A full symbol represents 20, so 5 symbols count 100. There is no additional fraction of an icon. The total is 100 items.

Common first error: Counting symbols rather than the key-scaled items gives only 5, not the represented frequency. A half symbol is half the key’s value, not automatically half an item.

Independent check: The icon-equivalent count is 5. Multiplying by 20 returns 100, the reported frequency.

Transfer: A second category has 5 full icons and represents 100. Compare the two by counts, not visual decoration. Their absolute difference is 0 items.

93. Pictogram 21: use the key to count

Question: A hypothetical pictogram key says each full symbol stands for 22 items. One category shows 2 full symbols. Find its represented frequency.

Work it out: A full symbol represents 22, so 2 symbols count 44. There is no additional fraction of an icon. The total is 44 items.

Common first error: Counting symbols rather than the key-scaled items gives only 2, not the represented frequency. A half symbol is half the key’s value, not automatically half an item.

Independent check: The icon-equivalent count is 2. Multiplying by 22 returns 44, the reported frequency.

Transfer: A second category has 3 full icons and represents 66. Compare the two by counts, not visual decoration. Their absolute difference is 22 items.

94. Pictogram 22: use the key to count

Question: A hypothetical pictogram key says each full symbol stands for 24 items. One category shows 5 full symbols and a half symbol. Find its represented frequency.

Work it out: A full symbol represents 24, so 5 symbols count 120. The half-symbol adds 12. The total is 132 items.

Common first error: Counting symbols rather than the key-scaled items gives only 5.5, not the represented frequency. A half symbol is half the key’s value, not automatically half an item.

Independent check: The icon-equivalent count is 5.5. Multiplying by 24 returns 132, the reported frequency.

Transfer: A second category has 1 full icons and represents 24. Compare the two by counts, not visual decoration. Their absolute difference is 108 items.

95. Pictogram 23: use the key to count

Question: A hypothetical pictogram key says each full symbol stands for 4 items. One category shows 2 full symbols. Find its represented frequency.

Work it out: A full symbol represents 4, so 2 symbols count 8. There is no additional fraction of an icon. The total is 8 items.

Common first error: Counting symbols rather than the key-scaled items gives only 2, not the represented frequency. A half symbol is half the key’s value, not automatically half an item.

Independent check: The icon-equivalent count is 2. Multiplying by 4 returns 8, the reported frequency.

Transfer: A second category has 6 full icons and represents 24. Compare the two by counts, not visual decoration. Their absolute difference is 16 items.

96. Pictogram 24: use the key to count

Question: A hypothetical pictogram key says each full symbol stands for 6 items. One category shows 5 full symbols. Find its represented frequency.

Work it out: A full symbol represents 6, so 5 symbols count 30. There is no additional fraction of an icon. The total is 30 items.

Common first error: Counting symbols rather than the key-scaled items gives only 5, not the represented frequency. A half symbol is half the key’s value, not automatically half an item.

Independent check: The icon-equivalent count is 5. Multiplying by 6 returns 30, the reported frequency.

Transfer: A second category has 4 full icons and represents 24. Compare the two by counts, not visual decoration. Their absolute difference is 6 items.

97. Pictogram 25: use the key to count

Question: A hypothetical pictogram key says each full symbol stands for 8 items. One category shows 2 full symbols and a half symbol. Find its represented frequency.

Work it out: A full symbol represents 8, so 2 symbols count 16. The half-symbol adds 4. The total is 20 items.

Common first error: Counting symbols rather than the key-scaled items gives only 2.5, not the represented frequency. A half symbol is half the key’s value, not automatically half an item.

Independent check: The icon-equivalent count is 2.5. Multiplying by 8 returns 20, the reported frequency.

Transfer: A second category has 2 full icons and represents 16. Compare the two by counts, not visual decoration. Their absolute difference is 4 items.

98. Pictogram 26: use the key to count

Question: A hypothetical pictogram key says each full symbol stands for 10 items. One category shows 5 full symbols. Find its represented frequency.

Work it out: A full symbol represents 10, so 5 symbols count 50. There is no additional fraction of an icon. The total is 50 items.

Common first error: Counting symbols rather than the key-scaled items gives only 5, not the represented frequency. A half symbol is half the key’s value, not automatically half an item.

Independent check: The icon-equivalent count is 5. Multiplying by 10 returns 50, the reported frequency.

Transfer: A second category has 7 full icons and represents 70. Compare the two by counts, not visual decoration. Their absolute difference is 20 items.

99. A full pie chart is 360 degrees

Question: What angle corresponds to 100% of a complete pie?

Work it out: 360°. A full circle represents the one whole partition.

Common first error: Using 180°, the angle along a straight line, instead of a full turn.

Independent check: Four sectors of ninety degrees each fill the full circle.

100. Half a circle is not 180 percent

Question: A sector spans 180°. What percentage is it?

Work it out: 180/360=1/2=50%.

Common first error: Treating the degree number 180 as the percentage number.

Independent check: Two equal 180° halves fill a full 360°.

101. A right-angle sector

Question: A sector is 90°. Find its share of the whole.

Work it out: 90/360=1/4=25%.

Common first error: Reporting 90% because 90 appears in the picture.

Independent check: Four such sectors fill the whole.

102. A 72-degree sector

Question: Find the percentage represented by a 72° slice.

Work it out: 72/360=1/5=20%.

Common first error: Reporting 72%, which exceeds the actual share.

Independent check: Five equal slices of 72° cover 360°.

103. Equal categories

Question: An exhaustive pie has three categories of equal frequency. What angle for each?

Work it out: 360/3=120° for each.

Common first error: Using thirty-three degrees after mixing percentage with angle units.

Independent check: 120×3=360°.

104. Pie sector missing angle

Question: Three sectors are 90°,108°,72°. What is the fourth in a complete four-category pie?

Work it out: Total known is 270°, so missing angle is 90°.

Common first error: Subtracting from 180°, not 360°.

Independent check: All four angles add to 360°.

105. Category frequency sum

Question: A survey total is 60, with category counts 10,15,20 and unknown. Find the unknown.

Work it out: 60−10−15−20=15 observations.

Common first error: Assuming angles without first finding the missing count.

Independent check: The four counts sum to sixty.

106. Pie categories exceed one whole

Question: Shares are 45%,35%,30% in three supposedly disjoint exhaustive categories. Is the pie valid?

Work it out: No. They total 110%, which cannot describe one partition.

Common first error: Trusting a nice-looking circle rather than arithmetic.

Independent check: Recheck denominators and whether categories overlap.

107. Pictogram with zero symbols

Question: A row explicitly labelled B has zero symbols under key one icon=5. What is its frequency?

Work it out: Zero items in this represented dataset.

Common first error: Assuming an entirely omitted category also necessarily means zero.

Independent check: Distinguish a labelled empty row from missing data.

108. Half an icon for six people

Question: A key says one full icon means six people. What does a half icon mean?

Work it out: Three people.

Common first error: Calling it half a person because one half-symbol is visible.

Independent check: 3+3=6, reconstructing a full icon.

109. A questionable key for discrete people

Question: One icon represents five people, but the chart shows a half icon. Does this represent an exact whole-person count?

Work it out: No, not without some alternate annotation: half of five is 2.5 people, which is not a valid exact count of individuals.

Common first error: Using fractions of persons without considering the object being counted.

Independent check: Choose a compatible key or state that the pictogram is approximate.

110. Pie and histogram are different

Question: A display partitions a whole sample into favourite-subject sectors. Is it a histogram?

Work it out: No. It is a categorical pie chart. A histogram groups numerical measurements into intervals.

Common first error: Calling any visual about frequency a histogram.

Independent check: Read the kind of horizontal numerical or categorical variable.

111. One chart may have a smaller sample

Question: Two pies have 50% sectors, one over 40 people and the other over 100. Are category counts equal?

Work it out: No. Counts are 20 and 50, respectively.

Common first error: Equating equal percentages with equal absolute frequencies.

Independent check: Multiply each proportion by its own total.

112. Sample does not mean all residents

Question: Six of ten fictional survey respondents choose an option. Does this establish a 60% preference among all Singapore residents?

Work it out: No. It shows 60% of the ten respondents only; population generalisation needs appropriate sampling.

Common first error: Making unsupported broader claims from a small convenience sample.

Independent check: State the sample and the observation period.

113. Pie ‘Other’ hides detail

Question: A sector labelled Other is 30%. Can you name its most common subcategory?

Work it out: No. The pie shows only the combined share of Other, not its internal breakdown.

Common first error: Guessing a category from its overall sector size.

Independent check: Request the raw subgroup frequency table.

114. A pie sector cannot exceed one whole

Question: A category is labelled 140% of the same complete pie. Is that possible?

Work it out: No. In a nonoverlapping partition, one category cannot exceed the 100% whole.

Common first error: Confusing a percentage increase over 100% with a share of a fixed sample.

Independent check: Its angle would be 504°, exceeding 360°.

115. Quarter of a pictogram key

Question: If one icon is eight items, what does a quarter icon stand for?

Work it out: Two items.

Common first error: Calling it 0.25 item regardless of key value.

Independent check: Four quarters of two items reconstruct eight.

116. Pictogram rows require one scale

Question: Could two rows with different icon sizes be compared accurately with no legend?

Work it out: Not reliably. A shared key and consistent representation are required.

Common first error: Comparing visual area instead of encoded values.

Independent check: Read the legend before drawing conclusions.

117. A percentage can be used to draw a pie

Question: If 12.5% of a class is in category A, what is its angle?

Work it out: 0.125×360°=45°.

Common first error: Using 12.5° as a sector angle.

Independent check: 45/360=0.125.

118. Frequency from five percent

Question: If five percent of a fictional eighty-response sample belongs to A, how many responses are A?

Work it out: 0.05×80=4, corresponding to 18°.

Common first error: Reporting five people directly from a five percent label.

Independent check: Four of eighty is one twentieth, or five percent.

119. Pictogram with 1.5 full equivalents

Question: One full icon=12 books. What do 1.5 icons represent?

Work it out: 1.5×12=18 books.

Common first error: Reading 1.5 icons as 1.5 books.

Independent check: Eighteen divided by twelve gives 1.5 icon equivalents.

120. No total, no absolute count

Question: A category is 25% of a pie. Can its exact frequency be found without the sample total?

Work it out: No. One quarter of an unspecified total has no determined absolute frequency.

Common first error: Reporting twenty-five observations from a twenty-five percent share.

Independent check: For totals forty and eighty, the category counts would be ten and twenty.

121. Chart date is necessary for ‘today’

Question: A pie chart has no publication date or collection period. Can it tell us what happened today?

Work it out: No. The chart’s time window is not specified.

Common first error: Treating an old or undated chart as current evidence.

Independent check: Find source date and measurement period.

122. Two close angles

Question: Sector A is 71° and B is 72°. Which has more frequency in the same pie?

Work it out: B, by a small difference corresponding to one degree of the same total.

Common first error: Reading similar-looking wedges as exactly equal.

Independent check: Compare printed angles, not line thickness.

123. A key may encode tens

Question: A pictogram key is one symbol=20 objects. A row has four symbols. Find count.

Work it out: 4×20=80 objects.

Common first error: Treating four visible icons as four objects.

Independent check: 80÷20=4 icon equivalents.

124. The whole must be exclusive and exhaustive

Question: Why should all pie categories add to one whole?

Work it out: They form a partition: each observed item appears in exactly one displayed category, and all are included.

Common first error: Double-counting overlapping categories or omitting unlabeled groups.

Independent check: Sum counts to sample total and angles to 360°.

Five common G1 chart mistakes

Sector angle is read as a percentage

Pictogram key is ignored

A category is mistaken for the total

Categories do not sum to one whole

A chart is treated as a universal survey

How a small-group tutor can teach charts

10 minutes: retrieve one-whole fractions and percentage equivalents.
20 minutes: draw pie sectors from simple fractions and counts.
25 minutes: individually solve sector-to-frequency and reverse-total questions.
15 minutes: count pictogram full and half-symbol equivalents.
10 minutes: identify one conclusion a chart does not justify.
10 minutes: correct the first mistake and retest with new data.

Five manageable home practice sessions

Day one: change 90°,120°,180° into fractions of a circle.
Day two: find category counts using stated fictional survey totals.
Day three: reverse a pie sector to recover total sample size.
Day four: create a pictogram with a consistent symbol key and a valid half symbol.
Day five: explain a sample-size limitation or category labelling issue in one sentence.

Current G1 curriculum versus the SEC year

Frequently asked questions

Does 90° mean 90% in a pie chart?

No. It is ninety of the full 360 degrees, or one quarter = 25%.

How do you find total observations from one sector?

Divide its category count by the fraction angle/360°.

Is a half icon half a person?

No. It is half of the key’s represented count, which should yield a valid count or be labelled approximate.

Can pie charts and histograms replace each other?

No. Pie charts show shares of one whole; histograms show distributions across numerical intervals.

Can equal pie sectors represent different respondent counts?

Yes, if the pies have different sample sizes.

What should a tutor check?

Can the pupil move independently among sector angles, fractions, percentages and absolute counts, and use a pictogram key accurately?

Official syllabus and accuracy note

Bring a misread pie chart or pictogram worksheet to eduKate Singapore and ask how the tutor will test the learner’s understanding of the whole and the chart key.

Did you know? A pie chart sector of 90° is not 90% of the whole. A full circle is 360°, so the sector is only one quarter: 25%. And three icons in a pictogram might represent thirty items rather than three, if each icon means ten. These are two common reasons a student can correctly count what they see but give the wrong mathematical answer.

Quick answer: how to read pie charts and pictograms

For a pie chart, sector fraction = angle/360°. To find frequency, multiply the sector fraction by the sample total. To find sector angle from frequency, use frequency/total×360°. For a pictogram, count symbols using the stated key: three full icons with key one icon=8 items represent 24 items. A half-icon means half the key’s quantity. In both displays, check the whole, units and category labels before calculating.

This guide is for families seeking Secondary 2 G1 Mathematics tuition in Bukit Timah. MOE’s G1 Mathematics lower-secondary outline introduces pie charts and pictograms in foundational Data Handling and Analysis, while Secondary Two further develops dot diagrams and equal-width histograms. The published 2027 SEC K110 G1 syllabus covers these representation types across the graduating course. This guide is prerequisite consolidation and critical reading, not a guarantee of the next school test’s content.

A one-minute chart diagnostic

The 360-degree circle is a mathematical whole

Two identical sectors can represent different counts

How to recover the sample total

An icon key is a unit conversion

What a chart does not tell you

124 original pie-chart and pictogram clinics

1. Pie chart 1: sector to category frequency

Question: A fictional pie chart shows 16 observations. A category occupies 180 degrees of the full circle. Find how many observations belong to it.

Work it out: Share is 180/360=1/2. Multiply by total 16: 8 observations. This also represents 50% of the whole.

Common first error: Treating 180 degrees as 180% or as 180 individual observations confuses angle, proportion and frequency. A sector does not have an independent count without the total.

Independent check: The reverse calculation (8/16)×360°=180°. The category count is no more than the full sample 16.

Transfer: If the same share applied to 32 responses, the count would double to 16, but the angle would remain 180°.

2. Pie chart 1: recover the complete sample

Question: An imaginary category has 8 responses and occupies 180° of a pie chart representing every recorded response. Find the whole sample size.

Work it out: The category is 1/2 of its whole. Divide the count 8 by 1/2 to recover 16 responses.

Common first error: Multiplying the category count by 1/2 instead would yield less than the category and cannot represent the total population being counted.

Independent check: 8/16=1/2, and (8/16)×360°=180°, matching the supplied angle.

Transfer: Ask what happens to the total if the same category frequency instead occupies a smaller sector angle. The total must be larger if the recorded count stays fixed.

3. Pie chart 1: frequency to angle

Question: In a fictional survey of 16, category A has 8 observations. Work out the sector angle for A.

Work it out: Sector angle=(part/whole)×360° = (8/16)×360° = 180°. The fraction is 1/2 and share approximately 50%.

Common first error: Reversing total and count gives a fraction greater than one, which could produce an impossible sector larger than a full 360° circle.

Independent check: The rest of the pie occupies 180° and corresponds to 8 other observations. Together they reconstruct the whole.

Transfer: If another category has the same frequency, it must receive the same sector angle within this pie, even if its label is different.

4. Pie chart 2: sector to category frequency

Question: A fictional pie chart shows 27 observations. A category occupies 120 degrees of the full circle. Find how many observations belong to it.

Work it out: Share is 120/360=1/3. Multiply by total 27: 9 observations. This also represents 33.333% of the whole.

Common first error: Treating 120 degrees as 120% or as 120 individual observations confuses angle, proportion and frequency. A sector does not have an independent count without the total.

Independent check: The reverse calculation (9/27)×360°=120°. The category count is no more than the full sample 27.

Transfer: If the same share applied to 54 responses, the count would double to 18, but the angle would remain 120°.

5. Pie chart 2: recover the complete sample

Question: An imaginary category has 9 responses and occupies 120° of a pie chart representing every recorded response. Find the whole sample size.

Work it out: The category is 1/3 of its whole. Divide the count 9 by 1/3 to recover 27 responses.

Common first error: Multiplying the category count by 1/3 instead would yield less than the category and cannot represent the total population being counted.

Independent check: 9/27=1/3, and (9/27)×360°=120°, matching the supplied angle.

Transfer: Ask what happens to the total if the same category frequency instead occupies a smaller sector angle. The total must be larger if the recorded count stays fixed.

6. Pie chart 2: frequency to angle

Question: In a fictional survey of 27, category A has 9 observations. Work out the sector angle for A.

Work it out: Sector angle=(part/whole)×360° = (9/27)×360° = 120°. The fraction is 1/3 and share approximately 33.333%.

Common first error: Reversing total and count gives a fraction greater than one, which could produce an impossible sector larger than a full 360° circle.

Independent check: The rest of the pie occupies 240° and corresponds to 18 other observations. Together they reconstruct the whole.

Transfer: If another category has the same frequency, it must receive the same sector angle within this pie, even if its label is different.

7. Pie chart 3: sector to category frequency

Question: A fictional pie chart shows 40 observations. A category occupies 90 degrees of the full circle. Find how many observations belong to it.

Work it out: Share is 90/360=1/4. Multiply by total 40: 10 observations. This also represents 25% of the whole.

Common first error: Treating 90 degrees as 90% or as 90 individual observations confuses angle, proportion and frequency. A sector does not have an independent count without the total.

Independent check: The reverse calculation (10/40)×360°=90°. The category count is no more than the full sample 40.

Transfer: If the same share applied to 80 responses, the count would double to 20, but the angle would remain 90°.

8. Pie chart 3: recover the complete sample

Question: An imaginary category has 10 responses and occupies 90° of a pie chart representing every recorded response. Find the whole sample size.

Work it out: The category is 1/4 of its whole. Divide the count 10 by 1/4 to recover 40 responses.

Common first error: Multiplying the category count by 1/4 instead would yield less than the category and cannot represent the total population being counted.

Independent check: 10/40=1/4, and (10/40)×360°=90°, matching the supplied angle.

Transfer: Ask what happens to the total if the same category frequency instead occupies a smaller sector angle. The total must be larger if the recorded count stays fixed.

9. Pie chart 3: frequency to angle

Question: In a fictional survey of 40, category A has 10 observations. Work out the sector angle for A.

Work it out: Sector angle=(part/whole)×360° = (10/40)×360° = 90°. The fraction is 1/4 and share approximately 25%.

Common first error: Reversing total and count gives a fraction greater than one, which could produce an impossible sector larger than a full 360° circle.

Independent check: The rest of the pie occupies 270° and corresponds to 30 other observations. Together they reconstruct the whole.

Transfer: If another category has the same frequency, it must receive the same sector angle within this pie, even if its label is different.

10. Pie chart 4: sector to category frequency

Question: A fictional pie chart shows 55 observations. A category occupies 72 degrees of the full circle. Find how many observations belong to it.

Work it out: Share is 72/360=1/5. Multiply by total 55: 11 observations. This also represents 20% of the whole.

Common first error: Treating 72 degrees as 72% or as 72 individual observations confuses angle, proportion and frequency. A sector does not have an independent count without the total.

Independent check: The reverse calculation (11/55)×360°=72°. The category count is no more than the full sample 55.

Transfer: If the same share applied to 110 responses, the count would double to 22, but the angle would remain 72°.

11. Pie chart 4: recover the complete sample

Question: An imaginary category has 11 responses and occupies 72° of a pie chart representing every recorded response. Find the whole sample size.

Work it out: The category is 1/5 of its whole. Divide the count 11 by 1/5 to recover 55 responses.

Common first error: Multiplying the category count by 1/5 instead would yield less than the category and cannot represent the total population being counted.

Independent check: 11/55=1/5, and (11/55)×360°=72°, matching the supplied angle.

Transfer: Ask what happens to the total if the same category frequency instead occupies a smaller sector angle. The total must be larger if the recorded count stays fixed.

12. Pie chart 4: frequency to angle

Question: In a fictional survey of 55, category A has 11 observations. Work out the sector angle for A.

Work it out: Sector angle=(part/whole)×360° = (11/55)×360° = 72°. The fraction is 1/5 and share approximately 20%.

Common first error: Reversing total and count gives a fraction greater than one, which could produce an impossible sector larger than a full 360° circle.

Independent check: The rest of the pie occupies 288° and corresponds to 44 other observations. Together they reconstruct the whole.

Transfer: If another category has the same frequency, it must receive the same sector angle within this pie, even if its label is different.

13. Pie chart 5: sector to category frequency

Question: A fictional pie chart shows 72 observations. A category occupies 60 degrees of the full circle. Find how many observations belong to it.

Work it out: Share is 60/360=1/6. Multiply by total 72: 12 observations. This also represents 16.667% of the whole.

Common first error: Treating 60 degrees as 60% or as 60 individual observations confuses angle, proportion and frequency. A sector does not have an independent count without the total.

Independent check: The reverse calculation (12/72)×360°=60°. The category count is no more than the full sample 72.

Transfer: If the same share applied to 144 responses, the count would double to 24, but the angle would remain 60°.

14. Pie chart 5: recover the complete sample

Question: An imaginary category has 12 responses and occupies 60° of a pie chart representing every recorded response. Find the whole sample size.

Work it out: The category is 1/6 of its whole. Divide the count 12 by 1/6 to recover 72 responses.

Common first error: Multiplying the category count by 1/6 instead would yield less than the category and cannot represent the total population being counted.

Independent check: 12/72=1/6, and (12/72)×360°=60°, matching the supplied angle.

Transfer: Ask what happens to the total if the same category frequency instead occupies a smaller sector angle. The total must be larger if the recorded count stays fixed.

15. Pie chart 5: frequency to angle

Question: In a fictional survey of 72, category A has 12 observations. Work out the sector angle for A.

Work it out: Sector angle=(part/whole)×360° = (12/72)×360° = 60°. The fraction is 1/6 and share approximately 16.667%.

Common first error: Reversing total and count gives a fraction greater than one, which could produce an impossible sector larger than a full 360° circle.

Independent check: The rest of the pie occupies 300° and corresponds to 60 other observations. Together they reconstruct the whole.

Transfer: If another category has the same frequency, it must receive the same sector angle within this pie, even if its label is different.

16. Pie chart 6: sector to category frequency

Question: A fictional pie chart shows 65 observations. A category occupies 144 degrees of the full circle. Find how many observations belong to it.

Work it out: Share is 144/360=2/5. Multiply by total 65: 26 observations. This also represents 40% of the whole.

Common first error: Treating 144 degrees as 144% or as 144 individual observations confuses angle, proportion and frequency. A sector does not have an independent count without the total.

Independent check: The reverse calculation (26/65)×360°=144°. The category count is no more than the full sample 65.

Transfer: If the same share applied to 130 responses, the count would double to 52, but the angle would remain 144°.

17. Pie chart 6: recover the complete sample

Question: An imaginary category has 26 responses and occupies 144° of a pie chart representing every recorded response. Find the whole sample size.

Work it out: The category is 2/5 of its whole. Divide the count 26 by 2/5 to recover 65 responses.

Common first error: Multiplying the category count by 2/5 instead would yield less than the category and cannot represent the total population being counted.

Independent check: 26/65=2/5, and (26/65)×360°=144°, matching the supplied angle.

Transfer: Ask what happens to the total if the same category frequency instead occupies a smaller sector angle. The total must be larger if the recorded count stays fixed.

18. Pie chart 6: frequency to angle

Question: In a fictional survey of 65, category A has 26 observations. Work out the sector angle for A.

Work it out: Sector angle=(part/whole)×360° = (26/65)×360° = 144°. The fraction is 2/5 and share approximately 40%.

Common first error: Reversing total and count gives a fraction greater than one, which could produce an impossible sector larger than a full 360° circle.

Independent check: The rest of the pie occupies 216° and corresponds to 39 other observations. Together they reconstruct the whole.

Transfer: If another category has the same frequency, it must receive the same sector angle within this pie, even if its label is different.

19. Pie chart 7: sector to category frequency

Question: A fictional pie chart shows 70 observations. A category occupies 216 degrees of the full circle. Find how many observations belong to it.

Work it out: Share is 216/360=3/5. Multiply by total 70: 42 observations. This also represents 60% of the whole.

Common first error: Treating 216 degrees as 216% or as 216 individual observations confuses angle, proportion and frequency. A sector does not have an independent count without the total.

Independent check: The reverse calculation (42/70)×360°=216°. The category count is no more than the full sample 70.

Transfer: If the same share applied to 140 responses, the count would double to 84, but the angle would remain 216°.

20. Pie chart 7: recover the complete sample

Question: An imaginary category has 42 responses and occupies 216° of a pie chart representing every recorded response. Find the whole sample size.

Work it out: The category is 3/5 of its whole. Divide the count 42 by 3/5 to recover 70 responses.

Common first error: Multiplying the category count by 3/5 instead would yield less than the category and cannot represent the total population being counted.

Independent check: 42/70=3/5, and (42/70)×360°=216°, matching the supplied angle.

Transfer: Ask what happens to the total if the same category frequency instead occupies a smaller sector angle. The total must be larger if the recorded count stays fixed.

21. Pie chart 7: frequency to angle

Question: In a fictional survey of 70, category A has 42 observations. Work out the sector angle for A.

Work it out: Sector angle=(part/whole)×360° = (42/70)×360° = 216°. The fraction is 3/5 and share approximately 60%.

Common first error: Reversing total and count gives a fraction greater than one, which could produce an impossible sector larger than a full 360° circle.

Independent check: The rest of the pie occupies 144° and corresponds to 28 other observations. Together they reconstruct the whole.

Transfer: If another category has the same frequency, it must receive the same sector angle within this pie, even if its label is different.

22. Pie chart 8: sector to category frequency

Question: A fictional pie chart shows 45 observations. A category occupies 240 degrees of the full circle. Find how many observations belong to it.

Work it out: Share is 240/360=2/3. Multiply by total 45: 30 observations. This also represents 66.667% of the whole.

Common first error: Treating 240 degrees as 240% or as 240 individual observations confuses angle, proportion and frequency. A sector does not have an independent count without the total.

Independent check: The reverse calculation (30/45)×360°=240°. The category count is no more than the full sample 45.

Transfer: If the same share applied to 90 responses, the count would double to 60, but the angle would remain 240°.

23. Pie chart 8: recover the complete sample

Question: An imaginary category has 30 responses and occupies 240° of a pie chart representing every recorded response. Find the whole sample size.

Work it out: The category is 2/3 of its whole. Divide the count 30 by 2/3 to recover 45 responses.

Common first error: Multiplying the category count by 2/3 instead would yield less than the category and cannot represent the total population being counted.

Independent check: 30/45=2/3, and (30/45)×360°=240°, matching the supplied angle.

Transfer: Ask what happens to the total if the same category frequency instead occupies a smaller sector angle. The total must be larger if the recorded count stays fixed.

24. Pie chart 8: frequency to angle

Question: In a fictional survey of 45, category A has 30 observations. Work out the sector angle for A.

Work it out: Sector angle=(part/whole)×360° = (30/45)×360° = 240°. The fraction is 2/3 and share approximately 66.667%.

Common first error: Reversing total and count gives a fraction greater than one, which could produce an impossible sector larger than a full 360° circle.

Independent check: The rest of the pie occupies 120° and corresponds to 15 other observations. Together they reconstruct the whole.

Transfer: If another category has the same frequency, it must receive the same sector angle within this pie, even if its label is different.

25. Pie chart 9: sector to category frequency

Question: A fictional pie chart shows 64 observations. A category occupies 270 degrees of the full circle. Find how many observations belong to it.

Work it out: Share is 270/360=3/4. Multiply by total 64: 48 observations. This also represents 75% of the whole.

Common first error: Treating 270 degrees as 270% or as 270 individual observations confuses angle, proportion and frequency. A sector does not have an independent count without the total.

Independent check: The reverse calculation (48/64)×360°=270°. The category count is no more than the full sample 64.

Transfer: If the same share applied to 128 responses, the count would double to 96, but the angle would remain 270°.

26. Pie chart 9: recover the complete sample

Question: An imaginary category has 48 responses and occupies 270° of a pie chart representing every recorded response. Find the whole sample size.

Work it out: The category is 3/4 of its whole. Divide the count 48 by 3/4 to recover 64 responses.

Common first error: Multiplying the category count by 3/4 instead would yield less than the category and cannot represent the total population being counted.

Independent check: 48/64=3/4, and (48/64)×360°=270°, matching the supplied angle.

Transfer: Ask what happens to the total if the same category frequency instead occupies a smaller sector angle. The total must be larger if the recorded count stays fixed.

27. Pie chart 9: frequency to angle

Question: In a fictional survey of 64, category A has 48 observations. Work out the sector angle for A.

Work it out: Sector angle=(part/whole)×360° = (48/64)×360° = 270°. The fraction is 3/4 and share approximately 75%.

Common first error: Reversing total and count gives a fraction greater than one, which could produce an impossible sector larger than a full 360° circle.

Independent check: The rest of the pie occupies 90° and corresponds to 16 other observations. Together they reconstruct the whole.

Transfer: If another category has the same frequency, it must receive the same sector angle within this pie, even if its label is different.

28. Pie chart 10: sector to category frequency

Question: A fictional pie chart shows 136 observations. A category occupies 45 degrees of the full circle. Find how many observations belong to it.

Work it out: Share is 45/360=1/8. Multiply by total 136: 17 observations. This also represents 12.5% of the whole.

Common first error: Treating 45 degrees as 45% or as 45 individual observations confuses angle, proportion and frequency. A sector does not have an independent count without the total.

Independent check: The reverse calculation (17/136)×360°=45°. The category count is no more than the full sample 136.

Transfer: If the same share applied to 272 responses, the count would double to 34, but the angle would remain 45°.

29. Pie chart 10: recover the complete sample

Question: An imaginary category has 17 responses and occupies 45° of a pie chart representing every recorded response. Find the whole sample size.

Work it out: The category is 1/8 of its whole. Divide the count 17 by 1/8 to recover 136 responses.

Common first error: Multiplying the category count by 1/8 instead would yield less than the category and cannot represent the total population being counted.

Independent check: 17/136=1/8, and (17/136)×360°=45°, matching the supplied angle.

Transfer: Ask what happens to the total if the same category frequency instead occupies a smaller sector angle. The total must be larger if the recorded count stays fixed.

30. Pie chart 10: frequency to angle

Question: In a fictional survey of 136, category A has 17 observations. Work out the sector angle for A.

Work it out: Sector angle=(part/whole)×360° = (17/136)×360° = 45°. The fraction is 1/8 and share approximately 12.5%.

Common first error: Reversing total and count gives a fraction greater than one, which could produce an impossible sector larger than a full 360° circle.

Independent check: The rest of the pie occupies 315° and corresponds to 119 other observations. Together they reconstruct the whole.

Transfer: If another category has the same frequency, it must receive the same sector angle within this pie, even if its label is different.

31. Pie chart 11: sector to category frequency

Question: A fictional pie chart shows 144 observations. A category occupies 135 degrees of the full circle. Find how many observations belong to it.

Work it out: Share is 135/360=3/8. Multiply by total 144: 54 observations. This also represents 37.5% of the whole.

Common first error: Treating 135 degrees as 135% or as 135 individual observations confuses angle, proportion and frequency. A sector does not have an independent count without the total.

Independent check: The reverse calculation (54/144)×360°=135°. The category count is no more than the full sample 144.

Transfer: If the same share applied to 288 responses, the count would double to 108, but the angle would remain 135°.

32. Pie chart 11: recover the complete sample

Question: An imaginary category has 54 responses and occupies 135° of a pie chart representing every recorded response. Find the whole sample size.

Work it out: The category is 3/8 of its whole. Divide the count 54 by 3/8 to recover 144 responses.

Common first error: Multiplying the category count by 3/8 instead would yield less than the category and cannot represent the total population being counted.

Independent check: 54/144=3/8, and (54/144)×360°=135°, matching the supplied angle.

Transfer: Ask what happens to the total if the same category frequency instead occupies a smaller sector angle. The total must be larger if the recorded count stays fixed.

33. Pie chart 11: frequency to angle

Question: In a fictional survey of 144, category A has 54 observations. Work out the sector angle for A.

Work it out: Sector angle=(part/whole)×360° = (54/144)×360° = 135°. The fraction is 3/8 and share approximately 37.5%.

Common first error: Reversing total and count gives a fraction greater than one, which could produce an impossible sector larger than a full 360° circle.

Independent check: The rest of the pie occupies 225° and corresponds to 90 other observations. Together they reconstruct the whole.

Transfer: If another category has the same frequency, it must receive the same sector angle within this pie, even if its label is different.

34. Pie chart 12: sector to category frequency

Question: A fictional pie chart shows 228 observations. A category occupies 150 degrees of the full circle. Find how many observations belong to it.

Work it out: Share is 150/360=5/12. Multiply by total 228: 95 observations. This also represents 41.667% of the whole.

Common first error: Treating 150 degrees as 150% or as 150 individual observations confuses angle, proportion and frequency. A sector does not have an independent count without the total.

Independent check: The reverse calculation (95/228)×360°=150°. The category count is no more than the full sample 228.

Transfer: If the same share applied to 456 responses, the count would double to 190, but the angle would remain 150°.

35. Pie chart 12: recover the complete sample

Question: An imaginary category has 95 responses and occupies 150° of a pie chart representing every recorded response. Find the whole sample size.

Work it out: The category is 5/12 of its whole. Divide the count 95 by 5/12 to recover 228 responses.

Common first error: Multiplying the category count by 5/12 instead would yield less than the category and cannot represent the total population being counted.

Independent check: 95/228=5/12, and (95/228)×360°=150°, matching the supplied angle.

Transfer: Ask what happens to the total if the same category frequency instead occupies a smaller sector angle. The total must be larger if the recorded count stays fixed.

36. Pie chart 12: frequency to angle

Question: In a fictional survey of 228, category A has 95 observations. Work out the sector angle for A.

Work it out: Sector angle=(part/whole)×360° = (95/228)×360° = 150°. The fraction is 5/12 and share approximately 41.667%.

Common first error: Reversing total and count gives a fraction greater than one, which could produce an impossible sector larger than a full 360° circle.

Independent check: The rest of the pie occupies 210° and corresponds to 133 other observations. Together they reconstruct the whole.

Transfer: If another category has the same frequency, it must receive the same sector angle within this pie, even if its label is different.

37. Pie chart 13: sector to category frequency

Question: A fictional pie chart shows 240 observations. A category occupies 210 degrees of the full circle. Find how many observations belong to it.

Work it out: Share is 210/360=7/12. Multiply by total 240: 140 observations. This also represents 58.333% of the whole.

Common first error: Treating 210 degrees as 210% or as 210 individual observations confuses angle, proportion and frequency. A sector does not have an independent count without the total.

Independent check: The reverse calculation (140/240)×360°=210°. The category count is no more than the full sample 240.

Transfer: If the same share applied to 480 responses, the count would double to 280, but the angle would remain 210°.

38. Pie chart 13: recover the complete sample

Question: An imaginary category has 140 responses and occupies 210° of a pie chart representing every recorded response. Find the whole sample size.

Work it out: The category is 7/12 of its whole. Divide the count 140 by 7/12 to recover 240 responses.

Common first error: Multiplying the category count by 7/12 instead would yield less than the category and cannot represent the total population being counted.

Independent check: 140/240=7/12, and (140/240)×360°=210°, matching the supplied angle.

Transfer: Ask what happens to the total if the same category frequency instead occupies a smaller sector angle. The total must be larger if the recorded count stays fixed.

39. Pie chart 13: frequency to angle

Question: In a fictional survey of 240, category A has 140 observations. Work out the sector angle for A.

Work it out: Sector angle=(part/whole)×360° = (140/240)×360° = 210°. The fraction is 7/12 and share approximately 58.333%.

Common first error: Reversing total and count gives a fraction greater than one, which could produce an impossible sector larger than a full 360° circle.

Independent check: The rest of the pie occupies 150° and corresponds to 100 other observations. Together they reconstruct the whole.

Transfer: If another category has the same frequency, it must receive the same sector angle within this pie, even if its label is different.

40. Pie chart 14: sector to category frequency

Question: A fictional pie chart shows 80 observations. A category occupies 36 degrees of the full circle. Find how many observations belong to it.

Work it out: Share is 36/360=1/10. Multiply by total 80: 8 observations. This also represents 10% of the whole.

Common first error: Treating 36 degrees as 36% or as 36 individual observations confuses angle, proportion and frequency. A sector does not have an independent count without the total.

Independent check: The reverse calculation (8/80)×360°=36°. The category count is no more than the full sample 80.

Transfer: If the same share applied to 160 responses, the count would double to 16, but the angle would remain 36°.

41. Pie chart 14: recover the complete sample

Question: An imaginary category has 8 responses and occupies 36° of a pie chart representing every recorded response. Find the whole sample size.

Work it out: The category is 1/10 of its whole. Divide the count 8 by 1/10 to recover 80 responses.

Common first error: Multiplying the category count by 1/10 instead would yield less than the category and cannot represent the total population being counted.

Independent check: 8/80=1/10, and (8/80)×360°=36°, matching the supplied angle.

Transfer: Ask what happens to the total if the same category frequency instead occupies a smaller sector angle. The total must be larger if the recorded count stays fixed.

42. Pie chart 14: frequency to angle

Question: In a fictional survey of 80, category A has 8 observations. Work out the sector angle for A.

Work it out: Sector angle=(part/whole)×360° = (8/80)×360° = 36°. The fraction is 1/10 and share approximately 10%.

Common first error: Reversing total and count gives a fraction greater than one, which could produce an impossible sector larger than a full 360° circle.

Independent check: The rest of the pie occupies 324° and corresponds to 72 other observations. Together they reconstruct the whole.

Transfer: If another category has the same frequency, it must receive the same sector angle within this pie, even if its label is different.

43. Pie chart 15: sector to category frequency

Question: A fictional pie chart shows 90 observations. A category occupies 108 degrees of the full circle. Find how many observations belong to it.

Work it out: Share is 108/360=3/10. Multiply by total 90: 27 observations. This also represents 30% of the whole.

Common first error: Treating 108 degrees as 108% or as 108 individual observations confuses angle, proportion and frequency. A sector does not have an independent count without the total.

Independent check: The reverse calculation (27/90)×360°=108°. The category count is no more than the full sample 90.

Transfer: If the same share applied to 180 responses, the count would double to 54, but the angle would remain 108°.

44. Pie chart 15: recover the complete sample

Question: An imaginary category has 27 responses and occupies 108° of a pie chart representing every recorded response. Find the whole sample size.

Work it out: The category is 3/10 of its whole. Divide the count 27 by 3/10 to recover 90 responses.

Common first error: Multiplying the category count by 3/10 instead would yield less than the category and cannot represent the total population being counted.

Independent check: 27/90=3/10, and (27/90)×360°=108°, matching the supplied angle.

Transfer: Ask what happens to the total if the same category frequency instead occupies a smaller sector angle. The total must be larger if the recorded count stays fixed.

45. Pie chart 15: frequency to angle

Question: In a fictional survey of 90, category A has 27 observations. Work out the sector angle for A.

Work it out: Sector angle=(part/whole)×360° = (27/90)×360° = 108°. The fraction is 3/10 and share approximately 30%.

Common first error: Reversing total and count gives a fraction greater than one, which could produce an impossible sector larger than a full 360° circle.

Independent check: The rest of the pie occupies 252° and corresponds to 63 other observations. Together they reconstruct the whole.

Transfer: If another category has the same frequency, it must receive the same sector angle within this pie, even if its label is different.

46. Pie chart 16: sector to category frequency

Question: A fictional pie chart shows 90 observations. A category occupies 80 degrees of the full circle. Find how many observations belong to it.

Work it out: Share is 80/360=2/9. Multiply by total 90: 20 observations. This also represents 22.222% of the whole.

Common first error: Treating 80 degrees as 80% or as 80 individual observations confuses angle, proportion and frequency. A sector does not have an independent count without the total.

Independent check: The reverse calculation (20/90)×360°=80°. The category count is no more than the full sample 90.

Transfer: If the same share applied to 180 responses, the count would double to 40, but the angle would remain 80°.

47. Pie chart 16: recover the complete sample

Question: An imaginary category has 20 responses and occupies 80° of a pie chart representing every recorded response. Find the whole sample size.

Work it out: The category is 2/9 of its whole. Divide the count 20 by 2/9 to recover 90 responses.

Common first error: Multiplying the category count by 2/9 instead would yield less than the category and cannot represent the total population being counted.

Independent check: 20/90=2/9, and (20/90)×360°=80°, matching the supplied angle.

Transfer: Ask what happens to the total if the same category frequency instead occupies a smaller sector angle. The total must be larger if the recorded count stays fixed.

48. Pie chart 16: frequency to angle

Question: In a fictional survey of 90, category A has 20 observations. Work out the sector angle for A.

Work it out: Sector angle=(part/whole)×360° = (20/90)×360° = 80°. The fraction is 2/9 and share approximately 22.222%.

Common first error: Reversing total and count gives a fraction greater than one, which could produce an impossible sector larger than a full 360° circle.

Independent check: The rest of the pie occupies 280° and corresponds to 70 other observations. Together they reconstruct the whole.

Transfer: If another category has the same frequency, it must receive the same sector angle within this pie, even if its label is different.

49. Pie chart 17: sector to category frequency

Question: A fictional pie chart shows 99 observations. A category occupies 160 degrees of the full circle. Find how many observations belong to it.

Work it out: Share is 160/360=4/9. Multiply by total 99: 44 observations. This also represents 44.444% of the whole.

Common first error: Treating 160 degrees as 160% or as 160 individual observations confuses angle, proportion and frequency. A sector does not have an independent count without the total.

Independent check: The reverse calculation (44/99)×360°=160°. The category count is no more than the full sample 99.

Transfer: If the same share applied to 198 responses, the count would double to 88, but the angle would remain 160°.

50. Pie chart 17: recover the complete sample

Question: An imaginary category has 44 responses and occupies 160° of a pie chart representing every recorded response. Find the whole sample size.

Work it out: The category is 4/9 of its whole. Divide the count 44 by 4/9 to recover 99 responses.

Common first error: Multiplying the category count by 4/9 instead would yield less than the category and cannot represent the total population being counted.

Independent check: 44/99=4/9, and (44/99)×360°=160°, matching the supplied angle.

Transfer: Ask what happens to the total if the same category frequency instead occupies a smaller sector angle. The total must be larger if the recorded count stays fixed.

51. Pie chart 17: frequency to angle

Question: In a fictional survey of 99, category A has 44 observations. Work out the sector angle for A.

Work it out: Sector angle=(part/whole)×360° = (44/99)×360° = 160°. The fraction is 4/9 and share approximately 44.444%.

Common first error: Reversing total and count gives a fraction greater than one, which could produce an impossible sector larger than a full 360° circle.

Independent check: The rest of the pie occupies 200° and corresponds to 55 other observations. Together they reconstruct the whole.

Transfer: If another category has the same frequency, it must receive the same sector angle within this pie, even if its label is different.

52. Pie chart 18: sector to category frequency

Question: A fictional pie chart shows 108 observations. A category occupies 200 degrees of the full circle. Find how many observations belong to it.

Work it out: Share is 200/360=5/9. Multiply by total 108: 60 observations. This also represents 55.556% of the whole.

Common first error: Treating 200 degrees as 200% or as 200 individual observations confuses angle, proportion and frequency. A sector does not have an independent count without the total.

Independent check: The reverse calculation (60/108)×360°=200°. The category count is no more than the full sample 108.

Transfer: If the same share applied to 216 responses, the count would double to 120, but the angle would remain 200°.

53. Pie chart 18: recover the complete sample

Question: An imaginary category has 60 responses and occupies 200° of a pie chart representing every recorded response. Find the whole sample size.

Work it out: The category is 5/9 of its whole. Divide the count 60 by 5/9 to recover 108 responses.

Common first error: Multiplying the category count by 5/9 instead would yield less than the category and cannot represent the total population being counted.

Independent check: 60/108=5/9, and (60/108)×360°=200°, matching the supplied angle.

Transfer: Ask what happens to the total if the same category frequency instead occupies a smaller sector angle. The total must be larger if the recorded count stays fixed.

54. Pie chart 18: frequency to angle

Question: In a fictional survey of 108, category A has 60 observations. Work out the sector angle for A.

Work it out: Sector angle=(part/whole)×360° = (60/108)×360° = 200°. The fraction is 5/9 and share approximately 55.556%.

Common first error: Reversing total and count gives a fraction greater than one, which could produce an impossible sector larger than a full 360° circle.

Independent check: The rest of the pie occupies 160° and corresponds to 48 other observations. Together they reconstruct the whole.

Transfer: If another category has the same frequency, it must receive the same sector angle within this pie, even if its label is different.

55. Pie chart 19: sector to category frequency

Question: A fictional pie chart shows 78 observations. A category occupies 300 degrees of the full circle. Find how many observations belong to it.

Work it out: Share is 300/360=5/6. Multiply by total 78: 65 observations. This also represents 83.333% of the whole.

Common first error: Treating 300 degrees as 300% or as 300 individual observations confuses angle, proportion and frequency. A sector does not have an independent count without the total.

Independent check: The reverse calculation (65/78)×360°=300°. The category count is no more than the full sample 78.

Transfer: If the same share applied to 156 responses, the count would double to 130, but the angle would remain 300°.

56. Pie chart 19: recover the complete sample

Question: An imaginary category has 65 responses and occupies 300° of a pie chart representing every recorded response. Find the whole sample size.

Work it out: The category is 5/6 of its whole. Divide the count 65 by 5/6 to recover 78 responses.

Common first error: Multiplying the category count by 5/6 instead would yield less than the category and cannot represent the total population being counted.

Independent check: 65/78=5/6, and (65/78)×360°=300°, matching the supplied angle.

Transfer: Ask what happens to the total if the same category frequency instead occupies a smaller sector angle. The total must be larger if the recorded count stays fixed.

57. Pie chart 19: frequency to angle

Question: In a fictional survey of 78, category A has 65 observations. Work out the sector angle for A.

Work it out: Sector angle=(part/whole)×360° = (65/78)×360° = 300°. The fraction is 5/6 and share approximately 83.333%.

Common first error: Reversing total and count gives a fraction greater than one, which could produce an impossible sector larger than a full 360° circle.

Independent check: The rest of the pie occupies 60° and corresponds to 13 other observations. Together they reconstruct the whole.

Transfer: If another category has the same frequency, it must receive the same sector angle within this pie, even if its label is different.

58. Pie chart 20: sector to category frequency

Question: A fictional pie chart shows 112 observations. A category occupies 315 degrees of the full circle. Find how many observations belong to it.

Work it out: Share is 315/360=7/8. Multiply by total 112: 98 observations. This also represents 87.5% of the whole.

Common first error: Treating 315 degrees as 315% or as 315 individual observations confuses angle, proportion and frequency. A sector does not have an independent count without the total.

Independent check: The reverse calculation (98/112)×360°=315°. The category count is no more than the full sample 112.

Transfer: If the same share applied to 224 responses, the count would double to 196, but the angle would remain 315°.

59. Pie chart 20: recover the complete sample

Question: An imaginary category has 98 responses and occupies 315° of a pie chart representing every recorded response. Find the whole sample size.

Work it out: The category is 7/8 of its whole. Divide the count 98 by 7/8 to recover 112 responses.

Common first error: Multiplying the category count by 7/8 instead would yield less than the category and cannot represent the total population being counted.

Independent check: 98/112=7/8, and (98/112)×360°=315°, matching the supplied angle.

Transfer: Ask what happens to the total if the same category frequency instead occupies a smaller sector angle. The total must be larger if the recorded count stays fixed.

60. Pie chart 20: frequency to angle

Question: In a fictional survey of 112, category A has 98 observations. Work out the sector angle for A.

Work it out: Sector angle=(part/whole)×360° = (98/112)×360° = 315°. The fraction is 7/8 and share approximately 87.5%.

Common first error: Reversing total and count gives a fraction greater than one, which could produce an impossible sector larger than a full 360° circle.

Independent check: The rest of the pie occupies 45° and corresponds to 14 other observations. Together they reconstruct the whole.

Transfer: If another category has the same frequency, it must receive the same sector angle within this pie, even if its label is different.

61. Pie chart 21: sector to category frequency

Question: A fictional pie chart shows 180 observations. A category occupies 30 degrees of the full circle. Find how many observations belong to it.

Work it out: Share is 30/360=1/12. Multiply by total 180: 15 observations. This also represents 8.333% of the whole.

Common first error: Treating 30 degrees as 30% or as 30 individual observations confuses angle, proportion and frequency. A sector does not have an independent count without the total.

Independent check: The reverse calculation (15/180)×360°=30°. The category count is no more than the full sample 180.

Transfer: If the same share applied to 360 responses, the count would double to 30, but the angle would remain 30°.

62. Pie chart 21: recover the complete sample

Question: An imaginary category has 15 responses and occupies 30° of a pie chart representing every recorded response. Find the whole sample size.

Work it out: The category is 1/12 of its whole. Divide the count 15 by 1/12 to recover 180 responses.

Common first error: Multiplying the category count by 1/12 instead would yield less than the category and cannot represent the total population being counted.

Independent check: 15/180=1/12, and (15/180)×360°=30°, matching the supplied angle.

Transfer: Ask what happens to the total if the same category frequency instead occupies a smaller sector angle. The total must be larger if the recorded count stays fixed.

63. Pie chart 21: frequency to angle

Question: In a fictional survey of 180, category A has 15 observations. Work out the sector angle for A.

Work it out: Sector angle=(part/whole)×360° = (15/180)×360° = 30°. The fraction is 1/12 and share approximately 8.333%.

Common first error: Reversing total and count gives a fraction greater than one, which could produce an impossible sector larger than a full 360° circle.

Independent check: The rest of the pie occupies 330° and corresponds to 165 other observations. Together they reconstruct the whole.

Transfer: If another category has the same frequency, it must receive the same sector angle within this pie, even if its label is different.

64. Pie chart 22: sector to category frequency

Question: A fictional pie chart shows 192 observations. A category occupies 330 degrees of the full circle. Find how many observations belong to it.

Work it out: Share is 330/360=11/12. Multiply by total 192: 176 observations. This also represents 91.667% of the whole.

Common first error: Treating 330 degrees as 330% or as 330 individual observations confuses angle, proportion and frequency. A sector does not have an independent count without the total.

Independent check: The reverse calculation (176/192)×360°=330°. The category count is no more than the full sample 192.

Transfer: If the same share applied to 384 responses, the count would double to 352, but the angle would remain 330°.

65. Pie chart 22: recover the complete sample

Question: An imaginary category has 176 responses and occupies 330° of a pie chart representing every recorded response. Find the whole sample size.

Work it out: The category is 11/12 of its whole. Divide the count 176 by 11/12 to recover 192 responses.

Common first error: Multiplying the category count by 11/12 instead would yield less than the category and cannot represent the total population being counted.

Independent check: 176/192=11/12, and (176/192)×360°=330°, matching the supplied angle.

Transfer: Ask what happens to the total if the same category frequency instead occupies a smaller sector angle. The total must be larger if the recorded count stays fixed.

66. Pie chart 22: frequency to angle

Question: In a fictional survey of 192, category A has 176 observations. Work out the sector angle for A.

Work it out: Sector angle=(part/whole)×360° = (176/192)×360° = 330°. The fraction is 11/12 and share approximately 91.667%.

Common first error: Reversing total and count gives a fraction greater than one, which could produce an impossible sector larger than a full 360° circle.

Independent check: The rest of the pie occupies 30° and corresponds to 16 other observations. Together they reconstruct the whole.

Transfer: If another category has the same frequency, it must receive the same sector angle within this pie, even if its label is different.

67. Pie chart 23: sector to category frequency

Question: A fictional pie chart shows 153 observations. A category occupies 40 degrees of the full circle. Find how many observations belong to it.

Work it out: Share is 40/360=1/9. Multiply by total 153: 17 observations. This also represents 11.111% of the whole.

Common first error: Treating 40 degrees as 40% or as 40 individual observations confuses angle, proportion and frequency. A sector does not have an independent count without the total.

Independent check: The reverse calculation (17/153)×360°=40°. The category count is no more than the full sample 153.

Transfer: If the same share applied to 306 responses, the count would double to 34, but the angle would remain 40°.

68. Pie chart 23: recover the complete sample

Question: An imaginary category has 17 responses and occupies 40° of a pie chart representing every recorded response. Find the whole sample size.

Work it out: The category is 1/9 of its whole. Divide the count 17 by 1/9 to recover 153 responses.

Common first error: Multiplying the category count by 1/9 instead would yield less than the category and cannot represent the total population being counted.

Independent check: 17/153=1/9, and (17/153)×360°=40°, matching the supplied angle.

Transfer: Ask what happens to the total if the same category frequency instead occupies a smaller sector angle. The total must be larger if the recorded count stays fixed.

69. Pie chart 23: frequency to angle

Question: In a fictional survey of 153, category A has 17 observations. Work out the sector angle for A.

Work it out: Sector angle=(part/whole)×360° = (17/153)×360° = 40°. The fraction is 1/9 and share approximately 11.111%.

Common first error: Reversing total and count gives a fraction greater than one, which could produce an impossible sector larger than a full 360° circle.

Independent check: The rest of the pie occupies 320° and corresponds to 136 other observations. Together they reconstruct the whole.

Transfer: If another category has the same frequency, it must receive the same sector angle within this pie, even if its label is different.

70. Pie chart 24: sector to category frequency

Question: A fictional pie chart shows 162 observations. A category occupies 280 degrees of the full circle. Find how many observations belong to it.

Work it out: Share is 280/360=7/9. Multiply by total 162: 126 observations. This also represents 77.778% of the whole.

Common first error: Treating 280 degrees as 280% or as 280 individual observations confuses angle, proportion and frequency. A sector does not have an independent count without the total.

Independent check: The reverse calculation (126/162)×360°=280°. The category count is no more than the full sample 162.

Transfer: If the same share applied to 324 responses, the count would double to 252, but the angle would remain 280°.

71. Pie chart 24: recover the complete sample

Question: An imaginary category has 126 responses and occupies 280° of a pie chart representing every recorded response. Find the whole sample size.

Work it out: The category is 7/9 of its whole. Divide the count 126 by 7/9 to recover 162 responses.

Common first error: Multiplying the category count by 7/9 instead would yield less than the category and cannot represent the total population being counted.

Independent check: 126/162=7/9, and (126/162)×360°=280°, matching the supplied angle.

Transfer: Ask what happens to the total if the same category frequency instead occupies a smaller sector angle. The total must be larger if the recorded count stays fixed.

72. Pie chart 24: frequency to angle

Question: In a fictional survey of 162, category A has 126 observations. Work out the sector angle for A.

Work it out: Sector angle=(part/whole)×360° = (126/162)×360° = 280°. The fraction is 7/9 and share approximately 77.778%.

Common first error: Reversing total and count gives a fraction greater than one, which could produce an impossible sector larger than a full 360° circle.

Independent check: The rest of the pie occupies 80° and corresponds to 36 other observations. Together they reconstruct the whole.

Transfer: If another category has the same frequency, it must receive the same sector angle within this pie, even if its label is different.

73. Pictogram 1: use the key to count

Question: A hypothetical pictogram key says each full symbol stands for 4 items. One category shows 2 full symbols and a half symbol. Find its represented frequency.

Work it out: A full symbol represents 4, so 2 symbols count 8. The half-symbol adds 2. The total is 10 items.

Common first error: Counting symbols rather than the key-scaled items gives only 2.5, not the represented frequency. A half symbol is half the key’s value, not automatically half an item.

Independent check: The icon-equivalent count is 2.5. Multiplying by 4 returns 10, the reported frequency.

Transfer: A second category has 1 full icons and represents 4. Compare the two by counts, not visual decoration. Their absolute difference is 6 items.

74. Pictogram 2: use the key to count

Question: A hypothetical pictogram key says each full symbol stands for 6 items. One category shows 5 full symbols. Find its represented frequency.

Work it out: A full symbol represents 6, so 5 symbols count 30. There is no additional fraction of an icon. The total is 30 items.

Common first error: Counting symbols rather than the key-scaled items gives only 5, not the represented frequency. A half symbol is half the key’s value, not automatically half an item.

Independent check: The icon-equivalent count is 5. Multiplying by 6 returns 30, the reported frequency.

Transfer: A second category has 6 full icons and represents 36. Compare the two by counts, not visual decoration. Their absolute difference is 6 items.

75. Pictogram 3: use the key to count

Question: A hypothetical pictogram key says each full symbol stands for 8 items. One category shows 2 full symbols. Find its represented frequency.

Work it out: A full symbol represents 8, so 2 symbols count 16. There is no additional fraction of an icon. The total is 16 items.

Common first error: Counting symbols rather than the key-scaled items gives only 2, not the represented frequency. A half symbol is half the key’s value, not automatically half an item.

Independent check: The icon-equivalent count is 2. Multiplying by 8 returns 16, the reported frequency.

Transfer: A second category has 4 full icons and represents 32. Compare the two by counts, not visual decoration. Their absolute difference is 16 items.

76. Pictogram 4: use the key to count

Question: A hypothetical pictogram key says each full symbol stands for 10 items. One category shows 5 full symbols and a half symbol. Find its represented frequency.

Work it out: A full symbol represents 10, so 5 symbols count 50. The half-symbol adds 5. The total is 55 items.

Common first error: Counting symbols rather than the key-scaled items gives only 5.5, not the represented frequency. A half symbol is half the key’s value, not automatically half an item.

Independent check: The icon-equivalent count is 5.5. Multiplying by 10 returns 55, the reported frequency.

Transfer: A second category has 2 full icons and represents 20. Compare the two by counts, not visual decoration. Their absolute difference is 35 items.

77. Pictogram 5: use the key to count

Question: A hypothetical pictogram key says each full symbol stands for 12 items. One category shows 2 full symbols. Find its represented frequency.

Work it out: A full symbol represents 12, so 2 symbols count 24. There is no additional fraction of an icon. The total is 24 items.

Common first error: Counting symbols rather than the key-scaled items gives only 2, not the represented frequency. A half symbol is half the key’s value, not automatically half an item.

Independent check: The icon-equivalent count is 2. Multiplying by 12 returns 24, the reported frequency.

Transfer: A second category has 7 full icons and represents 84. Compare the two by counts, not visual decoration. Their absolute difference is 60 items.

78. Pictogram 6: use the key to count

Question: A hypothetical pictogram key says each full symbol stands for 14 items. One category shows 5 full symbols. Find its represented frequency.

Work it out: A full symbol represents 14, so 5 symbols count 70. There is no additional fraction of an icon. The total is 70 items.

Common first error: Counting symbols rather than the key-scaled items gives only 5, not the represented frequency. A half symbol is half the key’s value, not automatically half an item.

Independent check: The icon-equivalent count is 5. Multiplying by 14 returns 70, the reported frequency.

Transfer: A second category has 5 full icons and represents 70. Compare the two by counts, not visual decoration. Their absolute difference is 0 items.

79. Pictogram 7: use the key to count

Question: A hypothetical pictogram key says each full symbol stands for 16 items. One category shows 2 full symbols and a half symbol. Find its represented frequency.

Work it out: A full symbol represents 16, so 2 symbols count 32. The half-symbol adds 8. The total is 40 items.

Common first error: Counting symbols rather than the key-scaled items gives only 2.5, not the represented frequency. A half symbol is half the key’s value, not automatically half an item.

Independent check: The icon-equivalent count is 2.5. Multiplying by 16 returns 40, the reported frequency.

Transfer: A second category has 3 full icons and represents 48. Compare the two by counts, not visual decoration. Their absolute difference is 8 items.

80. Pictogram 8: use the key to count

Question: A hypothetical pictogram key says each full symbol stands for 18 items. One category shows 5 full symbols. Find its represented frequency.

Work it out: A full symbol represents 18, so 5 symbols count 90. There is no additional fraction of an icon. The total is 90 items.

Common first error: Counting symbols rather than the key-scaled items gives only 5, not the represented frequency. A half symbol is half the key’s value, not automatically half an item.

Independent check: The icon-equivalent count is 5. Multiplying by 18 returns 90, the reported frequency.

Transfer: A second category has 1 full icons and represents 18. Compare the two by counts, not visual decoration. Their absolute difference is 72 items.

81. Pictogram 9: use the key to count

Question: A hypothetical pictogram key says each full symbol stands for 20 items. One category shows 2 full symbols. Find its represented frequency.

Work it out: A full symbol represents 20, so 2 symbols count 40. There is no additional fraction of an icon. The total is 40 items.

Common first error: Counting symbols rather than the key-scaled items gives only 2, not the represented frequency. A half symbol is half the key’s value, not automatically half an item.

Independent check: The icon-equivalent count is 2. Multiplying by 20 returns 40, the reported frequency.

Transfer: A second category has 6 full icons and represents 120. Compare the two by counts, not visual decoration. Their absolute difference is 80 items.

82. Pictogram 10: use the key to count

Question: A hypothetical pictogram key says each full symbol stands for 22 items. One category shows 5 full symbols and a half symbol. Find its represented frequency.

Work it out: A full symbol represents 22, so 5 symbols count 110. The half-symbol adds 11. The total is 121 items.

Common first error: Counting symbols rather than the key-scaled items gives only 5.5, not the represented frequency. A half symbol is half the key’s value, not automatically half an item.

Independent check: The icon-equivalent count is 5.5. Multiplying by 22 returns 121, the reported frequency.

Transfer: A second category has 4 full icons and represents 88. Compare the two by counts, not visual decoration. Their absolute difference is 33 items.

83. Pictogram 11: use the key to count

Question: A hypothetical pictogram key says each full symbol stands for 24 items. One category shows 2 full symbols. Find its represented frequency.

Work it out: A full symbol represents 24, so 2 symbols count 48. There is no additional fraction of an icon. The total is 48 items.

Common first error: Counting symbols rather than the key-scaled items gives only 2, not the represented frequency. A half symbol is half the key’s value, not automatically half an item.

Independent check: The icon-equivalent count is 2. Multiplying by 24 returns 48, the reported frequency.

Transfer: A second category has 2 full icons and represents 48. Compare the two by counts, not visual decoration. Their absolute difference is 0 items.

84. Pictogram 12: use the key to count

Question: A hypothetical pictogram key says each full symbol stands for 4 items. One category shows 5 full symbols. Find its represented frequency.

Work it out: A full symbol represents 4, so 5 symbols count 20. There is no additional fraction of an icon. The total is 20 items.

Common first error: Counting symbols rather than the key-scaled items gives only 5, not the represented frequency. A half symbol is half the key’s value, not automatically half an item.

Independent check: The icon-equivalent count is 5. Multiplying by 4 returns 20, the reported frequency.

Transfer: A second category has 7 full icons and represents 28. Compare the two by counts, not visual decoration. Their absolute difference is 8 items.

85. Pictogram 13: use the key to count

Question: A hypothetical pictogram key says each full symbol stands for 6 items. One category shows 2 full symbols and a half symbol. Find its represented frequency.

Work it out: A full symbol represents 6, so 2 symbols count 12. The half-symbol adds 3. The total is 15 items.

Common first error: Counting symbols rather than the key-scaled items gives only 2.5, not the represented frequency. A half symbol is half the key’s value, not automatically half an item.

Independent check: The icon-equivalent count is 2.5. Multiplying by 6 returns 15, the reported frequency.

Transfer: A second category has 5 full icons and represents 30. Compare the two by counts, not visual decoration. Their absolute difference is 15 items.

86. Pictogram 14: use the key to count

Question: A hypothetical pictogram key says each full symbol stands for 8 items. One category shows 5 full symbols. Find its represented frequency.

Work it out: A full symbol represents 8, so 5 symbols count 40. There is no additional fraction of an icon. The total is 40 items.

Common first error: Counting symbols rather than the key-scaled items gives only 5, not the represented frequency. A half symbol is half the key’s value, not automatically half an item.

Independent check: The icon-equivalent count is 5. Multiplying by 8 returns 40, the reported frequency.

Transfer: A second category has 3 full icons and represents 24. Compare the two by counts, not visual decoration. Their absolute difference is 16 items.

87. Pictogram 15: use the key to count

Question: A hypothetical pictogram key says each full symbol stands for 10 items. One category shows 2 full symbols. Find its represented frequency.

Work it out: A full symbol represents 10, so 2 symbols count 20. There is no additional fraction of an icon. The total is 20 items.

Common first error: Counting symbols rather than the key-scaled items gives only 2, not the represented frequency. A half symbol is half the key’s value, not automatically half an item.

Independent check: The icon-equivalent count is 2. Multiplying by 10 returns 20, the reported frequency.

Transfer: A second category has 1 full icons and represents 10. Compare the two by counts, not visual decoration. Their absolute difference is 10 items.

88. Pictogram 16: use the key to count

Question: A hypothetical pictogram key says each full symbol stands for 12 items. One category shows 5 full symbols and a half symbol. Find its represented frequency.

Work it out: A full symbol represents 12, so 5 symbols count 60. The half-symbol adds 6. The total is 66 items.

Common first error: Counting symbols rather than the key-scaled items gives only 5.5, not the represented frequency. A half symbol is half the key’s value, not automatically half an item.

Independent check: The icon-equivalent count is 5.5. Multiplying by 12 returns 66, the reported frequency.

Transfer: A second category has 6 full icons and represents 72. Compare the two by counts, not visual decoration. Their absolute difference is 6 items.

89. Pictogram 17: use the key to count

Question: A hypothetical pictogram key says each full symbol stands for 14 items. One category shows 2 full symbols. Find its represented frequency.

Work it out: A full symbol represents 14, so 2 symbols count 28. There is no additional fraction of an icon. The total is 28 items.

Common first error: Counting symbols rather than the key-scaled items gives only 2, not the represented frequency. A half symbol is half the key’s value, not automatically half an item.

Independent check: The icon-equivalent count is 2. Multiplying by 14 returns 28, the reported frequency.

Transfer: A second category has 4 full icons and represents 56. Compare the two by counts, not visual decoration. Their absolute difference is 28 items.

90. Pictogram 18: use the key to count

Question: A hypothetical pictogram key says each full symbol stands for 16 items. One category shows 5 full symbols. Find its represented frequency.

Work it out: A full symbol represents 16, so 5 symbols count 80. There is no additional fraction of an icon. The total is 80 items.

Common first error: Counting symbols rather than the key-scaled items gives only 5, not the represented frequency. A half symbol is half the key’s value, not automatically half an item.

Independent check: The icon-equivalent count is 5. Multiplying by 16 returns 80, the reported frequency.

Transfer: A second category has 2 full icons and represents 32. Compare the two by counts, not visual decoration. Their absolute difference is 48 items.

91. Pictogram 19: use the key to count

Question: A hypothetical pictogram key says each full symbol stands for 18 items. One category shows 2 full symbols and a half symbol. Find its represented frequency.

Work it out: A full symbol represents 18, so 2 symbols count 36. The half-symbol adds 9. The total is 45 items.

Common first error: Counting symbols rather than the key-scaled items gives only 2.5, not the represented frequency. A half symbol is half the key’s value, not automatically half an item.

Independent check: The icon-equivalent count is 2.5. Multiplying by 18 returns 45, the reported frequency.

Transfer: A second category has 7 full icons and represents 126. Compare the two by counts, not visual decoration. Their absolute difference is 81 items.

92. Pictogram 20: use the key to count

Question: A hypothetical pictogram key says each full symbol stands for 20 items. One category shows 5 full symbols. Find its represented frequency.

Work it out: A full symbol represents 20, so 5 symbols count 100. There is no additional fraction of an icon. The total is 100 items.

Common first error: Counting symbols rather than the key-scaled items gives only 5, not the represented frequency. A half symbol is half the key’s value, not automatically half an item.

Independent check: The icon-equivalent count is 5. Multiplying by 20 returns 100, the reported frequency.

Transfer: A second category has 5 full icons and represents 100. Compare the two by counts, not visual decoration. Their absolute difference is 0 items.

93. Pictogram 21: use the key to count

Question: A hypothetical pictogram key says each full symbol stands for 22 items. One category shows 2 full symbols. Find its represented frequency.

Work it out: A full symbol represents 22, so 2 symbols count 44. There is no additional fraction of an icon. The total is 44 items.

Common first error: Counting symbols rather than the key-scaled items gives only 2, not the represented frequency. A half symbol is half the key’s value, not automatically half an item.

Independent check: The icon-equivalent count is 2. Multiplying by 22 returns 44, the reported frequency.

Transfer: A second category has 3 full icons and represents 66. Compare the two by counts, not visual decoration. Their absolute difference is 22 items.

94. Pictogram 22: use the key to count

Question: A hypothetical pictogram key says each full symbol stands for 24 items. One category shows 5 full symbols and a half symbol. Find its represented frequency.

Work it out: A full symbol represents 24, so 5 symbols count 120. The half-symbol adds 12. The total is 132 items.

Common first error: Counting symbols rather than the key-scaled items gives only 5.5, not the represented frequency. A half symbol is half the key’s value, not automatically half an item.

Independent check: The icon-equivalent count is 5.5. Multiplying by 24 returns 132, the reported frequency.

Transfer: A second category has 1 full icons and represents 24. Compare the two by counts, not visual decoration. Their absolute difference is 108 items.

95. Pictogram 23: use the key to count

Question: A hypothetical pictogram key says each full symbol stands for 4 items. One category shows 2 full symbols. Find its represented frequency.

Work it out: A full symbol represents 4, so 2 symbols count 8. There is no additional fraction of an icon. The total is 8 items.

Common first error: Counting symbols rather than the key-scaled items gives only 2, not the represented frequency. A half symbol is half the key’s value, not automatically half an item.

Independent check: The icon-equivalent count is 2. Multiplying by 4 returns 8, the reported frequency.

Transfer: A second category has 6 full icons and represents 24. Compare the two by counts, not visual decoration. Their absolute difference is 16 items.

96. Pictogram 24: use the key to count

Question: A hypothetical pictogram key says each full symbol stands for 6 items. One category shows 5 full symbols. Find its represented frequency.

Work it out: A full symbol represents 6, so 5 symbols count 30. There is no additional fraction of an icon. The total is 30 items.

Common first error: Counting symbols rather than the key-scaled items gives only 5, not the represented frequency. A half symbol is half the key’s value, not automatically half an item.

Independent check: The icon-equivalent count is 5. Multiplying by 6 returns 30, the reported frequency.

Transfer: A second category has 4 full icons and represents 24. Compare the two by counts, not visual decoration. Their absolute difference is 6 items.

97. Pictogram 25: use the key to count

Question: A hypothetical pictogram key says each full symbol stands for 8 items. One category shows 2 full symbols and a half symbol. Find its represented frequency.

Work it out: A full symbol represents 8, so 2 symbols count 16. The half-symbol adds 4. The total is 20 items.

Common first error: Counting symbols rather than the key-scaled items gives only 2.5, not the represented frequency. A half symbol is half the key’s value, not automatically half an item.

Independent check: The icon-equivalent count is 2.5. Multiplying by 8 returns 20, the reported frequency.

Transfer: A second category has 2 full icons and represents 16. Compare the two by counts, not visual decoration. Their absolute difference is 4 items.

98. Pictogram 26: use the key to count

Question: A hypothetical pictogram key says each full symbol stands for 10 items. One category shows 5 full symbols. Find its represented frequency.

Work it out: A full symbol represents 10, so 5 symbols count 50. There is no additional fraction of an icon. The total is 50 items.

Common first error: Counting symbols rather than the key-scaled items gives only 5, not the represented frequency. A half symbol is half the key’s value, not automatically half an item.

Independent check: The icon-equivalent count is 5. Multiplying by 10 returns 50, the reported frequency.

Transfer: A second category has 7 full icons and represents 70. Compare the two by counts, not visual decoration. Their absolute difference is 20 items.

99. A full pie chart is 360 degrees

Question: What angle corresponds to 100% of a complete pie?

Work it out: 360°. A full circle represents the one whole partition.

Common first error: Using 180°, the angle along a straight line, instead of a full turn.

Independent check: Four sectors of ninety degrees each fill the full circle.

100. Half a circle is not 180 percent

Question: A sector spans 180°. What percentage is it?

Work it out: 180/360=1/2=50%.

Common first error: Treating the degree number 180 as the percentage number.

Independent check: Two equal 180° halves fill a full 360°.

101. A right-angle sector

Question: A sector is 90°. Find its share of the whole.

Work it out: 90/360=1/4=25%.

Common first error: Reporting 90% because 90 appears in the picture.

Independent check: Four such sectors fill the whole.

102. A 72-degree sector

Question: Find the percentage represented by a 72° slice.

Work it out: 72/360=1/5=20%.

Common first error: Reporting 72%, which exceeds the actual share.

Independent check: Five equal slices of 72° cover 360°.

103. Equal categories

Question: An exhaustive pie has three categories of equal frequency. What angle for each?

Work it out: 360/3=120° for each.

Common first error: Using thirty-three degrees after mixing percentage with angle units.

Independent check: 120×3=360°.

104. Pie sector missing angle

Question: Three sectors are 90°,108°,72°. What is the fourth in a complete four-category pie?

Work it out: Total known is 270°, so missing angle is 90°.

Common first error: Subtracting from 180°, not 360°.

Independent check: All four angles add to 360°.

105. Category frequency sum

Question: A survey total is 60, with category counts 10,15,20 and unknown. Find the unknown.

Work it out: 60−10−15−20=15 observations.

Common first error: Assuming angles without first finding the missing count.

Independent check: The four counts sum to sixty.

106. Pie categories exceed one whole

Question: Shares are 45%,35%,30% in three supposedly disjoint exhaustive categories. Is the pie valid?

Work it out: No. They total 110%, which cannot describe one partition.

Common first error: Trusting a nice-looking circle rather than arithmetic.

Independent check: Recheck denominators and whether categories overlap.

107. Pictogram with zero symbols

Question: A row explicitly labelled B has zero symbols under key one icon=5. What is its frequency?

Work it out: Zero items in this represented dataset.

Common first error: Assuming an entirely omitted category also necessarily means zero.

Independent check: Distinguish a labelled empty row from missing data.

108. Half an icon for six people

Question: A key says one full icon means six people. What does a half icon mean?

Work it out: Three people.

Common first error: Calling it half a person because one half-symbol is visible.

Independent check: 3+3=6, reconstructing a full icon.

109. A questionable key for discrete people

Question: One icon represents five people, but the chart shows a half icon. Does this represent an exact whole-person count?

Work it out: No, not without some alternate annotation: half of five is 2.5 people, which is not a valid exact count of individuals.

Common first error: Using fractions of persons without considering the object being counted.

Independent check: Choose a compatible key or state that the pictogram is approximate.

110. Pie and histogram are different

Question: A display partitions a whole sample into favourite-subject sectors. Is it a histogram?

Work it out: No. It is a categorical pie chart. A histogram groups numerical measurements into intervals.

Common first error: Calling any visual about frequency a histogram.

Independent check: Read the kind of horizontal numerical or categorical variable.

111. One chart may have a smaller sample

Question: Two pies have 50% sectors, one over 40 people and the other over 100. Are category counts equal?

Work it out: No. Counts are 20 and 50, respectively.

Common first error: Equating equal percentages with equal absolute frequencies.

Independent check: Multiply each proportion by its own total.

112. Sample does not mean all residents

Question: Six of ten fictional survey respondents choose an option. Does this establish a 60% preference among all Singapore residents?

Work it out: No. It shows 60% of the ten respondents only; population generalisation needs appropriate sampling.

Common first error: Making unsupported broader claims from a small convenience sample.

Independent check: State the sample and the observation period.

113. Pie ‘Other’ hides detail

Question: A sector labelled Other is 30%. Can you name its most common subcategory?

Work it out: No. The pie shows only the combined share of Other, not its internal breakdown.

Common first error: Guessing a category from its overall sector size.

Independent check: Request the raw subgroup frequency table.

114. A pie sector cannot exceed one whole

Question: A category is labelled 140% of the same complete pie. Is that possible?

Work it out: No. In a nonoverlapping partition, one category cannot exceed the 100% whole.

Common first error: Confusing a percentage increase over 100% with a share of a fixed sample.

Independent check: Its angle would be 504°, exceeding 360°.

115. Quarter of a pictogram key

Question: If one icon is eight items, what does a quarter icon stand for?

Work it out: Two items.

Common first error: Calling it 0.25 item regardless of key value.

Independent check: Four quarters of two items reconstruct eight.

116. Pictogram rows require one scale

Question: Could two rows with different icon sizes be compared accurately with no legend?

Work it out: Not reliably. A shared key and consistent representation are required.

Common first error: Comparing visual area instead of encoded values.

Independent check: Read the legend before drawing conclusions.

117. A percentage can be used to draw a pie

Question: If 12.5% of a class is in category A, what is its angle?

Work it out: 0.125×360°=45°.

Common first error: Using 12.5° as a sector angle.

Independent check: 45/360=0.125.

118. Frequency from five percent

Question: If five percent of a fictional eighty-response sample belongs to A, how many responses are A?

Work it out: 0.05×80=4, corresponding to 18°.

Common first error: Reporting five people directly from a five percent label.

Independent check: Four of eighty is one twentieth, or five percent.

119. Pictogram with 1.5 full equivalents

Question: One full icon=12 books. What do 1.5 icons represent?

Work it out: 1.5×12=18 books.

Common first error: Reading 1.5 icons as 1.5 books.

Independent check: Eighteen divided by twelve gives 1.5 icon equivalents.

120. No total, no absolute count

Question: A category is 25% of a pie. Can its exact frequency be found without the sample total?

Work it out: No. One quarter of an unspecified total has no determined absolute frequency.

Common first error: Reporting twenty-five observations from a twenty-five percent share.

Independent check: For totals forty and eighty, the category counts would be ten and twenty.

121. Chart date is necessary for ‘today’

Question: A pie chart has no publication date or collection period. Can it tell us what happened today?

Work it out: No. The chart’s time window is not specified.

Common first error: Treating an old or undated chart as current evidence.

Independent check: Find source date and measurement period.

122. Two close angles

Question: Sector A is 71° and B is 72°. Which has more frequency in the same pie?

Work it out: B, by a small difference corresponding to one degree of the same total.

Common first error: Reading similar-looking wedges as exactly equal.

Independent check: Compare printed angles, not line thickness.

123. A key may encode tens

Question: A pictogram key is one symbol=20 objects. A row has four symbols. Find count.

Work it out: 4×20=80 objects.

Common first error: Treating four visible icons as four objects.

Independent check: 80÷20=4 icon equivalents.

124. The whole must be exclusive and exhaustive

Question: Why should all pie categories add to one whole?

Work it out: They form a partition: each observed item appears in exactly one displayed category, and all are included.

Common first error: Double-counting overlapping categories or omitting unlabeled groups.

Independent check: Sum counts to sample total and angles to 360°.

Five common G1 chart mistakes

Sector angle is read as a percentage

Pictogram key is ignored

A category is mistaken for the total

Categories do not sum to one whole

A chart is treated as a universal survey

How a small-group tutor can teach charts

10 minutes: retrieve one-whole fractions and percentage equivalents.
20 minutes: draw pie sectors from simple fractions and counts.
25 minutes: individually solve sector-to-frequency and reverse-total questions.
15 minutes: count pictogram full and half-symbol equivalents.
10 minutes: identify one conclusion a chart does not justify.
10 minutes: correct the first mistake and retest with new data.

Five manageable home practice sessions

Day one: change 90°,120°,180° into fractions of a circle.
Day two: find category counts using stated fictional survey totals.
Day three: reverse a pie sector to recover total sample size.
Day four: create a pictogram with a consistent symbol key and a valid half symbol.
Day five: explain a sample-size limitation or category labelling issue in one sentence.

Current G1 curriculum versus the SEC year

Frequently asked questions

Does 90° mean 90% in a pie chart?

No. It is ninety of the full 360 degrees, or one quarter = 25%.

How do you find total observations from one sector?

Divide its category count by the fraction angle/360°.

Is a half icon half a person?

No. It is half of the key’s represented count, which should yield a valid count or be labelled approximate.

Can pie charts and histograms replace each other?

No. Pie charts show shares of one whole; histograms show distributions across numerical intervals.

Can equal pie sectors represent different respondent counts?

Yes, if the pies have different sample sizes.

What should a tutor check?

Can the pupil move independently among sector angles, fractions, percentages and absolute counts, and use a pictogram key accurately?

Official syllabus and accuracy note

Bring a misread pie chart or pictogram worksheet to eduKate Singapore and ask how the tutor will test the learner’s understanding of the whole and the chart key.

Secondary 2 G1 Mathematics Tutor Bukit Timah | Pie Charts and Pictograms Explained

Official syllabus and accuracy note

Official syllabus and accuracy note

Current G1 curriculum versus the SEC year

Frequently asked questions

Does 90° mean 90% in a pie chart?

How do you find total observations from one sector?

Is a half icon half a person?

Can pie charts and histograms replace each other?

Can equal pie sectors represent different respondent counts?

What should a tutor check?

Continue the Secondary 2 G1 data guides

Official syllabus and accuracy note

Current G1 curriculum versus the SEC year

Frequently asked questions

Does 90° mean 90% in a pie chart?

How do you find total observations from one sector?

Is a half icon half a person?

Can pie charts and histograms replace each other?

Can equal pie sectors represent different respondent counts?

What should a tutor check?

Continue the Secondary 2 G1 data guides

Official syllabus and accuracy note

Five manageable home practice sessions

Current G1 curriculum versus the SEC year

Frequently asked questions

Does 90° mean 90% in a pie chart?

How do you find total observations from one sector?

Is a half icon half a person?

Can pie charts and histograms replace each other?

Can equal pie sectors represent different respondent counts?

What should a tutor check?

Continue the Secondary 2 G1 data guides

Official syllabus and accuracy note

Five manageable home practice sessions

Current G1 curriculum versus the SEC year

Frequently asked questions

Does 90° mean 90% in a pie chart?

How do you find total observations from one sector?

Is a half icon half a person?

Can pie charts and histograms replace each other?

Can equal pie sectors represent different respondent counts?

What should a tutor check?

Continue the Secondary 2 G1 data guides

Official syllabus and accuracy note

A one-minute chart diagnostic

The 360-degree circle is a mathematical whole

Two identical sectors can represent different counts

How to recover the sample total

An icon key is a unit conversion

What a chart does not tell you

124 original pie-chart and pictogram clinics

1. Pie chart 1: sector to category frequency

2. Pie chart 1: recover the complete sample

3. Pie chart 1: frequency to angle

4. Pie chart 2: sector to category frequency

5. Pie chart 2: recover the complete sample

6. Pie chart 2: frequency to angle

7. Pie chart 3: sector to category frequency

8. Pie chart 3: recover the complete sample

9. Pie chart 3: frequency to angle

10. Pie chart 4: sector to category frequency

11. Pie chart 4: recover the complete sample

12. Pie chart 4: frequency to angle

13. Pie chart 5: sector to category frequency

14. Pie chart 5: recover the complete sample

15. Pie chart 5: frequency to angle

16. Pie chart 6: sector to category frequency

17. Pie chart 6: recover the complete sample

18. Pie chart 6: frequency to angle

19. Pie chart 7: sector to category frequency

20. Pie chart 7: recover the complete sample

21. Pie chart 7: frequency to angle

22. Pie chart 8: sector to category frequency

23. Pie chart 8: recover the complete sample

24. Pie chart 8: frequency to angle

25. Pie chart 9: sector to category frequency

26. Pie chart 9: recover the complete sample

27. Pie chart 9: frequency to angle

28. Pie chart 10: sector to category frequency