PSLE Mathematics Learning Guide · Guide 36
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One pie chart gives Reading a 40% share. Another gives Reading a 30% share. Which group contains more readers?
The larger percentage does not settle the question. Forty percent of 240 is 96. Thirty percent of 360 is 108. The second group has the smaller Reading share but the larger number of readers.
A pie-chart sector describes a share of its own whole. To compare actual quantities across charts, recover each sector’s quantity using that chart’s total before comparing the results. Do not let the visual size of a slice replace the missing total.
This guide concentrates on different-whole comparisons, recovering an unknown total and combining groups without averaging their percentages blindly. For the wider conversion route between fractions, percentages and central angles, use the established Reading Pie Charts Through Fractions, Percentages and Angles. The current guide adds a different learning job rather than replacing that resource.
Curriculum boundary: Reading and interpreting pie charts appears in Primary 4 Statistics in the MOE syllabus, page 40. Here it is revisited for PSLE preparation. All groups and figures below are invented teaching data. The sector labels are supplied in tables so no unprovided picture needs to be measured.
Choose the comparison you need to repair
Begin with share versus count when a larger percentage is being treated as a larger number. Use the two-chart case to see the complete calculation. Go to recovering the whole when a sector count is known but the total is hidden. Read combining groups before finding an overall percentage. Finish with the independent practice without looking at the answers first.
The question to keep asking is not only “What percentage?” It is “What percentage of which total?” That last phrase controls the size of the actual quantity.
A share and a count are different kinds of information
A share tells you how much of a whole belongs to a category. A count tells you how many objects or people belong to it. The same share can correspond to many different counts because the whole can vary.
One quarter of 40 pupils is 10 pupils. One quarter of 200 pupils is 50 pupils. Both groups have the same proportion in the category, but not the same number.
Within one chart representing one total, a larger exact share means a larger quantity. Across two charts with different totals, that implication no longer follows. The relationship must be rebuilt for each chart separately.
Use two labels in your working: share and actual number. Keeping them apart prevents a percentage comparison from quietly turning into a claim about people or items.
Establish the whole before reading any sector
A pie chart should identify the collection it represents: all survey participants, all books in a selection, or all of a specified budget. A full circle represents that complete collection, not automatically 100 people or 360 objects.
The 100 in 100% is the percentage scale. It does not supply the number of people. A whole chart can represent 80, 240 or 10,000 people while still representing 100% of its own group.
Similarly, a full turn has 360°, but a 360° circle does not imply a total of 360 people. Degrees describe the drawing’s angle measure. People describe the data being represented.
Before calculating, write “Chart A total = …” and “Chart B total = …”. If a total is missing, leave it explicitly unknown until another relationship supplies it.
Main case: two groups choose one activity each
Group A has 240 pupils. Group B has 360 different pupils. Each pupil chooses exactly one of Reading, Sports or Art. Everyone is included once, and the two groups do not overlap.
The following table supplies the exact sector labels for the two teaching pie charts. Each column of percentages adds to 100%.
| Activity | Group A: 240 pupils | Group B: 360 pupils |
|---|---|---|
| Reading | 40% | 30% |
| Sports | 35% | 45% |
| Art | 25% | 25% |
| Whole | 100% | 100% |
For Reading, Group A has 40% of 240 = 96 pupils. Group B has 30% of 360 = 108 pupils. Group B therefore has 108 − 96 = 12 more readers, despite its smaller percentage.
For Sports, Group A has 35% of 240 = 84 pupils. Group B has 45% of 360 = 162 pupils. Group B has the larger share and the larger count in this category, but that agreement should be established through the totals rather than assumed as a universal rule.
For Art, each chart shows 25%. Yet A has 60 pupils and B has 90. Equal sector shares do not guarantee equal counts when the totals differ.
Convert the chart labels into an actual-count table
A short table makes the two meanings visible side by side. It also reduces the chance of using Group A’s total with Group B’s percentage.
| Activity | Group A count | Group B count | Combined count |
|---|---|---|---|
| Reading | 96 | 108 | 204 |
| Sports | 84 | 162 | 246 |
| Art | 60 | 90 | 150 |
| Total | 240 | 360 | 600 |
Check each original column. For A, 96 + 84 + 60 = 240. For B, 108 + 162 + 90 = 360. The recovered counts exhaust each stated whole.
Then check the combined column: 204 + 246 + 150 = 600. This agrees with 240 + 360 = 600 because every pupil belongs to one group and one activity.
These checks establish internal consistency. They do not turn the invented example into an actual survey or provide evidence about real pupils’ preferences.
Different percentages can also describe the same count
Suppose a category contains one third of a 180-person group and one quarter of a 240-person group.
First count = 180 ÷ 3 = 60. Second count = 240 ÷ 4 = 60. The counts are equal even though one third is larger than one quarter.
That comparison is not an exception to the fraction rules. One third remains the larger fraction. The fractions are simply acting on different wholes.
When a learner objects that “one third must be bigger”, ask what is being compared: the fraction itself, or the number obtained after taking that fraction of a stated total?
Recover a missing total from a known sector
A chart says Reading is 40% and represents 96 pupils. What is the whole group?
Forty percent corresponds to 96. One percent corresponds to 96 ÷ 40 = 2.4. One hundred percent corresponds to 2.4 × 100 = 240 pupils.
Equivalently, 40% = 2/5. If two fifths represent 96, one fifth represents 48 and five fifths represent 240. Another valid calculation is 96 ÷ 0.4 = 240.
The 2.4 in the one-percent route is a proportional calculation unit, not a claim that 2.4 actual pupils form a separate group. The final pupil counts in this exact example are whole numbers.
Check forward: 40% of 240 = 96. A reverse-whole answer should recreate the sector count that was given.
A fraction label gives the same reverse relationship
A sector represents 3/8 of a collection and contains 45 items. Three equal fraction-units represent 45, so one eighth represents 15. The total is 8 × 15 = 120 items.
Do not calculate 3/8 of 45. That would take a fraction of the sector itself rather than recovering the whole from which the sector came.
The direction of the question matters. “Find the part from the whole” and “find the whole from the part” use the same relationship in opposite directions.
For a broader repair of the reference quantity, return to Guide 1: Find the Reference Whole.
A percentage alone may not determine an actual comparison
A chart gives Reading as 30%, but its group size is not stated. Another chart gives Reading as 40% of 240, which is 96 readers. Can the reader count in the first chart be compared with 96?
Not uniquely. If the first whole were 200, it would contain 60 readers. If it were 400, it would contain 120. Both totals fit a 30% sector, but they lead to opposite count comparisons.
Those two possible examples demonstrate why the information is insufficient. The missing whole is not a minor detail that can be guessed from the circle’s printed size.
What extra information would help? The group total, a known sector count and share, or a relationship between the two group sizes could fix the scale.
Combine counts and totals before finding an overall percentage
Return to Groups A and B. Together they have 600 pupils and 204 readers. The overall Reading percentage is 204/600 × 100% = 34%.
The calculation (40% + 30%) ÷ 2 = 35% is not correct for the combined group. It treats the two group percentages as though they represent equally sized groups. Here Group B contains more pupils.
Recovering the counts automatically preserves each group’s size: 96 readers from A and 108 from B. After adding the people, use the combined 600 as the new whole.
For the other activities, Sports = 246/600 × 100% = 41%; Art = 150/600 × 100% = 25%. Check 34% + 41% + 25% = 100%.
A combined chart represents a new whole. It is not formed by copying one original chart’s total or averaging sector labels without their counts.
When averaging two percentages does work
Suppose two separate groups each contain 100 pupils. Reading is 40% in one and 30% in the other. Their reader counts are 40 and 30, so the combined share is 70 out of 200 = 35%.
Here the simple mean of 40% and 30% gives the right result because the groups have equal size. Each percentage represents the same number of pupils per percentage point.
The useful rule is not “never average percentages”. It is “do not average them equally unless equal weighting is justified”. For a primary learner, rebuilding counts is often the clearest way to keep that condition visible.
A share can fall even when the count stays unchanged
A collection contains 200 books, of which 50 are adventure books. The adventure share is 50/200 = 25%. Fifty other books are added, none of them adventure books.
The adventure count remains 50, but the total becomes 250. The new share is 50/250 = 20%. The percentage fell even though no adventure books were removed.
Similarly, Group A in the main case has 96 readers out of 240 pupils. If 60 new pupils join and all choose activities other than Reading, the Reading count stays 96 while its share becomes 96/300 = 32%.
“The sector became smaller” can therefore mean the category occupies less of a larger whole. It does not automatically mean the actual category lost people or objects.
The printed circle size is not a second data scale unless stated
Two pie charts may be printed at the same size for convenient reading even when they represent different totals. The 240-pupil and 360-pupil charts could have identical diameters on the page.
Alternatively, a publisher might choose differently sized circles. Unless the task explicitly defines a relationship between drawn size and total quantity, do not calculate pupil counts from the diameters or areas of the printed circles.
The labelled sector proportion and the stated data total are the information used in this guide. Keep the physical drawing size separate from the size of the collection represented.
Exact percentages and rounded labels carry different information
If a chart represents exactly 250 people and a category is stated as exactly 25%, the calculation gives 62.5 people. That cannot be an exact count of whole people.
Do not simply round to 63 and declare the exact information satisfied. Instead, inspect whether the percentage was rounded, whether the total or category was read correctly, or whether the quantity is actually a divisible measure rather than a count of people.
A label of “about 25%” can reasonably describe an approximate share. It does not uniquely determine the exact count from a total of 250. The task’s precision instructions decide what answer is supported.
Likewise, independently rounded sector percentages may total 99% or 101%. That is a reason to check precision, not permission to change a category arbitrarily. When exact counts are supplied, use them for exact calculations.
Worked workshop: twelve pie-chart comparison problems
1. Compare shares within one chart
In Group A, Reading is 40% and Sports is 35% of the same 240 pupils. Reading therefore has more pupils. The counts are 96 and 84, giving a difference of 12 pupils. Within this one whole, the larger share does give the larger count.
2. Compare the same category across two charts
Group A has 40% readers out of 240; B has 30% out of 360. Decode separately: 96 and 108. B has 12 more readers. Do not carry the within-one-chart comparison rule across unequal totals without recalculating.
3. Equal shares, unequal quantities
Art is 25% in both groups. Counts are 60 in A and 90 in B. B has 30 more Art pupils. The equality is in proportions, not people.
4. Different shares, equal quantities
One third of 180 and one quarter of 240 each give 60. The second sector is a smaller fraction but acts on a larger total. Both parts can be equal without the fractions being equal.
5. Recover a whole from a percentage and count
A chart says 30% represents 108 pupils. Total = 108 ÷ 0.3 = 360 pupils. Check: 30% of 360 = 108. Multiplying 108 by 0.3 would answer a different question.
6. Recover a whole from a fraction and count
A 3/8 sector contains 45 items. One eighth is 15, so the whole is 120 items. The fraction is of the original collection, not of the known 45-item sector.
7. Compare actual-count ratios rather than percentage ratios
The reader counts in A and B are 96 and 108. Their ratio A:B is 96:108 = 8:9. The ratio of percentage labels, 40:30 = 4:3, is not their count ratio because the chart totals differ.
8. Find a combined percentage
A and B together have 204 readers among 600 pupils. Reading share = 34%. The arithmetic mean 35% ignores the unequal group sizes. Reconstructing counts avoids that error.
9. Find the comparison boundary
Thirty percent of Group X’s 200 people is 60. Reading is 20% of Group Y. Y would have the same 60 readers if its total were 300, because 20% of 300 = 60. With a larger total than 300 it would have more readers; with a smaller total it would have fewer, provided the exact counts are valid.
10. Interpret a falling share with a fixed count
Fifty adventure books out of 200 is 25%. The same 50 out of 250 is 20%. The count is unchanged. The falling share is explained by the larger denominator, not a loss of adventure books.
11. Find a remaining share only when categories exhaust the whole
A chart has exactly three non-overlapping categories. Two are 35% and 40%. The third is 100% − 35% − 40% = 25%. For a total of 160 items, it represents 40 items. This requires the statement that those categories cover the entire collection.
12. Identify when a unique count cannot be found
A sector is labelled 30%, but neither the total nor another actual count is supplied. The percentage identifies a share, not a unique count. More information is needed to say how many people or objects are in that sector.
Check whether the categories form one complete partition
A pie chart’s sectors should represent non-overlapping parts that together make the specified whole. In the activity examples, each pupil chooses exactly one activity. That condition makes the three counts add to the pupil total.
A different survey might allow each pupil to choose several hobbies. Adding the numbers selecting Reading, Sports and Art could then count some pupils more than once. Those overlapping selections are not automatically suitable as three slices of one pupil-total pie chart.
Before combining or subtracting categories, read how the records were defined. A complete-looking circle does not remove the need to check what is counted.
Repair the precise comparison error
Draft: “40% is larger than 30%, so A has more readers.”
Repair: A has the larger reader share. To compare reader counts, calculate 40% of A’s total and 30% of B’s total.
Draft: “Both Art sectors are 25%, so there are the same number of Art pupils.”
Repair: Equal shares give equal counts only when the totals also match. Here the counts are 60 and 90.
Draft: “The combined percentage is the mean of the two chart percentages.”
Repair: Add category counts and group totals first. Then divide the combined category count by the combined total.
Draft: “The category percentage fell, so some objects must have been removed.”
Repair: Check whether the whole grew. A fixed category count can occupy a smaller fraction of a larger total.
Draft: “The chart is printed bigger, so it represents more pupils.”
Repair: Use the stated total. Drawing size carries no extra count information unless a size scale is explicitly defined.
Independent practice: two learning groups
Group C contains 200 pupils and Group D contains 300 different pupils. Every pupil chooses exactly one activity. These are exact sector percentages, not rounded estimates.
| Activity | Group C: 200 pupils | Group D: 300 pupils |
|---|---|---|
| Robotics | 30% | 20% |
| Music | 45% | 50% |
| Reading | 25% | 30% |
| Whole | 100% | 100% |
- Find the Robotics count in each group.
- Which group has the larger Robotics percentage? Which has the larger Robotics count?
- Find the Music count in each group and their difference.
- Find the Reading count in each group and their difference.
- Check each group’s three counts add to its stated total.
- Find the combined number of pupils and combined Robotics count.
- Find the Robotics percentage of the combined group.
- Find the combined Music and Reading percentages.
- Explain why the simple mean of 30% and 20% does not give the combined Robotics percentage.
- Suppose Group D’s total is hidden, but 90 readers represent 30%. Recover the total.
- Group C gains 50 pupils, none choosing Robotics. Find its new Robotics count and percentage.
- If Group D’s total and all its actual counts were unknown, could its 20% Robotics sector alone establish whether it contained more than C’s original 60 Robotics pupils?
Independent practice: answers with reasons
1. C Robotics = 30% of 200 = 60. D Robotics = 20% of 300 = 60.
2. C has the larger percentage, 30% rather than 20%. Their actual Robotics counts are equal. Different totals make the two statements compatible.
3. C Music = 45% of 200 = 90; D Music = 50% of 300 = 150. D has 60 more Music pupils.
4. C Reading = 25% of 200 = 50; D Reading = 30% of 300 = 90. D has 40 more readers.
5. C: 60 + 90 + 50 = 200. D: 60 + 150 + 90 = 300. Both sets of category counts exhaust their own wholes.
6. Combined pupils = 200 + 300 = 500. Combined Robotics = 60 + 60 = 120.
7. Combined Robotics percentage = 120/500 × 100% = 24%.
8. Music = (90 + 150)/500 × 100% = 48%. Reading = (50 + 90)/500 × 100% = 28%. Check 24% + 48% + 28% = 100%.
9. The simple mean is 25%, but the two percentages represent unequal group sizes. Recovering counts gives 120 out of 500, or 24%. Group D’s larger total must not receive the same weight as C’s smaller total.
10. D total = 90 ÷ 0.3 = 300 pupils. Check 30% of 300 recreates the known 90 readers.
11. Robotics count stays 60. New whole = 250, so the new share is 60/250 × 100% = 24%. The share falls without any Robotics pupils leaving.
12. No. Twenty percent of 200 would be 40, whereas 20% of 400 would be 80. The unknown total could support either a smaller or larger count than 60.
Use reverse questions to test understanding
After solving a direct count question, reverse one part of it. Instead of “Find 30% of 300”, ask “A 30% sector contains 90 people; find the total.” Then ask “What total would make a 20% sector contain the same 90 people?”
The last total is 90 ÷ 0.2 = 450. The sector share became smaller while its required count stayed fixed, so the whole had to become larger. That direction check supports the result before multiplication confirms it.
This three-way movement—whole to part, part to whole, and a changed share with a fixed part—tests the relationship more thoroughly than repeating one percentage multiplication.
Diagnose whether the missing piece is fraction, whole or comparison
If a learner cannot calculate 25% of 240, fraction or percentage arithmetic may need practice. If that calculation is easy but the learner uses 240 for both charts, the problem is attaching each share to its own whole.
If both counts are correct but the wrong group is named in the final answer, the comparison labels were lost. Write the group names beside the recovered counts before subtracting.
If an overall percentage is wrong, inspect the new denominator. A combined chart requires the combined whole, not one original group’s total. If a falling share is misread as a falling count, compare the before-and-after totals explicitly.
These errors can produce similar wrong answers but need different next tasks. Preserve the first attempt long enough to identify where the meaning changed.
Parent and tutor teaching guide
Begin with two charts having the same total. Let the learner see why the larger exact share then gives the larger count. Next keep the percentages but change one total. Ask whether the earlier comparison still follows.
Use an equal-share example after that: 25% of 240 and 25% of 360. The percentages match, so any difference in count must come from the wholes. Finally use different shares that give equal counts, such as one third of 180 and one quarter of 240.
For longer tasks, encourage a two-column count table. This is useful working when it preserves which total belongs to which chart. A bare expression such as 0.4 × 240 − 0.3 × 360 may be efficient later, but it should not replace the learner’s understanding of the groups.
Give small prompts rather than the whole method: “What is A’s whole?”, “Are these counts or percentages?”, or “What is the combined total?” After the child succeeds with a prompt, use a changed example without the prompt to test independent control.
When a dataset is incomplete or inconsistent, reward a precise explanation of the missing condition. Inventing a total or rounding an impossible exact person-count is not a stronger solution than recognising the information boundary.
The learner’s final card
What whole does each chart represent? Is the question comparing shares or actual quantities? Have I calculated each part from its own total? If groups are combined, what is the new whole? Could a changed percentage come from a changed denominator rather than a changed count?
A bigger slice is a bigger share of its own circle. It is not automatically a bigger number than a slice belonging to another whole.
Return through the data-reading sequence
Use Guide 33: Tables and Missing Values, Guide 34: Graph Scales and Bar Graphs, and Guide 35: Line-Graph Values, Changes and Cumulative Totals.
For fractions, percentages and angle conversions within a chart, return to the existing pie-chart conversion guide.
Return to the PSLE Learning Guide Mathematics route.
Sources and teaching boundaries
MOE Primary Mathematics syllabus, updated October 2025, page 40; SEAB 2026 examination formats. Checked 5 September 2026.
All datasets, comparisons and suggested solutions are original teaching material. They are not official examination questions, marking schemes or actual pupil records. Combined-group percentages, changed wholes and rounded-label checks explain the underlying proportional reasoning; they are not presented as separately named syllabus requirements. Use the official documents for the learner’s examination year and course level.