Did you know? The same ratio, 3:7, can lead to two very different starting calculations. If the total is fifty, divide it into ten parts. If the difference is twenty, divide it into four parts. Neither calculation is a trick; each describes a different relationship.
This Secondary 1 G2 Mathematics guide for Bukit Timah students explains ratio tables, part–whole comparisons, sharing a total, using a given difference, fractional ratios and mixed ratio word problems. The most useful question to ask is: what exactly does one ratio part represent?
The content builds on the MOE G2 Mathematics syllabus and the unchanged eduKateSG Secondary 1 small-group tutorial reference, which describes three-pupil weekly 90-minute learning near Sixth Avenue MRT. Schools may order topics differently; practise those already taught.
The answer in one minute
- Keep the order of quantities and label every ratio part.
- Equivalent ratios scale every entry by the same factor.
- A given total means add the ratio parts.
- A given difference means subtract the relevant ratio parts.
- A known single share means divide by its own ratio entry.
- Fractional ratio entries can be cleared with a common denominator.
- Always verify both the original ratio and the total or difference.
Why part–whole diagrams work
Imagine two piles of identical counters with ratio 3:7. Draw three equal boxes for the first pile and seven equal boxes for the second. If all ten boxes represent fifty counters, each holds five. The piles are fifteen and thirty-five. But if the second pile is twenty counters larger than the first, its four extra boxes represent twenty. Each box still holds five. The amounts are again fifteen and thirty-five—this time arrived at from a difference.
A student can memorise either shortcut and still choose the wrong one on another question. Drawing parts and asking what is known trains the decision itself. In a 3-pax tutorial, each learner can explain a different story, but all should then solve a changed version independently.
Twenty first-principles ratio ideas
1. A ratio is a comparison in a fixed order
A ratio such as red:blue = 2:3 means two red parts for every three blue parts. Changing it to 3:2 describes a different relationship. The numbers by themselves are incomplete without labels.
Worked connection: For red:blue = 2:3 and 25 objects in total, there are five parts; one part is five, so ten red and fifteen blue.
Self-check: Write the quantity names above the parts before performing any division.
2. A ratio table keeps corresponding quantities aligned
A two-column ratio table is a simple way to see equivalent comparisons. The same scale factor must apply to both columns.
Worked connection: If two notebooks cost $6, four cost $12 and eight cost $24 at the same unit price. Both the notebook count and cost double each time.
Self-check: Check the ratio of each corresponding row rather than adding unrelated entries.
3. Equivalent ratios require a common scale factor
Multiplying both parts by the same positive factor preserves a ratio. Multiplying one part without the other changes it.
Worked connection: 3:4 = 6:8 = 9:12. Each ratio has first part three quarters of the second.
Self-check: Reduce the proposed ratio back to lowest terms to verify equivalence.
4. Simplest form uses a common divisor
Divide every part by its greatest common factor. For ratios involving three or more quantities, the divisor must work for all parts.
Worked connection: 18:30 becomes 3:5 after dividing both by six. And 12:18:24 becomes 2:3:4.
Self-check: Make sure none of the terms is left behind during the division.
5. A total is different from a comparison part
For a ratio A:B = 2:3, the total is five parts. A is two fifths of the total but two thirds of B. These two fractions answer different questions.
Worked connection: In a class with 12 boys and 18 girls, boys:girls = 2:3 while the share who are boys is 12/30 = 2/5.
Self-check: Circle the exact whole named in the question before choosing a denominator.
6. Sharing a given total requires the sum of the parts
Add the ratio entries to find the total number of equal parts. Divide the total quantity by that sum to obtain the value of one part.
Worked connection: Share $180 in ratio 2:3:4. There are nine parts; each equals $20, giving $40, $60 and $80.
Self-check: Check that shares add to the known total and preserve the ratio.
7. A given difference uses the gap between ratio parts
When the question supplies how much larger one quantity is than another, subtract the ratio entries and compare that part difference with the real difference.
Worked connection: If A:B = 3:7 and B exceeds A by 28, then four parts equal 28; one part equals seven. Thus A = 21 and B = 49.
Self-check: Dividing 28 by ten total parts would confuse a difference with a whole.
8. One known share can set the scale
Sometimes the total is unknown but one share is known. Divide that share by its number of ratio parts, then multiply to find the others.
Worked connection: If apples:oranges = 4:5 and apples = 32, one part equals eight; oranges = 40; total = 72.
Self-check: State which group the known value describes before calculating.
9. Fractional ratios can be made whole-number ratios
Multiplying all parts by a common denominator removes fractional entries without changing the comparison.
Worked connection: (1/2):(3/4) becomes 2:3 after multiplying both entries by four.
Self-check: Multiplying only one side by the denominator destroys equivalence.
10. Decimal ratios use the same principle
Ratios such as 0.6:1.2 can be simplified by multiplying to remove decimals and then dividing common factors.
Worked connection: 0.6:1.2 = 6:12 = 1:2.
Self-check: Treat 0.6 and 1.2 as quantities, not as arbitrary strings of digits.
11. Three-part ratios must keep all three labels
When A:B:C = 2:3:5, all three quantities share the same unit-part. Leaving out the third part changes the total.
Worked connection: If a mixture has ten total parts and weighs 200 g, the three components weigh 40 g, 60 g and 100 g.
Self-check: Use a table with headings A, B and C to keep each quantity attached to its share.
12. Combining pairwise ratios requires a shared middle quantity
When A:B and B:C are supplied separately, the B entries must first be scaled to agree before deriving A:B:C.
Worked connection: If A:B = 2:3 and B:C = 4:5, make B twelve in both: A:B = 8:12 and B:C = 12:15, so A:B:C = 8:12:15.
Self-check: Simply writing 2:3:5 would ignore the unequal B values.
13. An amount added to one group changes the ratio
Ratios do not always remain constant after a story’s quantities change. Model the original counts first, then apply the stated addition or removal.
Worked connection: If red:blue = 2:3 and there are ten red and fifteen blue objects, adding five red gives 15:15 = 1:1.
Self-check: Do not apply the old 2:3 relationship after only one group changes.
14. The final ratio must be checked against the new totals
A problem may ask for a ratio after both quantities change. Apply the changes to each actual count, not directly to simplified ratio digits unless the scale is known.
Worked connection: If 12 boys and 18 girls are joined by three boys and two girls, the new ratio is 15:20 = 3:4.
Self-check: Always recalculate the final counts before simplifying.
15. Ratio and fraction questions are closely connected
A ratio compares parts, while a fraction can describe one part of the combined total. Translating between them is a powerful word-problem check.
Worked connection: If A:B = 4:1, A is 4/5 of the combined whole, and B is 1/5.
Self-check: Use a strip of five equal blocks to make the total visible.
16. A fixed rate can be displayed as a ratio table
At a constant rate, corresponding values remain proportional. A table supports reasoning without immediately introducing formal direct-proportion notation.
Worked connection: If six identical items cost $15, the price per item is $2.50 and ten items cost $25.
Self-check: State the constant unit rate and check every table row against it.
17. A difference table needs a different reference column
When amounts are described as ‘seven more than’ or ‘twice plus three’, the relationship may not be a simple constant ratio. The table reveals the distinction.
Worked connection: For costs y = 2x + 3, when x = 1, y = 5; when x = 2, y = 7. The ratio y:x changes, because of the fixed extra three.
Self-check: Do not assume every linear-looking table describes a proportional relationship.
18. Ratios with units need compatible quantities
Comparing a length measured in centimetres with another in metres requires converting to the same unit before simplifying.
Worked connection: Compare 50 cm to 2 m: convert 2 m to 200 cm, so the ratio is 50:200 = 1:4.
Self-check: Avoid writing 50:2 before dealing with the unit mismatch.
19. A ratio answer can be verified in two directions
A sound solution should reproduce the given ratio and whichever total or difference the problem supplied. These are independent checks.
Worked connection: For shares 21 and 49, simplify 21:49 to 3:7 and check 49 – 21 = 28.
Self-check: A single correct sum does not prove the ratio is correct.
20. A short explanation beats a blind shortcut
A student who can explain which value is a total, difference or individual share can solve new stories without memorising every wording variation.
Worked connection: Compare two questions using 3:7: one gives a total of fifty, the other a difference of twenty. The same ratio requires different divisors: ten parts versus four-part difference.
Self-check: Ask the student to name the supplied relationship before doing the arithmetic.
36 worked ratio clinics
Cover each answer first. Write the quantity labels and decide whether the question gives a total, a difference or a single share before performing any calculation.
Clinic 1: Simplify 12:18
Answer: 2:3. Reason: Divide both entries by six.
After a correction, ask the learner to restate the reference amount and verify the final ratio. Use a new example a few days later to check whether the method can be chosen without a chapter heading.
Clinic 2: Simplify 15:35
Answer: 3:7. Reason: Divide both entries by five.
After a correction, ask the learner to restate the reference amount and verify the final ratio. Use a new example a few days later to check whether the method can be chosen without a chapter heading.
Clinic 3: Simplify 10:15:25
Answer: 2:3:5. Reason: All three entries have common factor five.
After a correction, ask the learner to restate the reference amount and verify the final ratio. Use a new example a few days later to check whether the method can be chosen without a chapter heading.
Clinic 4: Complete 2:5 = 8:x
Answer: x=20. Reason: Multiply both parts by four.
After a correction, ask the learner to restate the reference amount and verify the final ratio. Use a new example a few days later to check whether the method can be chosen without a chapter heading.
Clinic 5: Complete 3:7 = x:35
Answer: x=15. Reason: Multiply by five.
After a correction, ask the learner to restate the reference amount and verify the final ratio. Use a new example a few days later to check whether the method can be chosen without a chapter heading.
Clinic 6: Convert 0.6:1.2 to simplest form
Answer: 1:2. Reason: Multiply by ten, then simplify 6:12.
After a correction, ask the learner to restate the reference amount and verify the final ratio. Use a new example a few days later to check whether the method can be chosen without a chapter heading.
Clinic 7: Simplify (1/2):(3/4)
Answer: 2:3. Reason: Multiply both parts by four.
After a correction, ask the learner to restate the reference amount and verify the final ratio. Use a new example a few days later to check whether the method can be chosen without a chapter heading.
Clinic 8: Simplify (2/3):(5/6)
Answer: 4:5. Reason: Multiply both parts by six.
After a correction, ask the learner to restate the reference amount and verify the final ratio. Use a new example a few days later to check whether the method can be chosen without a chapter heading.
Clinic 9: Boys:girls = 2:3. Boys’ fraction of class?
Answer: 2/5. Reason: There are five parts in the whole.
After a correction, ask the learner to restate the reference amount and verify the final ratio. Use a new example a few days later to check whether the method can be chosen without a chapter heading.
Clinic 10: A:B=4:1. B’s fraction of total?
Answer: 1/5. Reason: B occupies one of five total parts.
After a correction, ask the learner to restate the reference amount and verify the final ratio. Use a new example a few days later to check whether the method can be chosen without a chapter heading.
Clinic 11: Share $90 in ratio 2:3
Answer: $36 and $54. Reason: One of five parts is eighteen.
After a correction, ask the learner to restate the reference amount and verify the final ratio. Use a new example a few days later to check whether the method can be chosen without a chapter heading.
Clinic 12: Share $120 in ratio 1:2:3
Answer: $20,$40,$60. Reason: Six total parts, twenty each.
After a correction, ask the learner to restate the reference amount and verify the final ratio. Use a new example a few days later to check whether the method can be chosen without a chapter heading.
Clinic 13: Share 270 in ratio 2:3:4
Answer: 60,90,120. Reason: Nine parts, thirty each.
After a correction, ask the learner to restate the reference amount and verify the final ratio. Use a new example a few days later to check whether the method can be chosen without a chapter heading.
Clinic 14: A:B=3:5, total 64. Find A
Answer: 24. Reason: Eight total parts, eight per part.
After a correction, ask the learner to restate the reference amount and verify the final ratio. Use a new example a few days later to check whether the method can be chosen without a chapter heading.
Clinic 15: A:B=3:5, total 64. Find B
Answer: 40. Reason: Five times eight.
After a correction, ask the learner to restate the reference amount and verify the final ratio. Use a new example a few days later to check whether the method can be chosen without a chapter heading.
Clinic 16: A:B=3:7, difference 28. Find A
Answer: 21. Reason: Four parts equal twenty-eight.
After a correction, ask the learner to restate the reference amount and verify the final ratio. Use a new example a few days later to check whether the method can be chosen without a chapter heading.
Clinic 17: A:B=3:7, difference 28. Find B
Answer: 49. Reason: One part seven, B seven parts.
After a correction, ask the learner to restate the reference amount and verify the final ratio. Use a new example a few days later to check whether the method can be chosen without a chapter heading.
Clinic 18: A:B=2:5, B exceeds A by 21. Find total
Answer: 49. Reason: Three parts equal twenty-one, one part seven; seven total parts.
After a correction, ask the learner to restate the reference amount and verify the final ratio. Use a new example a few days later to check whether the method can be chosen without a chapter heading.
Clinic 19: Red:blue=4:5, red=32. Find blue
Answer: 40. Reason: One part equals eight.
After a correction, ask the learner to restate the reference amount and verify the final ratio. Use a new example a few days later to check whether the method can be chosen without a chapter heading.
Clinic 20: Red:blue=4:5, red=32. Find total
Answer: 72. Reason: Nine parts at eight each.
After a correction, ask the learner to restate the reference amount and verify the final ratio. Use a new example a few days later to check whether the method can be chosen without a chapter heading.
Clinic 21: A:B:C=2:3:5, total 100. Find C
Answer: 50. Reason: Ten parts of ten, C five parts.
After a correction, ask the learner to restate the reference amount and verify the final ratio. Use a new example a few days later to check whether the method can be chosen without a chapter heading.
Clinic 22: A:B=2:3 and B:C=4:5. Find A:B:C
Answer: 8:12:15. Reason: Scale middle B entries to twelve.
After a correction, ask the learner to restate the reference amount and verify the final ratio. Use a new example a few days later to check whether the method can be chosen without a chapter heading.
Clinic 23: A:B=1:2 and B:C=3:4. Find A:B:C
Answer: 3:6:8. Reason: Scale middle B entries to six.
After a correction, ask the learner to restate the reference amount and verify the final ratio. Use a new example a few days later to check whether the method can be chosen without a chapter heading.
Clinic 24: Red:blue=2:3, red=10. Add five red. New ratio?
Answer: 1:1. Reason: Counts become fifteen red and fifteen blue.
After a correction, ask the learner to restate the reference amount and verify the final ratio. Use a new example a few days later to check whether the method can be chosen without a chapter heading.
Clinic 25: 12 boys and 18 girls, add three boys and two girls. New ratio?
Answer: 3:4. Reason: Fifteen to twenty simplifies to three to four.
After a correction, ask the learner to restate the reference amount and verify the final ratio. Use a new example a few days later to check whether the method can be chosen without a chapter heading.
Clinic 26: Compare 50 cm:2 m
Answer: 1:4. Reason: Convert two metres to two hundred centimetres.
After a correction, ask the learner to restate the reference amount and verify the final ratio. Use a new example a few days later to check whether the method can be chosen without a chapter heading.
Clinic 27: Six items cost $15 at a constant rate. Ten items cost?
Answer: $25. Reason: Each item costs $2.50.
After a correction, ask the learner to restate the reference amount and verify the final ratio. Use a new example a few days later to check whether the method can be chosen without a chapter heading.
Clinic 28: Five kilograms cost $20. Price for two kilograms?
Answer: $8. Reason: Unit cost four dollars per kilogram.
After a correction, ask the learner to restate the reference amount and verify the final ratio. Use a new example a few days later to check whether the method can be chosen without a chapter heading.
Clinic 29: Recipe sugar:flour=2:3, flour=450 g. Sugar?
Answer: 300 g. Reason: One part of flour is 150 g.
After a correction, ask the learner to restate the reference amount and verify the final ratio. Use a new example a few days later to check whether the method can be chosen without a chapter heading.
Clinic 30: Ratio 3:7, total 50. Find larger part
Answer: 35. Reason: Ten parts total, five each.
After a correction, ask the learner to restate the reference amount and verify the final ratio. Use a new example a few days later to check whether the method can be chosen without a chapter heading.
Clinic 31: Ratio 3:7, difference 20. Find larger part
Answer: 35. Reason: Four parts difference, five each.
After a correction, ask the learner to restate the reference amount and verify the final ratio. Use a new example a few days later to check whether the method can be chosen without a chapter heading.
Clinic 32: Boys:girls=2:3 and there are 12 boys. Percentage boys?
Answer: 40%. Reason: Boys are two of five equal parts.
After a correction, ask the learner to restate the reference amount and verify the final ratio. Use a new example a few days later to check whether the method can be chosen without a chapter heading.
Clinic 33: A:B=5:2, A is 15 more than B. Find total
Answer: 35. Reason: Three parts are fifteen; seven parts at five.
After a correction, ask the learner to restate the reference amount and verify the final ratio. Use a new example a few days later to check whether the method can be chosen without a chapter heading.
Clinic 34: A:B=7:3, total 150. Find difference
Answer: 60. Reason: Ten parts at fifteen, difference four parts.
After a correction, ask the learner to restate the reference amount and verify the final ratio. Use a new example a few days later to check whether the method can be chosen without a chapter heading.
Clinic 35: Simplify 75 m:1.5 km
Answer: 1:20. Reason: Convert 1.5 km to 1500 m, then 75:1500.
After a correction, ask the learner to restate the reference amount and verify the final ratio. Use a new example a few days later to check whether the method can be chosen without a chapter heading.
Clinic 36: A:B:C=1/2:1/3:1/6. Simplify
Answer: 3:2:1. Reason: Multiply all entries by six.
After a correction, ask the learner to restate the reference amount and verify the final ratio. Use a new example a few days later to check whether the method can be chosen without a chapter heading.
A three-week improvement plan
Week 1: read and simplify ratios
Use labelled ratios, equivalent tables and simple part–whole diagrams. Begin with two quantities, then introduce three-part comparisons only after the first representations are secure.
A few independent explanations are more valuable than many repeated solutions. Record the first wrong decision, not merely the final mark.
Week 2: totals, differences and known shares
Compare one story of each kind, deliberately using similar numbers. Highlight why the divisor is the sum, difference or known entry of ratio parts. Check results against the given story.
A few independent explanations are more valuable than many repeated solutions. Record the first wrong decision, not merely the final mark.
Week 3: fractional ratios and mixed problems
Clear fractional ratios with a common denominator, combine linked pairwise ratios, then work on short practical examples involving recipes, item counts and lengths with compatible units. Finish with an unseen mixed task.
A few independent explanations are more valuable than many repeated solutions. Record the first wrong decision, not merely the final mark.
Frequently asked questions
When should I add ratio parts?
Add parts when the known quantity is the combined total. For a difference, use the gap between the relevant parts instead.
How do I explain a ratio table?
Each row shows the same relationship under a common scale factor. Multiply both associated quantities together, not just one entry.
Why are fractions and ratios related?
A ratio compares amounts. A fraction of the whole describes one amount divided by the combined total.
Can ratio entries be fractions?
Yes. Multiply each fractional entry by a suitable common denominator to obtain an equivalent whole-number ratio.
Can a ratio involve three quantities?
Yes. Keep labels in the same order and use one consistent part value across all entries.
How do pairwise ratios combine?
Make the shared quantity equal in both ratios, then place the three parts together.
Is every table a proportional table?
No. A fixed fee plus a changing unit rate is generally not a direct ratio through zero. Check whether the output/input ratio remains constant.
Why does my child use HCF or LCM incorrectly in ratios?
They may calculate mechanically without knowing whether the problem asks for equivalent units, a common comparison or equal sharing. Start by naming the meaning of one part.
How can a parent check a ratio answer?
Reconstruct the supplied total or difference and simplify the final amounts back to the original ratio.
How does small-group tuition support this?
A tutor can observe whether the child mislabels quantities, chooses the wrong part count or loses a common scale factor, then test a fresh problem independently.
Official syllabus and related lessons
Read the MOE G2 Mathematics syllabus for curriculum scope and the eduKateSG tutorial reference for the learning environment. These connected guides provide additional practice:
- How ratio, percentage and speed connect
- How to solve non-routine word problems
- Comparing fractions on number lines
- Percentage change and original values
The central lesson is to identify the relationship before choosing the arithmetic. Once a ratio part has a clear meaning, the question becomes easier to explain and verify.