PSLE Mathematics Learning Guide · Guide 3
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A ratio tells you how quantities are related. It does not, by itself, tell you how large the quantities are.
If red beads : blue beads = 3 : 5, the collection could contain 3 red and 5 blue beads, 6 red and 10 blue beads, 30 red and 50 blue beads, or many other scaled pairs. The ratio fixes the relationship. Another piece of information—such as a total, a difference or one actual quantity—is needed to determine the scale.
This guide teaches the bridge between ratio units and actual quantities. The core habit is: identify what the ratio parts represent, identify the information that fixes the scale, find the value of one ratio unit, then return to the quantity the question asks for.
The term “ratio unit” here is an eduKate teaching label for one equal part in a ratio model. It should not be confused with a measurement unit such as centimetres or kilograms.
The examples and suggested solutions are original eduKate teaching material. They are not official PSLE questions or marking schemes. For current examination context, the SEAB 2026 PSLE examination-format page links the Mathematics syllabus. The MOE Primary Mathematics syllabus updated October 2025 applies the 2021 syllabus to Primary 6 from 2026.
Choose the difficulty you recognise
If you read 3 : 5 as the actual numbers 3 and 5, begin with Ratio is not quantity. If a total is given, use When the total fixes the scale. If a difference is given, use When the difference fixes the scale. If one group is known, use When one actual quantity fixes the scale. If a ratio changes after adding or removing something, go to Do not carry the old unit value into a new ratio automatically.
Ratio and quantity · One ratio unit · Total · Difference · Ratio and fraction · Changing ratio · Worked workshop · Independent check
Ratio is not quantity
Suppose the ratio of apples to oranges is 2 : 3. The ratio says that for every 2 equal parts assigned to apples, 3 equal parts are assigned to oranges. It does not say there are exactly 2 apples and 3 oranges.
If each ratio unit is worth 4 fruits, there are 8 apples and 12 oranges. If each ratio unit is worth 10 fruits, there are 20 apples and 30 oranges. Both collections preserve 2 : 3.
This is why a ratio problem normally contains one more condition. “There are 25 fruits altogether” fixes the scale because the five total ratio units correspond to 25 fruits. “There are 8 more oranges than apples” fixes the scale because the one-unit difference corresponds to 8 fruits. “There are 18 oranges” fixes the scale because three ratio units correspond to 18.
The ratio gives structure. The extra condition gives size.
Find the value of one ratio unit
For A : B = 3 : 5, draw or imagine 3 equal units for A and 5 equal units for B. Let one ratio unit have actual value u. Then:
A = 3u
B = 5u
Total = 8u
Difference = 2u.
A total of 64 means 8u = 64, so u = 8. A difference of 18 means 2u = 18, so u = 9. If B = 45, then 5u = 45, so u = 9.
This one-unit representation unifies several problem types. The visible numbers change, but the mathematical job is the same: identify which combination of ratio units corresponds to the given actual quantity.
When the total fixes the scale
Red : blue = 3 : 5. There are 64 beads altogether.
Total ratio units = 3 + 5 = 8 units. If 8 units = 64 beads, 1 unit = 8 beads. Red = 3 × 8 = 24. Blue = 5 × 8 = 40.
The mistake 64 ÷ 5 = 12.8 begins with the larger group rather than the total ratio. The 64 describes both colours together, so it must correspond to all 8 units.
Read the noun attached to the actual number. “64 beads altogether” means total units. “40 blue beads” means blue units only. The ratio model should match that noun before division.
When the difference fixes the scale
Books : magazines = 7 : 4. There are 21 more books than magazines.
Difference in ratio units = 7 − 4 = 3 units. If 3 units = 21 items, 1 unit = 7 items. Books = 49. Magazines = 28. Total = 77.
A common error is to use 7 + 4 = 11 units for the given 21. But the 21 is not the total number of items; it is the difference. The ratio expression must match the relationship named by the given quantity.
A useful card is:
Given total → add ratio units.
Given difference → subtract ratio units.
Given one group → use that group’s ratio units.
When one actual quantity fixes the scale
Sam : Dina = 4 : 7. Dina has 56 stickers.
Dina’s 7 ratio units = 56 stickers. One unit = 8. Sam = 4 × 8 = 32.
Do not add the ratio units first unless the given actual number is a total. The 56 belongs to Dina alone, so it corresponds to 7 units alone.
If instead Sam had 32 stickers, 4 units = 32, so one unit is again 8. The same ratio scale can be fixed from either group.
Equivalent ratios preserve the same relationship
6 : 10 and 3 : 5 are equivalent ratios because both terms have been divided by 2. Likewise 3 : 5 and 12 : 20 are equivalent because both terms have been multiplied by 4.
Simplifying can make a relationship easier to see, but it does not change the actual quantities in a given situation. If 12 red and 20 blue beads are present, simplifying their ratio to 3 : 5 does not mean the collection now has only 8 beads. The simplified terms are ratio units, not a recount of the objects.
Always distinguish actual count from simplified relationship.
A ratio can be converted to a fraction of the combined whole
If boys : girls = 2 : 3, the total has 2 + 3 = 5 ratio units. Boys therefore form 2/5 of the class and girls form 3/5.
But boys are not 2/3 of the class. The fraction 2/3 compares boys to girls, not boys to the combined class. The denominator for a fraction of the whole must represent the total number of ratio units.
Conversely, if 3/8 of a group are in Team A and the rest are in Team B, Team B forms 5/8. Therefore A : B = 3 : 5.
This conversion is useful because some problems move between ratio language and fraction language. The reference whole must remain clear.
Three-part ratios use the same unit-value logic
A : B : C = 2 : 3 : 5. There are 80 items altogether.
Total units = 2 + 3 + 5 = 10. One unit = 80 ÷ 10 = 8. Therefore A = 16, B = 24, C = 40.
If instead C has 24 more items than A, the difference is 5 − 2 = 3 units. One unit = 24 ÷ 3 = 8, leading to the same scale.
Nothing fundamentally new happens when a third group appears. The challenge is keeping track of which combination of units corresponds to the given quantity.
Equal ratio units mean equal actual amounts only within the same ratio state
If A : B = 3 : 5 and one unit is 8, then A = 24 and B = 40. If the problem then adds 16 to A and states a new ratio, the old “one unit = 8” does not automatically apply to the new ratio. The quantities have changed and the new ratio may use a different unit value.
This is one of the most important protections in before-and-after ratio problems. A ratio unit is not a physical object that survives every change. It is a scale for a particular ratio state.
Do not carry the old unit value into a new ratio automatically
Example: A : B = 2 : 3. There are 20 A-items and 30 B-items, so the old unit value is 10. Ten A-items are added, producing 30 A-items and 30 B-items. The new ratio is 1 : 1. The simplest new ratio unit value is 30, not 10.
The actual quantities connect the two states; the ratio units are representations of each state. When a problem gives before and after ratios, use a conserved quantity, known change, total or difference to connect the states. Do not assume the unit bars have identical value just because they are drawn with the same visual width.
For a focused advanced treatment, see Before-and-After Ratio Problems With Changing Quantities, Constant-Total Problems and Constant-Difference Problems.
A scale factor multiplies every term in the ratio
If juice concentrate : water = 1 : 4 and you use 3 cups of concentrate, the scale factor from 1 to 3 is 3. Water must also be multiplied by 3, giving 12 cups.
Changing only one term changes the ratio. For example, 1 : 4 scaled by 3 is 3 : 12, not 3 : 4.
This is why ratio tables can help:
| Concentrate | Water |
|---|---|
| 1 | 4 |
| 2 | 8 |
| 3 | 12 |
| 5 | 20 |
Each row preserves the same relationship.
Main worked workshop: twelve original ratio-unit problems
Problem 1: total given
The ratio of red to yellow counters is 3 : 7. There are 80 counters altogether. How many are red?
Reasoning: Total units = 10. One unit = 80 ÷ 10 = 8. Red = 3 × 8 = 24.
Problem 2: difference given
Ali : Ben = 5 : 8. Ben has 27 more tokens than Ali. How many tokens does Ali have?
Reasoning: Difference = 3 units = 27. One unit = 9. Ali = 5 × 9 = 45.
Problem 3: one group given
Small boxes : large boxes = 4 : 9. There are 63 large boxes. How many small boxes are there?
Reasoning: 9 units = 63, so 1 unit = 7. Small boxes = 4 × 7 = 28.
Problem 4: ratio to fraction
Blue : green marbles = 2 : 5. What fraction of all the marbles are blue?
Reasoning: Total units = 7. Blue fraction = 2/7.
Problem 5: fraction to ratio
Three eighths of the pupils choose Art. The rest choose Music. Find the ratio Art : Music.
Reasoning: Art = 3/8, Music = 5/8. Ratio = 3 : 5.
Problem 6: three-part total
A : B : C = 2 : 5 : 3. Their total is 120. Find B.
Reasoning: Total units = 10. One unit = 12. B = 5 × 12 = 60.
Problem 7: three-part difference
P : Q : R = 4 : 6 : 9. R has 35 more items than P. Find Q.
Reasoning: R − P = 5 units = 35, so 1 unit = 7. Q = 6 × 7 = 42.
Problem 8: simplify without changing quantities
A basket contains 24 pears and 36 apples. Find the simplest ratio pears : apples.
Reasoning: 24 : 36 divide both by 12 = 2 : 3. This describes the relationship; the basket still contains 60 fruits.
Problem 9: actual quantity from a part-to-part ratio
Boys : girls = 7 : 9. There are 32 pupils altogether. How many girls are there?
Reasoning: Total units = 16. One unit = 2 pupils. Girls = 18.
Problem 10: unit rate inside a ratio context
Flour : sugar = 5 : 2. A recipe uses 350 g of flour. How much sugar is needed to preserve the ratio?
Reasoning: 5 units = 350 g, so 1 unit = 70 g. Sugar = 2 × 70 = 140 g.
Problem 11: changing one group
There are 24 red beads and 36 blue beads. Twelve red beads are added. What is the new ratio red : blue in simplest form?
Reasoning: New red = 36; blue remains 36. New ratio = 36 : 36 = 1 : 1. Do not preserve the old ratio 2 : 3 after the quantity changes.
Problem 12: find the scale from a combined subgroup
A : B : C = 3 : 4 : 5. A and B together contain 42 items. How many items are in C?
Reasoning: A + B = 7 units = 42. One unit = 6. C = 5 × 6 = 30.
Match the given information to the correct ratio-unit expression
| Given information | For A : B = 3 : 5 | Meaning |
|---|---|---|
| A + B = 64 | 8 units = 64 | Total fixes scale |
| B − A = 18 | 2 units = 18 | Difference fixes scale |
| A = 24 | 3 units = 24 | A fixes scale |
| B = 40 | 5 units = 40 | B fixes scale |
This table is more useful than memorising “add the ratio” or “subtract the ratio” as isolated instructions. The operation depends on what the actual number describes.
A bar model makes unit correspondence visible
For A : B = 3 : 5, draw three equal boxes for A and five equal boxes for B. If the total is 64, bracket all eight boxes and label 64. If the difference is 18, bracket only the extra two boxes on B’s side and label 18. If B is 40, bracket B’s five boxes and label 40.
The same visual ratio can therefore support different calculations depending on where the given actual number is attached. The model does not choose the method automatically; the label does.
Read the order of the ratio literally
If cats : dogs = 2 : 5, then cats correspond to 2 and dogs to 5. Reversing the ratio gives dogs : cats = 5 : 2.
When a question asks for girls : boys but the sentence gives boys : girls, rewrite the order before calculating. Many ratio errors are not arithmetic failures; they are mapping failures between labels and terms.
Use names beside the ratio terms, especially in three-part ratios. “A:B:C = 2:3:5” is safer than carrying “2:3:5” alone through several paragraphs of a problem.
Do not add ratios from different populations without rebuilding the quantities
Class X has boys : girls = 2 : 3. Class Y has boys : girls = 4 : 5. The combined ratio is not automatically 6 : 8 = 3 : 4. The two classes may have different sizes and therefore different unit values.
Example: Class X could have 20 boys and 30 girls. Class Y could have 8 boys and 10 girls. Combined boys : girls = 28 : 40 = 7 : 10, not 3 : 4.
To combine ratios from separate groups, recover compatible actual quantities or enough information about their scales first.
Repair the smallest reasoning defect
Draft: Ratio 3 : 5, total 64, so one unit = 64 ÷ 5.
Problem: 64 is the combined total, not the second group.
Repair: 3 + 5 = 8 units = 64.
Draft: Ratio 7 : 4, difference 21, so 11 units = 21.
Problem: 21 describes the difference, not the total.
Repair: 7 − 4 = 3 units = 21.
Draft: 24 : 36 simplifies to 2 : 3, so there are 2 pears and 3 apples.
Problem: simplified ratio terms were mistaken for actual counts.
Repair: Keep 24 and 36 as the actual quantities; 2 : 3 describes their relationship.
Draft: Before ratio 2 : 3 has unit value 10, so after the quantities change the new ratio units must also be 10.
Problem: ratio unit value was carried across states without justification.
Repair: Connect states through actual conserved quantities or known changes, then determine the new scale.
Four checks for a ratio answer
- Order check: Did each ratio term stay attached to the correct group?
- Scale check: What actual information fixed one ratio unit?
- Relationship check: Does the final pair simplify back to the given ratio?
- Condition check: If quantities changed, did you update the ratio state rather than preserve an old scale automatically?
Example: If a solution says A = 36 and B = 60 for A : B = 3 : 5, simplify 36 : 60 = 3 : 5. The ratio check passes. If the total was supposed to be 88, however, 36 + 60 = 96, so the scale condition fails. Both checks matter.
Independent transfer check: the activity packs
A school prepares Art, Science and Games packs in the ratio 3 : 5 : 4. There are 144 packs altogether. Later, 12 additional Art packs are made while the numbers of Science and Games packs stay unchanged.
- How many ratio units are there altogether at first?
- What is the value of one ratio unit at first?
- How many Art packs are there at first?
- How many Science packs are there at first?
- How many Games packs are there at first?
- After 12 Art packs are added, what is the new ratio Art : Science : Games in simplest form?
- Explain why the original ratio-unit value cannot simply be attached to the terms of the new simplified ratio.
- If instead the problem had said Science had 24 more packs than Art at first, which ratio-unit expression would correspond to 24?
Independent-check answers
1. 3 + 5 + 4 = 12 units.
2. 144 ÷ 12 = 12 packs per original ratio unit.
3. Art = 3 × 12 = 36.
4. Science = 5 × 12 = 60.
5. Games = 4 × 12 = 48.
6. New Art = 48; Science = 60; Games = 48. New ratio = 48 : 60 : 48. Divide by 12 = 4 : 5 : 4.
7. The simplified new ratio describes the relationship after the change. Its terms are not a continuation of the old bars. In the new ratio 4 : 5 : 4, one simplified ratio unit corresponds to 12 actual packs here, but that equality must be established from the new actual quantities; it is not guaranteed merely because the old unit value happened to be 12.
8. Science − Art = 5 − 3 = 2 ratio units. So 2 units would correspond to 24.
Build ratio control in layers
Begin with ratio and one known group. Then use total-given problems. Next use difference-given problems. Add three-part ratios only after the learner can state which units match the given quantity. Finally introduce before-and-after states where the unit value may change.
Do not rush to algebraic-looking shortcuts if the learner cannot explain what one unit means. A compact calculation is useful after the relationship is stable, not before.
For each correction, record the exact mismatch: “used total units for a difference”, “reversed ratio order”, “treated simplified terms as actual counts”, or “carried old unit value into new ratio”. These are different errors and require different practice.
Parent and tutor guide: ask what one unit stands for
When a learner is stuck, ask, “What does one equal part represent?” and “Which parts does the number in the question describe?” If the problem gives a total, point to all ratio units. If it gives a difference, point only to the unmatched units. If it gives one group, point to that group’s units.
Avoid saying only “add the ratio” or “subtract the ratio”. Those instructions can become brittle rules. Tie the operation to the meaning: add because the given number is a total; subtract because the given number is a difference.
Use new contexts for transfer. A child who solves bead ratios should later meet money, pupils, mass, ingredients or distances, provided the quantities remain appropriate to the syllabus. The surface changes while the unit-value logic stays the same.
What progress looks like
Early progress appears when the learner writes labels beside ratio terms. Stronger progress appears when the learner can identify whether a given actual number corresponds to a total, difference, subgroup or combined subgroup before calculating.
Independent control appears when the learner rejects an invalid method in words: “I cannot divide the total by 5 because 5 units are only the blue group; the total is 8 units.” That explanation shows the scale relationship has become explicit.
The learner’s final card
What do the ratio terms stand for? What actual information fixes the scale? Which ratio units match that information? What is one unit worth? Which actual quantity does the question ask for?
Keep ratios as relationships until the scale is known. Keep actual quantities as quantities. Do not let simplified parts pretend to be counts.
Continue through the PSLE Mathematics Learning Guide
Use Guide 1: Find the Reference Whole Before You Use a Fraction or Percentage and Guide 2: Make Units Agree Before You Calculate when the difficulty is reference or measurement. Continue with Guide 4: Rebuild Average Problems From Total ÷ Number of Items.
Return to the PSLE Learning Guide for the wider English, Mathematics and Science pathway.
Sources and boundaries
Official assessment reference: Singapore Examinations and Assessment Board: PSLE formats examined in 2026 and its linked Mathematics syllabus. Checked 5 September 2026.
Curriculum reference: Ministry of Education: Primary Mathematics syllabus, updated October 2025.
Teaching-material boundary: The ratio-unit language, worked examples, practice tasks and suggested solutions are original eduKate teaching material. They are not reproduced SEAB questions or official marking schemes. Equivalent correct methods may exist. Current official examination information and school instructions remain authoritative.