PRIMARY 5 MATHEMATICS LEARNING GUIDE · BATCH 6 · GUIDE 21
Comparison problems become easier when the learner separates two different questions: how much more, and how many times as much. One is additive. The other is multiplicative. Both can appear in bar models, and confusing them is one of the fastest ways to turn a correct diagram into a wrong solution.
This guide develops comparison bar models as a reasoning system. It covers equal starts, differences, unknown smaller or larger quantities, multiplicative comparison, fractions, percentages, mixed relationships and reverse problems. The goal is not to force bar models onto every question. It is to use them when aligned bars make the relationship visible.
For the official curriculum framework, see the MOE Primary Mathematics Syllabus.
Series route: return to the Primary 5 Mathematics Learning Hub. Continue to Before–After Models, Change Unknowns & Working Backwards, Assumption Method, Fixed Totals, Equalisation & Difference Reasoning and Systematic Listing, Case Organisation, Tables & Invariant Thinking.
1. Comparison begins with a reference quantity
Suppose Mei has 84 stickers and Ravi has 27 fewer. Mei is the reference larger quantity. Ravi’s quantity is:
84 − 27 = 57 stickers.
A comparison bar aligns the shared 57 with an extra segment of 27 on Mei’s bar. The extra segment is the additive difference.
2. Additive comparison asks for a difference
If A = 72 and B = 45, then A is 27 more than B.
The relationship is:
A = B + 27.
Reverse it and B = A − 27.
The bar model makes the common part and extra part visible.
3. Multiplicative comparison asks for a scale factor
If A = 72 and B = 24, then A is 3 times B.
The relationship is:
A = 3 × B.
The difference is 48, but that is a different description.
“48 more” and “3 times as many” answer different questions about the same pair.
4. The words can look similar while the mathematics differs
“Ali has 20 more marbles than Ben” means addition.
“Ali has 20 times as many marbles as Ben” means multiplication.
“Ali has 20% more marbles than Ben” means Ali has 120% of Ben’s amount.
Do not use the word “more” as an operation keyword. Read the whole relationship.
5. Find the smaller quantity from a difference
Jia has 156 books, which is 38 more than Hana. Hana has:
156 − 38 = 118 books.
The bar model shows one bar equal to the other plus a 38-book extension.
6. Find the larger quantity from a difference
Hana has 118 books. Jia has 38 more. Jia has:
118 + 38 = 156 books.
The same model is read in the opposite direction.
7. Find both quantities from total and difference
Two classes have 74 students altogether. Class A has 8 more than Class B.
Remove the extra 8: 74 − 8 = 66. Split the equal part: 66 ÷ 2 = 33.
Class B = 33. Class A = 41.
This “remove the difference, then split” method comes directly from aligned bars.
8. Find both quantities from total and multiplicative comparison
A is 3 times B. Together they total 120.
Represent B as 1 unit and A as 3 units. Total = 4 units = 120.
One unit = 30. So B = 30 and A = 90.
This is equal-unit reasoning, not additive difference reasoning.
9. Difference between multiplicatively related quantities
A is 5 times B and exceeds B by 72.
Represent A as 5 units and B as 1 unit. Difference = 4 units = 72.
One unit = 18. B = 18 and A = 90.
The difference reveals the gap between the unit counts.
10. Fractions can encode comparison units
If A is 3/5 of B and B = 100, then A = 60.
If instead A = 60 and A is 3/5 of B, then B = 100.
The same fraction can be used forward or backward depending on which quantity is known.
11. Fraction of another quantity is not fraction of total
If red beads are 3/4 of blue beads, represent red = 3 units and blue = 4 units.
Red is then 3/7 of the combined total, not 3/4.
The phrase “of blue” defines the reference quantity.
12. Percentage comparison uses the reference amount as 100%
Hana has 240 beads. Jia has 35% more than Hana.
Hana = 100% = 240. Jia = 135%.
Jia = 1.35 × 240 = 324 beads.
The extra 35% is measured against Hana’s quantity.
13. “20% more” and “20 more” are not interchangeable
If B = 100, then 20 more gives 120 and 20% more also gives 120.
If B = 250, then 20 more gives 270 while 20% more gives 300.
One is a fixed additive change; the other scales with the reference quantity.
14. Percentage decrease uses the original as 100%
If A is 20% less than B, then A = 80% of B.
If A = 160, then B = 160 ÷ 0.8 = 200.
Subtracting 20% of 160 would use the wrong whole.
15. Comparison with changing states
Suppose A has 40 more than B. Then both receive 15.
The difference remains 40 because the same amount was added to both quantities.
This is difference conservation.
16. Same additive change preserves difference
If A − B = 27, then (A + 10) − (B + 10) = 27.
Likewise, subtracting the same amount from both preserves the difference.
This invariant is useful in before–after comparison problems.
17. Same multiplicative scaling preserves ratio, not difference
If A:B = 3:2 and both are doubled, the new comparison is 6:4, which preserves the ratio 3:2.
But the additive difference doubles.
Know which relationship stays constant under the transformation.
18. Comparison bar models can reveal hidden totals
A has 24 more than B. C has 10 fewer than B. The total of A, B and C is 206.
Represent B as x. Then A = x + 24 and C = x − 10.
3x + 14 = 206, so 3x = 192 and x = 64.
Therefore A = 88, B = 64, C = 54.
A bar model and equation encode the same comparison system.
19. Equal-unit models support multiplicative comparison
If A:B = 4:3 in a transitional comparison model, the bars can be divided into equal units. If the difference is 25, one unit = 25, so A = 100 and B = 75.
Formal ratio belongs later in the current primary sequence, but equal-unit comparison is already useful as a bridge from fractions and rate.
20. Use a bar model only when it reduces uncertainty
For 72 − 27, a bar may be unnecessary. For a two-stage comparison involving total, difference and percentage, a bar can prevent reference quantities from being lost.
The model should earn its place by making the relationship clearer.
21. Label bars with quantity names and units
An unlabeled long rectangle and short rectangle do not explain anything. Mark “Hana”, “Jia”, total values, unknowns and difference segments.
Labels turn geometry into a mathematical model.
22. Comparison with money
Plan A costs $36 more than Plan B. Together they cost $204.
Remove difference: 204 − 36 = 168. Half = 84.
Plan B = $84; Plan A = $120.
23. Comparison with rates
Machine A makes 42 items/min and Machine B makes 30 items/min. A makes 12 more items each minute.
Over 10 minutes at constant rates, A makes 120 more items.
A per-unit difference can scale across repeated units.
24. Comparison with area
Triangle A has area 48 cm². Triangle B has 12 cm² less. B has 36 cm².
If instead A has twice B’s area, then B = 24 cm².
The same visible quantities can support additive or multiplicative comparison depending on the statement.
25. Error map
| Visible error | Likely cause | Repair question |
|---|---|---|
| “3 times as many” solved by +3 | Multiplicative comparison read additively | How many equal copies of the reference quantity? |
| “20% more” solved by +20 | Percentage scale lost | What quantity is 100%? |
| Total and difference split equally without removing difference | Extra segment ignored | Which part is not shared? |
| 3/4 of B treated as 3/4 of total | Reference quantity lost | What does “of B” refer to? |
| Bars drawn but unlabelled | Representation not semantic | Which bar represents which quantity? |
26. Practice laboratory
- A has 92 stickers and B has 27 fewer. Find B.
- A and B total 156. A has 24 more. Find both.
- A is 4 times B. Together they total 150. Find both.
- A is 5 times B and exceeds B by 64. Find both.
- A is 3/5 of B. If A = 72, find B.
- Red is 3/4 of blue. What fraction of the total is red?
- Jia has 25% more than Hana. Hana has 240. Find Jia.
- An item A costs 20% less than item B. A costs $160. Find B.
- Two quantities differ by 38. Both increase by 12. What is the new difference?
- Machine A makes 15 more items/min than B. Over 8 min, how many more items does A make?
27. Explained answers
1. 92 − 27 = 65.
2. (156 − 24) ÷ 2 = 66; values 90 and 66.
3. 5 units = 150; one = 30; values 120 and 30.
4. 4 units = 64; one = 16; values 80 and 16.
5. 3 parts = 72; one = 24; 5 parts = 120.
6. Red 3 units, blue 4; red is 3/7 of total.
7. 125% × 240 = 300.
8. A = 80% of B; B = 160 ÷ 0.8 = $200.
9. 38.
10. 15 × 8 = 120.
28. Full mixed comparison problem
Mei has 40% more beads than Ravi. Together they have 360 beads. Find each amount.
Ravi = 100%; Mei = 140%. Together = 240%.
1% = 360 ÷ 240 = 1.5. Ravi = 150. Mei = 210.
Check: Mei has 60 more, and 60/150 = 40%.
29. Final checkpoint
A strong Primary 5 comparison learner can distinguish additive difference from multiplicative comparison, use aligned bars, recover quantities from total and difference, interpret fractions and percentages relative to the correct reference, recognise invariants and choose a bar model only when it makes the relationship easier to inspect.
Continue to Primary 5 Mathematics Learning Guide | Before–After Models, Change Unknowns & Working Backwards.
Editorial approach: Wintour House V1.0 · CivDJ · eduKate Publishing. Align the reference quantities, separate additive gaps from multiplicative scale, preserve the comparison invariant, and test the model by reconstructing both quantities.
