A and B are in the ratio 2 : 3.
Then A receives 12 more units while B does not change.
The new ratio becomes 4 : 5.
What were the original quantities?
This is not a routine equivalent-ratio question.
The ratio has changed because the quantities did not scale together.
Before-and-after ratio problems are solved by linking two states through the quantities that remain unchanged and the changes that are explicitly known.
The essential skill is not memorising one model.
It is identifying the bridge between the “before” world and the “after” world.
That bridge might be:
- a quantity that did not change;
- a total that stayed constant;
- a known amount transferred from one person to another;
- a fixed increase or decrease;
- a known difference;
- or a quantity that can be represented in both ratio states.
Singapore’s current Primary 6 syllabus explicitly includes equivalent ratios and the relationship between fraction and ratio. It does not need to name every composite “before-and-after ratio” problem as a separate syllabus item for the reasoning to be educationally useful. This article therefore distinguishes the formal syllabus content from a broader upper-primary problem-solving extension built from those concepts.
The first rule: do not assume the ratio units stay the same size
Suppose A : B = 2 : 3.
A beginner may draw two units for A and three for B.
After the change, if A : B = 4 : 5, the learner may draw four units and five units and assume every unit in both diagrams has the same value.
That assumption is usually unjustified.
“2:3” and “4:5” describe different relationships.
The size of one ratio unit in the before-state may differ from the size of one ratio unit in the after-state.
The diagrams can only be linked when a real quantity provides a common anchor.
Ratio units are local to the relationship they describe unless a common unchanged quantity lets us calibrate them across states.
Find the invariant before doing arithmetic
An invariant is something that remains unchanged while other parts of the problem change.
In before-and-after ratio work, common invariants include:
- B stays unchanged while A changes;
- the combined total stays unchanged during a transfer;
- one population stays fixed while another increases;
- the amount of one ingredient stays fixed while another is added;
- the original quantity can be recovered because the amount of change is known.
If the learner cannot name what stayed fixed, the models often remain disconnected sketches.
Worked example 1: one quantity stays unchanged
Before:
A : B = 2 : 3.
A receives 12.
After:
A : B = 4 : 5.
B is unchanged.
Use B as the anchor.
Before, B = 3 old ratio units.
After, B = 5 new ratio units.
To compare the two states using a common unit, scale both ratio descriptions so B has the same number of parts.
LCM of 3 and 5 = 15.
Before:
A : B = 10 : 15.
After:
A : B = 12 : 15.
B now has 15 common comparison units in both states.
A increased from 10 to 12 units.
Increase = 2 units = 12.
1 unit = 6.
Original A = 10×6 = 60.
B = 15×6 = 90.
Check:
60:90 = 2:3.
After A receives 12, A = 72.
72:90 = 4:5.
The key step was not “find LCM” by itself.
The key was recognising that B represented the same actual quantity before and after.
Why the LCM method works
When one quantity is unchanged, we can rescale both ratio diagrams until that unchanged quantity uses the same number of ratio units.
This creates a common measurement unit across the two states.
It is exactly the same conceptual move as creating common denominators for fractions:
we are not changing the value of the ratios.
We are renaming them so corresponding quantities become directly comparable.
Worked example 2: the total stays constant during a transfer
Before:
A : B = 3 : 5.
B gives 16 to A.
After:
A : B = 5 : 3.
No amount entered or left the two-person system.
Total is invariant.
Before total ratio units:
3 + 5 = 8 units.
After total ratio units:
5 + 3 = 8 units.
Here the total-unit count already matches.
Before A = 3 units.
After A = 5 units.
A gained 2 units.
2 units = 16.
1 unit = 8.
Original A = 24.
Original B = 40.
After transfer:
A = 40.
B = 24.
Ratio = 5:3.
The transfer changed each individual quantity but preserved the total.
Transfers change the difference twice as fast as the transferred amount
Suppose B gives x to A.
A increases by x.
B decreases by x.
The gap between them changes by 2x.
This is useful when a problem gives differences rather than totals.
Example:
B is 50 more than A.
B gives 12 to A.
New difference:
50 − 24 = 26.
The total stayed fixed.
The difference did not.
One invariant never guarantees that every other relationship stays invariant.
Worked example 3: both quantities change, but by known amounts
Before:
A : B = 4 : 7.
A gains 18.
B gains 6.
After:
A : B = 5 : 8.
Neither A nor B is invariant.
The total is not invariant because 24 units entered the system.
But the changes are known.
Let the original common ratio unit be x.
Original:
A = 4x.
B = 7x.
After:
A = 4x + 18.
B = 7x + 6.
New ratio:
(4x + 18) : (7x + 6) = 5 : 8.
Using equality of ratios:
(4x + 18)/(7x + 6) = 5/8.
8(4x + 18) = 5(7x + 6).
32x + 144 = 35x + 30.
114 = 3x.
x = 38.
Original A = 152.
Original B = 266.
After:
A = 170.
B = 272.
170:272 simplifies to 5:8.
This algebraic route is a broader extension beyond the simplest primary model method, but it reveals the same underlying logic: define the before-state, apply the known changes, then impose the after-state relationship.
The model-method version of the same idea
For primary learners, algebra is not always the best first representation.
A bar model can show:
- four old units for A;
- seven old units for B;
- an added 18 segment to A;
- an added 6 segment to B;
- a final relationship of five new units to eight new units.
The important teaching move is to resist pretending that old units and new units are automatically equal.
Use known actual changes to connect the diagrams.
Worked example 4: one quantity decreases
Before:
A : B = 5 : 8.
B loses 21.
A stays unchanged.
After:
A : B = 5 : 7.
A is the unchanged anchor.
Before A = 5 units.
After A = 5 units.
The matching ratio-unit count means one common unit can be used immediately.
B falls from 8 units to 7 units.
1 unit = 21.
Original A = 5×21 = 105.
Original B = 8×21 = 168.
After B loses 21:
B = 147.
105:147 = 5:7.
Worked example 5: a total is given instead of a change amount
Before A : B = 2 : 5.
After A receives some amount, A : B = 4 : 5.
The final total is 81.
B did not change.
Use the after-ratio first.
Total after units:
4 + 5 = 9.
1 after-unit = 81 ÷ 9 = 9.
B = 5×9 = 45.
B is unchanged, so original B = 45.
Original ratio A:B = 2:5.
If 5 units = 45, 1 original unit = 9.
Original A = 18.
After A = 36.
A therefore received 18.
Sometimes the cleanest route begins from the after-state rather than the before-state.
Choose the state with the strongest information
A common habit is to start solving from the first sentence simply because it appears first.
That is not always efficient.
If the final ratio and final total are known, the after-state may be directly solvable.
If one quantity is unchanged and its before-value can be found, the before-state may be better.
If a transfer preserves total, the total may be the most useful anchor.
Start where the information constrains the system most strongly, not necessarily where the story begins.
A before-and-after ratio checklist
- Write the before ratio clearly.
- Write the after ratio clearly.
- List exactly what changed for each quantity.
- Identify what stayed unchanged.
- Decide whether total, difference, one quantity or a known transfer can link the two states.
- Only then decide whether to use equivalent ratios, common units, unitary method, bars or algebra.
- After solving, reconstruct both states and check both ratios.
Common misconception 1: the same ratio unit can be carried across both states automatically
Different ratios can use different unit sizes.
Repair: link them through an unchanged actual quantity or another invariant before equating units.
Common misconception 2: equal addition preserves ratio
If A:B = 2:3 and both gain 5, the ratio usually changes.
Example:
10:15 = 2:3.
After adding 5 to both:
15:20 = 3:4.
Difference stayed 5.
Ratio changed.
Common misconception 3: a transfer changes the total
If an amount moves from A to B inside the same system, the combined total stays fixed.
Repair: write the total before and after explicitly.
Common misconception 4: all ratio problems should be solved by LCM
LCM is useful when an unchanged quantity appears with different ratio-unit counts across states.
It is not a universal ratio-problem button.
Repair: identify the invariant first, then choose the method.
Common misconception 5: before-and-after means work forwards
The after-state may contain the stronger numerical information.
Working backwards can be cleaner.
A diagnostic ladder
- Can the learner solve a static ratio problem?
- Can the learner generate equivalent ratios?
- Can the learner distinguish ratio from difference?
- Can the learner identify an unchanged quantity across two states?
- Can the learner rescale two ratios around that common quantity?
- Can the learner recognise a constant total during a transfer?
- Can the learner distinguish transfer, addition and removal?
- Can the learner solve from the after-state when it is more constrained?
- Can the learner check both the original and final ratios?
- Can the learner explain why the chosen method preserves the relevant relationship?
How this fits current Singapore Primary Mathematics
The Primary Mathematics syllabus applicable to Primary 6 from 2026 explicitly includes equivalent ratios, simplifying ratios, finding missing terms in equivalent ratios, dividing quantities in a given ratio and the relationship between fraction and ratio.
The current syllabus does not require every classroom problem family to appear as a named sub-strand. “Before-and-after ratio problems with changing quantities” is best treated as a broader problem-solving application that draws on ratio, equivalence, additive change, invariants and model representation.
This distinction is important because older Singapore materials and teaching traditions may use “changing ratio” as a familiar label. For a 2026 reader, the official current syllabus should remain the authority for examinable scope, while the method itself remains mathematically valuable for reasoning and transfer.
The 2026 PSLE Mathematics assessment objectives include applying concepts in varied contexts and selecting appropriate strategies. That supports teaching learners to identify invariants and choose representations rather than memorising one template.
The deeper lesson: two snapshots need a bridge
A ratio describes one state.
A second ratio describes another.
The mathematics lives in the bridge between them.
What stayed fixed?
What entered?
What left?
What moved internally?
Which quantity can be represented in both states?
Before-and-after ratio problems become manageable when the learner stops comparing diagrams and starts tracking the conserved or known quantities that connect them.
Final thought
Do not begin a changing-ratio problem by asking which shortcut applies.
Begin by asking what reality remained the same while the ratio changed.
Once that anchor is found, the arithmetic usually becomes much quieter.