Before-and-After Ratio Problems are PSLE Mathematics questions in which a ratio changes after something is added, removed, transferred or redistributed. The hard part is not the ratio arithmetic. It is connecting two different states without assuming that one unit before the change is automatically equal to one unit after the change.

Students searching for before and after ratio, changing ratio problems, PSLE ratio heuristics or Primary 5 and 6 ratio problem sums often write the two ratios side by side and immediately equate units. That works only when a genuine invariant justifies it. The safe method is state → change → invariant → scaling → solve → check.

This page is the canonical Before-and-After Ratio owner under the PSLE Mathematics Heuristics hub. It works directly with Constant Total, Constant Difference and Constant Part and Units and Parts.

Quick answer: the six-step method

1. Write the before ratio.
2. Write the after ratio.
3. Describe exactly what changed.
4. Identify what stayed constant.
5. Scale the two ratios so the invariant represents the same number of units.
6. Use the changed units to find one unit and solve the required quantity.

Why before-units and after-units are not automatically equal

Suppose a ratio changes from 2:3 to 3:4. The symbol “1 unit” in the first ratio is defined by the first state. The symbol “1 unit” in the second ratio is defined by the second state. They are only equal if a constant quantity links them.

This is the central safety rule. Many ratio mistakes come from skipping it.

Case 1: constant total

Problem: Red and blue beads are in the ratio 3:7. After 20 blue beads are moved to the red group, the ratio becomes 1:1. How many beads were there altogether?

The transfer stays inside the same two groups, so the total is constant. When 20 moves from the larger group to the smaller group, the gap closes by 40. The original ratio difference is 4 units, so 4 units = 40, hence 1 unit = 10. The total is 10 units = 100 beads.

The ratio change is solved by noticing what the transfer does to the gap.

Case 2: constant part

Problem: Aisha and Ben have money in the ratio 2:5. Aisha’s amount remains unchanged. After Ben spends $45, the ratio becomes 2:3. How much did Ben have at first?

Aisha is the constant part. Her 2 units before and 2 units after represent the same actual amount, so the unit size can be aligned directly. Ben falls from 5 units to 3 units: 2 units = $45. Therefore 1 unit = $22.50 and Ben originally had 5 units = $112.50.

The important move is not subtraction; it is identifying Aisha as the bridge.

Case 3: constant difference

Problem structure: Two quantities are in one ratio. The same amount is added to both, producing a new ratio. Because both quantities increase equally, their actual difference remains constant.

Scale the before and after ratios so the difference in units represents the same actual gap. Then use the known addition or a known amount to find one unit.

Do not assume the ratio-unit size stays constant merely because the difference stays constant. Scale first.

Case 4: one side gains or loses, the other stays fixed

This is a constant-part problem even if the story does not use the words “remains unchanged”. If only one quantity changes and nothing happens to the other, the unchanged side can connect the ratios.

The state table

When a problem feels confusing, write a three-row table:

• Before: ratio and known values.
• Change: +, − or transfer action.
• After: new ratio and known values.

Then circle the quantity that is unchanged. The table prevents information from two states being mixed together.

A worked scaling example

Problem: The ratio of boys to girls is 3:4. After 6 boys join, the ratio becomes 9:10. How many girls are there?

Girls do not change, so girls are the constant part.

Before: boys:girls = 3:4.
After: boys:girls = 9:10.

Scale the before ratio so girls are 20 units: 15:20.
Scale the after ratio so girls are 20 units: 18:20.

Boys increase by 3 aligned units, and that increase equals 6 boys.
1 unit = 2 boys.
Girls = 20 units = 40 girls.

The key is aligning the unchanged girls, not equating the original 1 unit to the new 1 unit.

A transfer problem can change two quantities at once

If 8 items are transferred from A to B, A decreases by 8 while B increases by 8. The total stays constant, but the difference changes by 16. This “double movement” is why transfer problems often surprise pupils.

Always ask whether the question is better solved through total, difference or a direct aligned model.

When to use a bar model

Use bars when the before/after states are easier to compare visually than symbolically. Draw one state above the other, mark the constant quantity and show the added, removed or transferred section. The Model Method guide gives the broader representation system.

Common errors

• equating units across two ratios without justification
• missing the invariant
• calling total constant when items enter or leave the whole system
• forgetting that a transfer changes both groups
• scaling the wrong side
• mixing before values with after units
• finding one unit correctly but answering for the wrong state

The before-and-after diagnostic

1. Can the pupil state the two ratios?
2. Can the pupil describe the change in one sentence?
3. Can the pupil identify the invariant?
4. Can the pupil explain why the scaling is legitimate?
5. Can the pupil solve a similar problem when the invariant changes?

A transfer practice set

Practice 1

Structure: Ratio 2:5 becomes 3:5 after A gains 14 while B is unchanged.

First invariant to test: Constant part: B.

Practice 2

Structure: Ratio 4:7 becomes 5:6 after items are transferred from the second group to the first.

First invariant to test: Constant total.

Practice 3

Structure: Ratio 3:8 becomes 5:10 after both groups receive the same amount.

First invariant to test: Constant difference.

Practice 4

Structure: Ratio 5:9 becomes 5:7 after the second group loses 18.

First invariant to test: Constant part: first group.

Frequently asked questions

What is the main idea in before-and-after ratios?

Two ratios describe two different states. A constant quantity is needed to connect them safely.

Can I just cross-multiply the ratios?

Not unless the equations correctly represent the change. Primary heuristic methods usually make the invariant visible first.

What is the most common invariant?

Constant total, constant difference and constant part are common, but the actual story determines which applies.

Why does transfer change the difference by twice the transferred amount?

One group decreases while the other increases, so the gap changes from both sides.

Should I always draw bars?

No. Use bars when they clarify the state change.

Where does this sit in Atlas?

This is the canonical Before-and-After Ratio owner under PSLE Mathematics Heuristics.

The final ratio-state rule

Never connect two ratios merely because they belong to the same problem. Identify the unchanged real quantity first. The invariant tells you how the two unit systems can be aligned.

Future Master Publish Deepening | Before-and-After Ratio

Use this guide to build a method you can apply independently. The operating rule is to treat each ratio as a description of its own state and align the states only through a real unchanged quantity or known change. The route below turns public explanation into independent capability: recognition, representation, execution, fresh practice, verification, specialist handoff and return. One key boundary is that before and after unit sizes are not automatically equal.

1. Reader job

The focus at this stage is to move from a word problem to a defensible representation. For Before-and-After Ratio, the diagnostic question is: What relationship is hidden and what exactly is the target? The tutor should not treat a correct final number as enough evidence. Better evidence is that the student can state the target and relationship before calculating. This matters because topical worksheets often make the route obvious, while mixed PSLE problems require the learner to choose and monitor the route independently.

A practical way to train this stage is to classify three short problems without solving them. After the attempt, ask the learner what changed in their thinking, what information became useful, and what they would do if the first route failed. The purpose is to make the decision process visible long enough for accurate feedback, then gradually remove the verbal scaffolding as the student becomes more independent.

2. Structure recognition

The focus at this stage is to separate mathematical structure from surface vocabulary. For Before-and-After Ratio, the diagnostic question is: Which quantities are compared, combined, repeated, or changed? The tutor should not treat a correct final number as enough evidence. Better evidence is that method choice survives a change of names and story context. This matters because topical worksheets often make the route obvious, while mixed PSLE problems require the learner to choose and monitor the route independently.

A practical way to train this stage is to rewrite one familiar problem with different language. After the attempt, ask the learner what changed in their thinking, what information became useful, and what they would do if the first route failed. The purpose is to make the decision process visible long enough for accurate feedback, then gradually remove the verbal scaffolding as the student becomes more independent.

3. Boundary test

The focus at this stage is to recognise when the method should not be used. For Before-and-After Ratio, the diagnostic question is: What defining condition is missing from this problem? The tutor should not treat a correct final number as enough evidence. Better evidence is that the learner can reject the method for a structural reason. This matters because topical worksheets often make the route obvious, while mixed PSLE problems require the learner to choose and monitor the route independently.

A practical way to train this stage is to mix one valid and two false-positive examples. After the attempt, ask the learner what changed in their thinking, what information became useful, and what they would do if the first route failed. The purpose is to make the decision process visible long enough for accurate feedback, then gradually remove the verbal scaffolding as the student becomes more independent.

4. Dependency check

The focus at this stage is to find the earliest weak prerequisite. For Before-and-After Ratio, the diagnostic question is: Which lower skill fails before the method itself fails? The tutor should not treat a correct final number as enough evidence. Better evidence is that a smaller diagnostic task reproduces the error. This matters because topical worksheets often make the route obvious, while mixed PSLE problems require the learner to choose and monitor the route independently.

A practical way to train this stage is to repair the dependency before returning to the full problem. After the attempt, ask the learner what changed in their thinking, what information became useful, and what they would do if the first route failed. The purpose is to make the decision process visible long enough for accurate feedback, then gradually remove the verbal scaffolding as the student becomes more independent.

5. Target control

The focus at this stage is to keep the requested quantity visible. For Before-and-After Ratio, the diagnostic question is: What must the final answer describe and in what unit? The tutor should not treat a correct final number as enough evidence. Better evidence is that intermediate calculations remain connected to the target. This matters because topical worksheets often make the route obvious, while mixed PSLE problems require the learner to choose and monitor the route independently.

A practical way to train this stage is to write the target above the working. After the attempt, ask the learner what changed in their thinking, what information became useful, and what they would do if the first route failed. The purpose is to make the decision process visible long enough for accurate feedback, then gradually remove the verbal scaffolding as the student becomes more independent.

6. Representation choice

The focus at this stage is to choose the cheapest valid representation. For Before-and-After Ratio, the diagnostic question is: Does a bar, table, diagram, statement, or equation reduce uncertainty? The tutor should not treat a correct final number as enough evidence. Better evidence is that the representation preserves all important conditions. This matters because topical worksheets often make the route obvious, while mixed PSLE problems require the learner to choose and monitor the route independently.

A practical way to train this stage is to compare two possible representations. After the attempt, ask the learner what changed in their thinking, what information became useful, and what they would do if the first route failed. The purpose is to make the decision process visible long enough for accurate feedback, then gradually remove the verbal scaffolding as the student becomes more independent.

7. Label discipline

The focus at this stage is to keep every part of the representation meaningful. For Before-and-After Ratio, the diagnostic question is: What does each bar, unit, segment, or symbol represent? The tutor should not treat a correct final number as enough evidence. Better evidence is that the learner can explain every label without guessing. This matters because topical worksheets often make the route obvious, while mixed PSLE problems require the learner to choose and monitor the route independently.

A practical way to train this stage is to cover the question and read the representation back as a story. After the attempt, ask the learner what changed in their thinking, what information became useful, and what they would do if the first route failed. The purpose is to make the decision process visible long enough for accurate feedback, then gradually remove the verbal scaffolding as the student becomes more independent.

8. Invariant search

The focus at this stage is to identify what remains fixed across changes. For Before-and-After Ratio, the diagnostic question is: What stays the same while other quantities move? The tutor should not treat a correct final number as enough evidence. Better evidence is that the chosen bridge remains true in both states. This matters because topical worksheets often make the route obvious, while mixed PSLE problems require the learner to choose and monitor the route independently.

A practical way to train this stage is to circle the invariant before solving. After the attempt, ask the learner what changed in their thinking, what information became useful, and what they would do if the first route failed. The purpose is to make the decision process visible long enough for accurate feedback, then gradually remove the verbal scaffolding as the student becomes more independent.

9. Worked reasoning

The focus at this stage is to make each transformation inspectable. For Before-and-After Ratio, the diagnostic question is: Why is this step allowed by the relationship? The tutor should not treat a correct final number as enough evidence. Better evidence is that the learner can justify major steps in ordinary language. This matters because topical worksheets often make the route obvious, while mixed PSLE problems require the learner to choose and monitor the route independently.

A practical way to train this stage is to pause after each step and name its purpose. After the attempt, ask the learner what changed in their thinking, what information became useful, and what they would do if the first route failed. The purpose is to make the decision process visible long enough for accurate feedback, then gradually remove the verbal scaffolding as the student becomes more independent.

10. Alternative route

The focus at this stage is to compare valid methods rather than worship one. For Before-and-After Ratio, the diagnostic question is: Could another representation reach the same result more safely? The tutor should not treat a correct final number as enough evidence. Better evidence is that two routes agree and their trade-offs are understood. This matters because topical worksheets often make the route obvious, while mixed PSLE problems require the learner to choose and monitor the route independently.

A practical way to train this stage is to solve once visually and once arithmetically. After the attempt, ask the learner what changed in their thinking, what information became useful, and what they would do if the first route failed. The purpose is to make the decision process visible long enough for accurate feedback, then gradually remove the verbal scaffolding as the student becomes more independent.

11. Error taxonomy

The focus at this stage is to classify the first wrong decision. For Before-and-After Ratio, the diagnostic question is: Was the error recognition, representation, execution, or checking? The tutor should not treat a correct final number as enough evidence. Better evidence is that the same error category predicts mistakes on another problem. This matters because topical worksheets often make the route obvious, while mixed PSLE problems require the learner to choose and monitor the route independently.

A practical way to train this stage is to record mechanism rather than page number. After the attempt, ask the learner what changed in their thinking, what information became useful, and what they would do if the first route failed. The purpose is to make the decision process visible long enough for accurate feedback, then gradually remove the verbal scaffolding as the student becomes more independent.

12. Error ledger

The focus at this stage is to turn correction into future prevention. For Before-and-After Ratio, the diagnostic question is: What rule would prevent this error next time? The tutor should not treat a correct final number as enough evidence. Better evidence is that the error disappears on delayed retest. This matters because topical worksheets often make the route obvious, while mixed PSLE problems require the learner to choose and monitor the route independently.

A practical way to train this stage is to write one corrected rule and one fresh test. After the attempt, ask the learner what changed in their thinking, what information became useful, and what they would do if the first route failed. The purpose is to make the decision process visible long enough for accurate feedback, then gradually remove the verbal scaffolding as the student becomes more independent.

13. Near transfer

The focus at this stage is to remove exact-number memory. For Before-and-After Ratio, the diagnostic question is: Can the same reasoning survive different numbers? The tutor should not treat a correct final number as enough evidence. Better evidence is that the learner rebuilds rather than copies the prior layout. This matters because topical worksheets often make the route obvious, while mixed PSLE problems require the learner to choose and monitor the route independently.

A practical way to train this stage is to change all numbers while preserving structure. After the attempt, ask the learner what changed in their thinking, what information became useful, and what they would do if the first route failed. The purpose is to make the decision process visible long enough for accurate feedback, then gradually remove the verbal scaffolding as the student becomes more independent.

14. Language transfer

The focus at this stage is to remove phrase dependence. For Before-and-After Ratio, the diagnostic question is: Can the relationship survive different wording and sentence order? The tutor should not treat a correct final number as enough evidence. Better evidence is that recognition remains accurate with new vocabulary. This matters because topical worksheets often make the route obvious, while mixed PSLE problems require the learner to choose and monitor the route independently.

A practical way to train this stage is to paraphrase the problem before solving. After the attempt, ask the learner what changed in their thinking, what information became useful, and what they would do if the first route failed. The purpose is to make the decision process visible long enough for accurate feedback, then gradually remove the verbal scaffolding as the student becomes more independent.

15. Representation transfer

The focus at this stage is to move between visual and symbolic forms. For Before-and-After Ratio, the diagnostic question is: Can the same relationship be expressed another way? The tutor should not treat a correct final number as enough evidence. Better evidence is that the student translates without changing meaning. This matters because topical worksheets often make the route obvious, while mixed PSLE problems require the learner to choose and monitor the route independently.

A practical way to train this stage is to convert a model to an equation or table. After the attempt, ask the learner what changed in their thinking, what information became useful, and what they would do if the first route failed. The purpose is to make the decision process visible long enough for accurate feedback, then gradually remove the verbal scaffolding as the student becomes more independent.

16. Method discrimination

The focus at this stage is to choose between neighbouring strategies. For Before-and-After Ratio, the diagnostic question is: Why does this method fit better than the closest alternative? The tutor should not treat a correct final number as enough evidence. Better evidence is that the learner can explain both selection and rejection. This matters because topical worksheets often make the route obvious, while mixed PSLE problems require the learner to choose and monitor the route independently.

A practical way to train this stage is to mix two similar problem families. After the attempt, ask the learner what changed in their thinking, what information became useful, and what they would do if the first route failed. The purpose is to make the decision process visible long enough for accurate feedback, then gradually remove the verbal scaffolding as the student becomes more independent.

17. Counterexample

The focus at this stage is to learn the edge of the heuristic. For Before-and-After Ratio, the diagnostic question is: What looks familiar here but is structurally different? The tutor should not treat a correct final number as enough evidence. Better evidence is that the learner rejects keyword traps. This matters because topical worksheets often make the route obvious, while mixed PSLE problems require the learner to choose and monitor the route independently.

A practical way to train this stage is to design one problem containing a misleading cue. After the attempt, ask the learner what changed in their thinking, what information became useful, and what they would do if the first route failed. The purpose is to make the decision process visible long enough for accurate feedback, then gradually remove the verbal scaffolding as the student becomes more independent.

18. Reverse engineering

The focus at this stage is to reconstruct language from mathematics. For Before-and-After Ratio, the diagnostic question is: What story could generate this completed representation? The tutor should not treat a correct final number as enough evidence. Better evidence is that the invented story preserves every relationship. This matters because topical worksheets often make the route obvious, while mixed PSLE problems require the learner to choose and monitor the route independently.

A practical way to train this stage is to hide the original question and rebuild it. After the attempt, ask the learner what changed in their thinking, what information became useful, and what they would do if the first route failed. The purpose is to make the decision process visible long enough for accurate feedback, then gradually remove the verbal scaffolding as the student becomes more independent.

19. Problem design

The focus at this stage is to control the structure from the inside. For Before-and-After Ratio, the diagnostic question is: Can the learner write a valid new problem for this method? The tutor should not treat a correct final number as enough evidence. Better evidence is that another student can solve it unambiguously. This matters because topical worksheets often make the route obvious, while mixed PSLE problems require the learner to choose and monitor the route independently.

A practical way to train this stage is to author, exchange, solve, and revise a question. After the attempt, ask the learner what changed in their thinking, what information became useful, and what they would do if the first route failed. The purpose is to make the decision process visible long enough for accurate feedback, then gradually remove the verbal scaffolding as the student becomes more independent.

20. Contradiction test

The focus at this stage is to recognise impossible or insufficient data. For Before-and-After Ratio, the diagnostic question is: Do all stated conditions fit one solution? The tutor should not treat a correct final number as enough evidence. Better evidence is that the learner can demonstrate inconsistency instead of forcing an answer. This matters because topical worksheets often make the route obvious, while mixed PSLE problems require the learner to choose and monitor the route independently.

A practical way to train this stage is to alter one condition until the problem breaks. After the attempt, ask the learner what changed in their thinking, what information became useful, and what they would do if the first route failed. The purpose is to make the decision process visible long enough for accurate feedback, then gradually remove the verbal scaffolding as the student becomes more independent.

21. Spaced retrieval

The focus at this stage is to make the method survive time. For Before-and-After Ratio, the diagnostic question is: Can the learner recover the rule after several days? The tutor should not treat a correct final number as enough evidence. Better evidence is that recognition returns without reopening notes. This matters because topical worksheets often make the route obvious, while mixed PSLE problems require the learner to choose and monitor the route independently.

A practical way to train this stage is to schedule one-day, three-day, and seven-day retests. After the attempt, ask the learner what changed in their thinking, what information became useful, and what they would do if the first route failed. The purpose is to make the decision process visible long enough for accurate feedback, then gradually remove the verbal scaffolding as the student becomes more independent.

22. Timed transfer

The focus at this stage is to add pressure only after reliability. For Before-and-After Ratio, the diagnostic question is: Where is time being spent: recognition, setup, calculation, or checking? The tutor should not treat a correct final number as enough evidence. Better evidence is that speed improves without loss of structure. This matters because topical worksheets often make the route obvious, while mixed PSLE problems require the learner to choose and monitor the route independently.

A practical way to train this stage is to time a short mixed set and tag the delay source. After the attempt, ask the learner what changed in their thinking, what information became useful, and what they would do if the first route failed. The purpose is to make the decision process visible long enough for accurate feedback, then gradually remove the verbal scaffolding as the student becomes more independent.

23. Verification

The focus at this stage is to test the result against the original story. For Before-and-After Ratio, the diagnostic question is: Which independent condition can confirm or reject the answer? The tutor should not treat a correct final number as enough evidence. Better evidence is that the answer reproduces the given relationships. This matters because topical worksheets often make the route obvious, while mixed PSLE problems require the learner to choose and monitor the route independently.

A practical way to train this stage is to reconstruct the total, ratio, difference, or state. After the attempt, ask the learner what changed in their thinking, what information became useful, and what they would do if the first route failed. The purpose is to make the decision process visible long enough for accurate feedback, then gradually remove the verbal scaffolding as the student becomes more independent.

24. Return route

The focus at this stage is to reconnect the method to the larger heuristic map. For Before-and-After Ratio, the diagnostic question is: What different method should be compared next? The tutor should not treat a correct final number as enough evidence. Better evidence is that the learner can name one similarity and one decisive difference. This matters because topical worksheets often make the route obvious, while mixed PSLE problems require the learner to choose and monitor the route independently.

A practical way to train this stage is to return to the PSLE heuristic map and choose a neighbour. After the attempt, ask the learner what changed in their thinking, what information became useful, and what they would do if the first route failed. The purpose is to make the decision process visible long enough for accurate feedback, then gradually remove the verbal scaffolding as the student becomes more independent.

Worked example and verification

Problem: Leo:Mia = 7:4. Mia stays unchanged. Leo gives away 15 marbles and the ratio becomes 4:3. Find the original amounts. Align Mia to 12 parts. Before the ratio is 21:12 and after it is 16:12. Leo falls by 5 aligned units, representing 15, so one unit is 3. Leo originally had 63 and Mia 36.

Check: After Leo gives away 15 he has 48; 48:36 simplifies to 4:3. The answer is accepted because it satisfies the original relationships, not merely because the arithmetic produced a plausible number.

Specialist handoff and return

For deeper specialist Mathematics treatment, continue to How Representation Works in PSLE Mathematics Problem Solving, How PSLE Mathematics Works, and How Method Marks and Working Steps Work. Then return to PSLE Mathematics Heuristics and compare this method with another family.

The route receipt is an unseen problem solved without the method name, with a defensible representation, visible working and a verification step. The learner should also identify one problem where Before-and-After Ratio would be a poor choice. That comparison shows that the method has become part of a larger decision system rather than a memorised trick.