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Before and After Ratio Problems | The PSLE Mathematics State-Change Method

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.

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Choose the question that brought you here. Open one useful guide, try a small task, and stop when you have what you need.

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The same learning habit can travel across subjects, while each subject keeps its own methods. These routes help you notice a difficulty, understand one part of it, and return to something you can do.

A word is familiar, but using it is difficult.

Move from recognising a word to retrieving it in a new context. Understand vocabulary plateaus.

Try it without the guide: Choose one word you already know. Close the guide and use it in a new sentence. Explain why it fits; try another context tomorrow.

A piece of writing has ideas, but the reader loses the thread.

Make the order of events and the links between sentences clear. Explore composition writing.

Try it without the guide: Choose one short paragraph. Read the relevant explanation, close it, and revise the paragraph. Ask someone to tell you what happened and why.

The Mathematics seems familiar, but marks still disappear.

Find the first point where the working stops being reliable. Find Secondary 4 A-Math mark leakage.

Try it without the guide: For a Secondary 4 A-Math question you have attempted, locate the first uncertain line. Repair that step, then try a comparable question without the worked answer.

A Science fact is remembered, but the explanation is incomplete.

Connect the evidence to a scientific idea and the resulting change. Follow the Primary Science learning route.

Try it without the guide: Choose a familiar Primary Science example. Explain the evidence, the idea and the result without notes. Then change one condition and explain your prediction.

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Check the question, source, date and evidence before combining claims. Explore the World Knowledge research library.

Try it without the guide: Take one claim. Find the source best placed to support it, note its date, and state what remains uncertain. Return to your original question.

There is plenty of help, but independence is hard to see.

Check what the learner can understand and do after support is removed. Understand how education works.

Try it without the guide: Choose one small task the child has practised. Agree on a calm, brief attempt without prompts. Use what happens to choose one next step, then stop.

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