PSLE Mathematics Before-and-After Ratios: Use the Unchanged Part

PSLE Learner’s Guide · Vol 0305 · Before-and-after ratios with an invariant

Find the exact decision this guide helps you make

This guide helps a Primary 6 learner solve a before-and-after ratio by identifying the quantity that stays unchanged, equalising its ratio units and then attaching the stated change to the correct part. The short rule is find what stayed fixed, equalise its units, then let the real change determine one unit. You will learn the mechanism, see worked examples, repair common misconceptions, practise in stages and use a compact examination routine.

This is an examination-performance guide, not a replacement for full topic teaching. Use the PSLE Learning Guide for the wider sequence and the Mathematics Learning Hub for deeper subject instruction.

For complete teaching, worked practice and topic progression beyond this performance decision, continue through the Mathematics Learning Hub. This volume keeps a narrower job: it helps the learner recognise, execute and check one examination decision without competing with the wider topic owner.

A careful note about the current examination year

This article is written for the 2026 PSLE preparation context. SEAB’s official 2026 formats page lists English Language as subject 0001, revised Mathematics as 0008 and revised Science as 0009. Format details can change, so confirm any paper-specific timing, component or administrative instruction with the current official SEAB 2026 source and your school. The reasoning routine taught here is deliberately broader: it strengthens how you read evidence, make a decision and check an answer.

Why this mistake happens

A ratio describes relative parts at one moment. When one group gains, loses or transfers items, the ratio units after the change are not automatically the same size as before. The bridge is usually an invariant: one group stayed unchanged, the total stayed unchanged during a transfer, or a difference stayed unchanged when equal amounts were added to both sides. By naming that invariant first and equalising its ratio units, the learner creates comparable before-and-after models instead of subtracting unrelated ratio numbers.

The mind likes a fast label. A familiar word, operation, result or rule triggers an answer before the conditions have been checked. That shortcut sometimes works in easy practice, so it becomes a habit. In a demanding question, however, a nearby distractor preserves the surface pattern while changing one important condition. The learner then performs a correct-looking move on the wrong object. The repair is to slow only the decision point: identify what is being linked or measured, preserve its labels, test the conditions, and then continue at normal speed.

Think of the method as a chain with four locks. Object: what exact idea or quantity is under discussion? Relationship: how are the relevant parts connected? Condition: what must be true for the move to be valid? Claim: what is the strongest conclusion the evidence actually supports? A weak answer often skips from object to claim. A strong answer makes the two middle locks visible.

The one-line operating rule

Find what stayed fixed, equalise its units, then let the real change determine one unit. Say it before the first practice set. During timed work, shorten it to four cue words you can recall without stopping.

The step-by-step method

Step 1: Draw separate before and after ratio statements and name the actual quantities represented

Draw separate before and after ratio statements and name the actual quantities represented. Do this on the page, not only in your head. A brief arrow, label, unit, underline or condition note makes the decision inspectable. If you later find an error, you can repair the exact step instead of restarting the entire answer.

At this stage, do not decorate the answer with extra facts. Ask one diagnostic question: “What evidence would make this step wrong?” If you can name a possible failure, check it now. If the failure is absent, continue. This negative check prevents confidence from replacing proof.

Step 2: Identify the invariant quantity: one part, the total, the difference or another explicitly unchanged amount

Identify the invariant quantity: one part, the total, the difference or another explicitly unchanged amount. Do this on the page, not only in your head. A brief arrow, label, unit, underline or condition note makes the decision inspectable. If you later find an error, you can repair the exact step instead of restarting the entire answer.

Step 3: Equalise the ratio units belonging to that invariant across the two moments

Equalise the ratio units belonging to that invariant across the two moments. Do this on the page, not only in your head. A brief arrow, label, unit, underline or condition note makes the decision inspectable. If you later find an error, you can repair the exact step instead of restarting the entire answer.

Step 4: Compare the scaled models and identify which unit difference represents the stated gain, loss or transfer

Compare the scaled models and identify which unit difference represents the stated gain, loss or transfer. Do this on the page, not only in your head. A brief arrow, label, unit, underline or condition note makes the decision inspectable. If you later find an error, you can repair the exact step instead of restarting the entire answer.

Step 5: Find the value of one common unit and calculate only the quantity requested

Find the value of one common unit and calculate only the quantity requested. Do this on the page, not only in your head. A brief arrow, label, unit, underline or condition note makes the decision inspectable. If you later find an error, you can repair the exact step instead of restarting the entire answer.

Step 6: Replay the change with actual values and verify both ratios and the conserved quantity

Replay the change with actual values and verify both ratios and the conserved quantity. Do this on the page, not only in your head. A brief arrow, label, unit, underline or condition note makes the decision inspectable. If you later find an error, you can repair the exact step instead of restarting the entire answer.

Misconceptions to diagnose and repair

Misconception 1

Tempting move: Subtracting ratio terms before making their unit sizes comparable. This feels efficient because it removes a decision. It is unsafe because the question may have changed the reference, unit, interval, definition or strength of claim.

Repair: return to the line, number, setup or statement that supplies the condition. Mark it with a short label. Then rebuild only the affected link. Keep all earlier work that remains valid. The purpose of checking is targeted correction, not anxious repetition.

Misconception 2

Tempting move: Assuming the total stays constant when items enter or leave the system. This feels efficient because it removes a decision. It is unsafe because the question may have changed the reference, unit, interval, definition or strength of claim.

Misconception 3

Tempting move: Using the changed group as the scaling anchor. This feels efficient because it removes a decision. It is unsafe because the question may have changed the reference, unit, interval, definition or strength of claim.

Misconception 4

Tempting move: Treating a transfer as if only one group changed. This feels efficient because it removes a decision. It is unsafe because the question may have changed the reference, unit, interval, definition or strength of claim.

Misconception 5

Tempting move: Matching a stated change to the difference between totals rather than the correct ratio parts. This feels efficient because it removes a decision. It is unsafe because the question may have changed the reference, unit, interval, definition or strength of claim.

Eight worked cases

Worked case 1: One group gains while the other stays

Scenario. Boys:girls is 3:5. After 8 boys join, the ratio is 5:5. Find the original number of pupils.

The tempting error. Subtracting 5−3 and declaring one ratio unit equals four without checking the girls. The response may contain relevant knowledge, yet it does not answer the precise decision because one label or condition has been lost.

Work the mechanism. Girls are unchanged. Their five units already match in both ratios, so the unit size is common. Boys rise from three to five units; two units equal eight, so one unit is four. Write the decisive evidence beside the step where it is used. This makes the reasoning readable to another person and prevents a later sentence from quietly changing the meaning.

Answer form. Originally there were 3×4 + 5×4 = 32 pupils. Check that adding eight gives 20:20. Notice that the answer carries both the conclusion and the reason that licenses it. It is complete without becoming a list of everything you know.

Check. Re-read the command word, substitute the conclusion into the original situation and ask whether every noun, number and comparison still has the same referent or unit. If the answer is stronger than the evidence, narrow it.

Delayed transfer. Change the after ratio to 4:5 and choose a compatible number joining. Do this variation after a gap, not immediately, so retrieval rather than imitation drives the solution.

Worked case 2: One group leaves

Scenario. Red:blue beads is 7:4. After 18 red beads are removed, the ratio is 4:4.

The tempting error. Scaling the after ratio arbitrarily because both terms equal four. The response may contain relevant knowledge, yet it does not answer the precise decision because one label or condition has been lost.

Work the mechanism. Blue beads are unchanged, so match the blue four units before and after. Red falls from seven to four units; three units equal eighteen. Write the decisive evidence beside the step where it is used. This makes the reasoning readable to another person and prevents a later sentence from quietly changing the meaning.

Answer form. One unit is six; originally there were 42 red and 24 blue beads. Notice that the answer carries both the conclusion and the reason that licenses it. It is complete without becoming a list of everything you know.

Delayed transfer. Ask for the final total rather than the original red count. Do this variation after a gap, not immediately, so retrieval rather than imitation drives the solution.

Worked case 3: Transfer keeps the total

Scenario. Ali:Ben stamps is 5:3. Ali gives 12 stamps to Ben and the ratio becomes 3:5.

The tempting error. Anchoring either person even though both change. The response may contain relevant knowledge, yet it does not answer the precise decision because one label or condition has been lost.

Work the mechanism. A transfer within the pair keeps the total fixed. Both ratios total eight units, so the unit size is already comparable. Ali falls by two units and Ben rises by two; the transferred twelve equals two units. Write the decisive evidence beside the step where it is used. This makes the reasoning readable to another person and prevents a later sentence from quietly changing the meaning.

Answer form. One unit is six; originally Ali had 30 and Ben had 18 stamps. Notice that the answer carries both the conclusion and the reason that licenses it. It is complete without becoming a list of everything you know.

Delayed transfer. Use ratios with different total units and equalise the totals first. Do this variation after a gap, not immediately, so retrieval rather than imitation drives the solution.

Worked case 4: Transfer with unequal total units

Scenario. P:Q is 7:5. P gives some counters to Q and the ratio becomes 4:5. The total stays 72.

The tempting error. Comparing seven directly with four although the before total is twelve units and after total is nine. The response may contain relevant knowledge, yet it does not answer the precise decision because one label or condition has been lost.

Work the mechanism. Scale the before ratio to 21:15 and after ratio to 16:20 so each model totals thirty-six common units. The transfer is five units from P to Q. Since 36 units equal 72, one unit is two. Write the decisive evidence beside the step where it is used. This makes the reasoning readable to another person and prevents a later sentence from quietly changing the meaning.

Answer form. P gives 10 counters. Check original 42:30 and final 32:40 = 4:5. Notice that the answer carries both the conclusion and the reason that licenses it. It is complete without becoming a list of everything you know.

Delayed transfer. Use the known transfer instead of the total and recover the total. Do this variation after a gap, not immediately, so retrieval rather than imitation drives the solution.

Worked case 5: Equal additions preserve difference

Scenario. A:B is 2:5. Ten is added to both, making the ratio 4:7.

The tempting error. Saying the total is unchanged because both receive the same amount. The response may contain relevant knowledge, yet it does not answer the precise decision because one label or condition has been lost.

Work the mechanism. Equal additions change the total but preserve the difference B−A. The difference is three units before and three units after, so units align. A rises from two to four units, and two units equal ten. Write the decisive evidence beside the step where it is used. This makes the reasoning readable to another person and prevents a later sentence from quietly changing the meaning.

Answer form. One unit is five; the original quantities were 10 and 25. Notice that the answer carries both the conclusion and the reason that licenses it. It is complete without becoming a list of everything you know.

Delayed transfer. Try equal removals and check which difference remains fixed. Do this variation after a gap, not immediately, so retrieval rather than imitation drives the solution.

Worked case 6: External addition changes total

Scenario. Cats:dogs is 4:3. After 12 dogs arrive, the ratio is 4:5.

The tempting error. Using total as the invariant. The response may contain relevant knowledge, yet it does not answer the precise decision because one label or condition has been lost.

Work the mechanism. Cats are unchanged, not the total. Match the cat four units. Dogs rise by two units, so two units equal twelve. Write the decisive evidence beside the step where it is used. This makes the reasoning readable to another person and prevents a later sentence from quietly changing the meaning.

Answer form. One unit is six; there were originally 24 cats and 18 dogs. Notice that the answer carries both the conclusion and the reason that licenses it. It is complete without becoming a list of everything you know.

Delayed transfer. Ask how many animals there are after the arrival. Do this variation after a gap, not immediately, so retrieval rather than imitation drives the solution.

Worked case 7: Unknown which group changed

Scenario. A:B changes from 3:4 to 5:6 after 14 items are added to A only.

The tempting error. Treating B’s four and six units as the same size. The response may contain relevant knowledge, yet it does not answer the precise decision because one label or condition has been lost.

Work the mechanism. B is unchanged, so equalise its units using 12: before 9:12, after 10:12. A rises by one common unit, which equals fourteen. Write the decisive evidence beside the step where it is used. This makes the reasoning readable to another person and prevents a later sentence from quietly changing the meaning.

Answer form. Originally A=126 and B=168; after addition A=140 and 140:168 simplifies to 5:6. Notice that the answer carries both the conclusion and the reason that licenses it. It is complete without becoming a list of everything you know.

Delayed transfer. Use a smaller compatible change and ask for the new total. Do this variation after a gap, not immediately, so retrieval rather than imitation drives the solution.

Worked case 8: Rejecting a false invariant

Scenario. A:B is 2:3. Five objects are removed from A and seven from B.

The tempting error. Claiming the difference is unchanged because both groups lost objects. The response may contain relevant knowledge, yet it does not answer the precise decision because one label or condition has been lost.

Work the mechanism. The losses are unequal, so neither total nor difference nor either part is automatically invariant. Two ratios plus the two distinct changes may still be solvable, but a different equation or model is required. Write the decisive evidence beside the step where it is used. This makes the reasoning readable to another person and prevents a later sentence from quietly changing the meaning.

Answer form. Do not force the invariant-ratio method. State what changed and choose an algebraic or unit equation route. Notice that the answer carries both the conclusion and the reason that licenses it. It is complete without becoming a list of everything you know.

Delayed transfer. Change both losses to five and show that the difference then remains fixed. Do this variation after a gap, not immediately, so retrieval rather than imitation drives the solution.

Practice progression: from recognition to transfer

Round 1: mark only

Use six short items. Do not answer them. Mark the object, reference, unit, interval, condition or rule that controls the decision. The goal is accurate noticing without the burden of full writing.

Record the result as one of four codes: N for noticing error, M for method error, I for interpretation error, and C for checking error. Practise the weakest code with a fresh item. Do not repeat an entire worksheet when only one decision needs repair.

Round 2: explain the choice

Solve four items and add one sentence beginning “This is valid because…”. If the sentence merely repeats the answer, the condition is still hidden.

Round 3: repair a wrong answer

Take four plausible wrong answers. Identify the earliest faulty link, preserve the correct work and rewrite only what must change. This develops error diagnosis rather than answer copying.

Round 4: mixed questions

Mix this job with two older skills. Remove headings that reveal the topic. Decide which routine applies before solving. Selection is part of performance.

Round 5: timed set

Use a small timed set with a fixed check budget. Mark a cue at the decision point, finish the answer, then perform one positive check and one rejection check.

Round 6: delayed transfer

Return after two or three days with different surface details. Before seeing notes, write the one-line rule from memory and solve. Compare the new error pattern with the earlier one.

A compact practice log

Question source and date
Decision cue you marked
Your first answer in one line
Earliest wrong link, if any
Corrected rule in twelve words or fewer
Date for delayed transfer
Whether the new item was solved without notes

A useful log is small enough to maintain. It should help you predict the next mistake, not become a scrapbook of completed pages.

Checking that tests meaning, not neatness

A check should have a target. First run a positive test: identify the evidence, calculation or condition that supports your answer. Then run a rejection test: name the closest tempting alternative and the exact reason it fails. This pair is more powerful than rereading the same line while hoping to feel certain.

Use a reverse check where possible. Reconstruct the original amount, replace a connector with its expanded meaning, map a conclusion back to the data, or apply the repaired rule to the apparent exception. Reverse checking exposes lost labels and unsupported leaps because the answer has to return to its source.

Stop when the check has answered a real question. Repeatedly changing a sound answer because it feels unfamiliar is not checking. If the object, relationship, condition and claim all remain consistent, keep the answer and move on.

Examination control

When time is tight, use the cue ORCC: Object, Relationship, Condition, Claim. Write only the smallest useful mark—an arrow, a unit, a time interval, a boundary or a qualifying word. The mark should prevent the known error without slowing the rest of the paper.

Read the command word and identify the decision.
Mark the object or quantity that must keep its identity.
State the relationship or operation in plain language.
Check the condition that makes it valid.
Give the claim at the strength supported by evidence.
Use one positive and one rejection check, then stop.

If you become stuck, do not produce a longer answer immediately. Shorten the task: write what is known, what must be decided and what condition separates the two best options. A precise partial structure often unlocks the next step and also makes any partial credit more defensible.

A seven-day delayed-transfer plan

Day 1: learn the one-line rule and complete two worked cases with notes.
Day 2: solve four short items without notes; classify errors by N, M, I or C.
Day 3: mix two new items with two older PSLE skills and hide the topic labels.
Day 4: explain one case aloud to a parent, tutor or imaginary learner in under two minutes.
Day 5: complete a timed mini-set and use the ORCC marks only at decision points.
Day 6: rest this skill or practise a different subject so memory has to consolidate.
Day 7: attempt one unfamiliar transfer item, then write the smallest rule revision needed.

Success on Day 7 matters more than fluency immediately after reading. Delayed transfer shows whether you can retrieve the method, recognise its job under new surface details and control it without a model beside you.

Guidance for parents and tutors

Ask for the decision, not a performance speech. Useful prompts are: “What exactly are you deciding?”, “Which word, unit or condition controls it?”, “What is the closest wrong option?” and “How can you check without starting again?” These questions keep ownership with the learner while making thinking visible.

When an error occurs, locate its earliest point. Praise accurate diagnosis even before the final answer is corrected. If the learner repeatedly loses the same label, introduce one stable notation: arrows for reference, units beside remainders, interval brackets for rates, or boxes around claim conditions. Avoid adding several new routines at once.

Build variation gradually. Begin with clean examples, then add one plausible distractor, then mix the skill with older content, and finally delay the next attempt. A learner who succeeds only when the worksheet title names the method has learned recognition, not transfer.

Keep emotional temperature low. A surprising exception or remainder is information, not proof that the learner is “bad at” a subject. The practical question is which link changed. Once that link is named, practise a small set that isolates it and then return to mixed questions.

The independent-reader test

Before you call the skill secure, explain one solution to a reader who cannot see your thoughts. Name the starting information, the decision, the condition that controls it and the reason the rejected alternative fails. If the explanation depends on “you know what I mean”, a label is still missing. Add that label to the answer, then remove any sentence that does not help the reader follow the chain.

Next, cover the worked example and rebuild the method from a fresh prompt. Do not copy the surface details. Recover the sequence of decisions. This separates genuine control from familiarity with one page. A learner who can transfer the decision routine to new names, numbers or apparatus has a more dependable examination skill than a learner who can reproduce only the model answer.

Finally, compress the method without losing its safety condition. Write one sentence of no more than fifteen words, then test that sentence against an awkward case. If the short rule gives the wrong result, restore the missing condition. Good compression keeps the mechanism available under time pressure while the awkward case prevents the cue from becoming a careless shortcut.

A timed diagnosis drill

Set aside twelve minutes and choose three unfamiliar questions that could trigger this error. Spend the first minute identifying the decision in each question without solving it. Mark the object, relationship, condition and permitted claim. This preview is not wasted time: it prevents you from entering a long method before you know what the method must preserve.

Solve the three questions in the next eight minutes. After each answer, write one tiny prediction about what a correct answer should preserve: a reference word, a whole-number boundary, a pathway, a unit, a direction or an observable result. Then compare the finished answer with that prediction. If they disagree, repair the earliest link and keep the later work only when its meaning remains valid.

Use the final three minutes to classify the errors. A noticing error means the controlling detail was never marked. A method error means the right information was present but the wrong operation or reasoning chain was used. An interpretation error means the result was calculated or observed correctly but described at the wrong level. A checking error means the answer contained a detectable contradiction that the final audit missed.

Repeat the drill a week later with different surface details and without looking at the old corrections. Improvement is not merely a higher score. Look for a shorter pause at the decision point, fewer unnecessary restarts, more stable labels and a clearer reason for rejecting the nearest wrong answer. Those are signs that the performance routine is becoming portable.

Frequently asked questions

What is an invariant?

It is a quantity that remains unchanged across the before and after situations.

Why equalise ratio units?

Ratio units from different moments may have different values; aligning the invariant makes them comparable.

Does a transfer keep the total fixed?

Yes, if nothing enters or leaves the two-group system.

Do equal additions keep the ratio fixed?

Usually no. They preserve the difference, not the ratio.

Can I use equations instead?

Yes, but the invariant model often shows why the equation is valid and supports checking.

What if no quantity is unchanged?

Use the full change information to build equations rather than inventing an invariant.

What is the best check?

Substitute actual values, perform the stated change and simplify the resulting ratio.

What should I study next?

Use the Mathematics Learning Hub for ratio units, models, transfers and multi-stage comparison problems.

Your next route

Use this guide until you can state the rule, solve an unfamiliar case and explain why the closest alternative fails. Then move to a deeper subject route: the English Learning Hub, Mathematics Learning Hub, Science Learning Hub, or the full learning-hubs directory. Return to the PSLE Learning Guide when you want to place this skill inside a broader revision sequence.

Continue through this four-guide sequence

This batch begins by rebuilding direct questions as embedded clauses. It then preserves an unchanged quantity across ratio change, separates observation from explanation in Science, and finishes with a cross-subject pre-mortem that predicts the most likely failure before submission.

How to Perform in PSLE | Learner’s Guide Vol 0305 | Mathematics: Anchor a Before-and-After Ratio to the Quantity That Did Not Change