PSLE Mathematics Ratio Difference Units: Find One Unit

PSLE Learner’s Guide · Vol 0382 · Ratio language, unit values and differences

Find the exact decision this guide helps you make

This guide helps a Primary 6 learner solve a ratio problem when the given quantity is a difference by matching it to the gap between the two ratio terms rather than the total ratio units. The short rule is if the real quantity is a difference, match it to the difference between ratio terms; if it is a total, match it to their sum. 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. The Primary 5 ratios and problem-representation guide gives a focused canonical route for the underlying topic. This volume keeps a narrower job: it helps the learner recognise, execute and check one examination decision without competing with the wider owner. Use the complete PSLE Learner’s Guide volume directory as the permanent series route. For a nearby published performance routine, continue to Vol 0350 on using total ratio units when a whole is given.

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 such as 7:4 contains eleven total units but only three difference units. When the question says one group has 24 more, the 24 matches the three-unit gap, not all eleven units. Learners who divide by the total are using the method for a given whole on a given difference. The repair is to label what the real number represents before dividing.

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

If the real quantity is a difference, match it to the difference between ratio terms; if it is a total, match it to their sum. 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: Write the ratio with named quantities in the correct order

Write the ratio with named quantities in the correct order. 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 whether the given number is a total, one part, or a difference between parts

Identify whether the given number is a total, one part, or a difference between parts. 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: For a difference, subtract the smaller ratio term from the larger term

For a difference, subtract the smaller ratio term from the larger term. 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: Match the stated real difference to that ratio-unit gap and find the value of one unit

Match the stated real difference to that ratio-unit gap and find the value of one unit. 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: Scale each ratio term by the same unit value

Scale each ratio term by the same unit value. 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: Check both the ratio and the stated difference in the reconstructed amounts

Check both the ratio and the stated difference in the reconstructed amounts. 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: Dividing a given difference by the sum of ratio units. 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: Subtracting the real quantities before confirming the ratio order. 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: Treating “more than” as a request to add the ratio terms. 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: Scaling one part but not the other by the same unit value. 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: Checking only the total and forgetting the original difference. 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: Direct difference

Scenario. The ratio of red to blue counters is 7:4. There are 24 more red counters.

The tempting error. Dividing 24 by 11 total units. 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 phrase more red describes the gap. Seven units minus four units equals three units, so three units represent 24. 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 8; red is 56 and blue is 32. 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. Check 56:32 simplifies to 7:4 and the difference is 24. Do this variation after a gap, not immediately, so retrieval rather than imitation drives the solution.

Worked case 2: Smaller quantity named

Scenario. Boys:girls is 5:8, and there are 18 fewer boys.

The tempting error. Subtracting in the wrong direction and getting a negative unit. 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 exceed boys by three ratio units. The magnitude of the difference is 18, so three units equal 18. 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 6; boys 30, girls 48. 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. State the difference as girls minus boys. Do this variation after a gap, not immediately, so retrieval rather than imitation drives the solution.

Worked case 3: Given total instead

Scenario. Cats:dogs is 3:2 and there are 40 animals altogether.

The tempting error. Using the one-unit gap because the previous question involved a ratio. 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. Altogether signals a whole, so 3+2=5 units represent 40. 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 8; cats 24 and dogs 16. 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. Contrast the answer if the difference were 40. Do this variation after a gap, not immediately, so retrieval rather than imitation drives the solution.

Worked case 4: Unknown larger amount

Scenario. Ali:Bala is 9:5. Ali has 28 more stickers.

The tempting error. Adding 28 to nine ratio units. 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 real difference matches 9−5=4 units. Four units equal 28. 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 7; Ali has 63 and Bala 35. 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. Rebuild 63−35 and simplify the ratio. Do this variation after a gap, not immediately, so retrieval rather than imitation drives the solution.

Worked case 5: Three-part ratio

Scenario. P:Q:R is 6:4:3. P has 21 more than R.

The tempting error. Using all thirteen units. 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 comparison is only P against R. Their unit gap is 6−3=3, so three units represent 21. 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 7; P=42, Q=28, R=21. 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. Check that Q was scaled even though it was not in the difference statement. Do this variation after a gap, not immediately, so retrieval rather than imitation drives the solution.

Worked case 6: Difference after a change

Scenario. After equal amounts are added, A:B becomes 8:5 and A has 30 more than B.

The tempting error. Assuming the equal addition changed their difference. 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. Adding the same amount to both preserves the difference. In the final ratio, three units equal 30. 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. Final amounts are 80 and 50. 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. Explain why the earlier difference was also 30, while the earlier ratio need not be 8:5. Do this variation after a gap, not immediately, so retrieval rather than imitation drives the solution.

Worked case 7: Amount fewer as fraction

Scenario. C:D is 2:5 and C is 36 fewer than D.

The tempting error. Treating two units as the difference because C has two units. 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 gap is 5−2=3 units, and that gap represents 36. 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 12; C=24 and D=60. 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. Check that 24 is 36 fewer than 60. Do this variation after a gap, not immediately, so retrieval rather than imitation drives the solution.

Worked case 8: Ratio reversed in wording

Scenario. The number of pears is 15 fewer than apples. Pears:apples is 4:7.

The tempting error. Reading apples:pears because apples is named first in the sentence. 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. Keep the written ratio order. The gap is 7−4=3 units, matching 15. 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 5; pears 20 and apples 35. 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. Label the ratio before using the sentence. 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.

Graded independent practice with explained answers

Practice item 1: Direct gap

Prompt. A:B=9:4 and A is 35 more. Find both.

Answer. A=63, B=28

Why. Five ratio units represent 35, so one unit is 7.

Check or extension. Check ratio and difference.

Practice item 2: Fewer wording

Prompt. M:N=3:7 and M is 20 fewer.

Answer. M=15, N=35

Why. The four-unit gap represents 20.

Check or extension. Write N−M=20.

Practice item 3: Total contrast

Prompt. P:Q=5:3 and total is 64.

Answer. P=40, Q=24

Why. Eight total units represent 64.

Check or extension. Do not use the two-unit gap.

Practice item 4: Three parts

Prompt. X:Y:Z=8:5:2 and X is 24 more than Z.

Answer. X=32, Y=20, Z=8

Why. Six difference units represent 24, so one unit is 4.

Check or extension. Scale every term.

Practice item 5: Reverse construction

Prompt. Create a 7:3 problem whose difference is 28.

Answer. Amounts 49 and 21 are one valid pair.

Why. Four units equal 28, so one unit is 7.

Check or extension. Verify 49−21=28.

Practice item 6: Delayed transfer

Prompt. After two days, explain why a 6:5 ratio with a difference of 12 gives one unit 12.

Answer. The ratio terms differ by one unit, so that one unit is the whole stated difference.

Why. The sum is irrelevant because the given is not a total.

Check or extension. Rebuild both amounts.

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

When do I subtract ratio terms?

When the given quantity describes how much more or less one part is than another.

When do I add ratio terms?

When the given quantity is the combined total of those parts.

Can a difference be negative?

Use the positive magnitude for “how many more/fewer”, while keeping the direction in the sentence.

What if there are three ratio terms?

Subtract only the two terms named in the comparison, then scale all terms.

Does adding the same amount to both groups preserve the ratio?

Usually no, but it does preserve their difference.

What is the fastest check?

Rebuild the two amounts, subtract them and simplify their ratio.

Why label the ratio order?

The wording may name quantities in a different order and invite reversal.

What should I study next?

Use the Mathematics Learning Hub and ratio guide for full model and transfer work.

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.

The Mathematics guide continues the boundary theme: a stated difference belongs to the gap in ratio units, not automatically to the whole.

How to Perform in PSLE | Learner’s Guide Vol 0382 | Mathematics: Match the Given Difference to the Difference in Ratio Units