PSLE Mathematics Linked Ratios: Equalise the Shared Term

PSLE Learner’s Guide · Vol 0165 · Linked ratios and shared terms

Find the exact decision this guide helps you make

This guide helps a Primary 6 learner combine two linked ratios by identifying the shared quantity, equalising its units, and preserving each quantity’s meaning and time state. The short rule is name the shared quantity, match its ratio units, then read the combined ratio. 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.

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

In A:B=2:3 and B:C=4:5, the two appearances of B refer to the same quantity but not the same number of ratio units. Writing 2:3:5 quietly treats 3 units of B as equal to 4 units of B. The repair is to scale both ratios until the shared term matches. This works only when the shared label also refers to the same group and same moment.

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

Name the shared quantity, match its ratio units, then read the combined ratio. 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 both ratios with full quantity labels and the stated order

Write both ratios with full quantity labels and the stated 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: Circle the label that appears in both ratios

Circle the label that appears in both ratios. 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: Check that the shared label refers to the same group, unit and time state

Check that the shared label refers to the same group, unit and time state. 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: Find a common number of ratio units for the shared term

Find a common number of ratio units for the shared 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 5: Scale every term in each ratio by the same factor used on that ratio

Scale every term in each ratio by the same factor used on that ratio. 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: Join the ratios, then test totals or actual values against the original relationships

Join the ratios, then test totals or actual values against the original relationships. 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: Joining ratios without matching the shared term. 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: Scaling only the shared number and not the whole ratio. 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: Reversing one ratio to make labels appear aligned without tracking 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 4

Tempting move: Combining a before-transfer B with an after-transfer B. 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: Adding ratio numbers as though they were actual quantities. 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: Basic linked ratio

Scenario. A:B=2:3 and B:C=4:5.

The tempting error. Writing A:B:C=2:3:5. 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 shared B must match. LCM of 3 and 4 is 12. Scale 2:3 by 4 to 8:12 and 4:5 by 3 to 12:15. 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. A:B:C=8:12:15. Both original ratios can be recovered. 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. Use B=24 and find actual A and C. Do this variation after a gap, not immediately, so retrieval rather than imitation drives the solution.

Worked case 2: Shared term already equal

Scenario. P:Q=5:7 and Q:R=7:9.

The tempting error. Scaling again just because there are two ratios. 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. Both ratios already represent Q with 7 units, so they can join directly. 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:Q:R=5:7:9. 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. Multiply the first ratio by 2 and show why the combined ratio must then be adjusted. Do this variation after a gap, not immediately, so retrieval rather than imitation drives the solution.

Worked case 3: Reversed order

Scenario. M:N=3:8 and P:N=5:4.

The tempting error. Appending 5 because N appears second in both statements. 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. Reorder the second relationship as N:P=4:5. Match N values 8 and 4 by scaling the second ratio to 8:10. 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. M:N:P=3:8:10. 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. Recover P:N from the combined ratio to verify. Do this variation after a gap, not immediately, so retrieval rather than imitation drives the solution.

Worked case 4: Finding actual values

Scenario. Red:Blue=3:5 and Blue:Green=10:7. Total is 92.

The tempting error. Adding 3+5+10+7 as if all units share one scale. 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. Match Blue: first ratio becomes 6:10, so Red:Blue:Green=6:10:7, with 23 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 4; quantities are 24, 40 and 28. 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 both original ratios using actual values. Do this variation after a gap, not immediately, so retrieval rather than imitation drives the solution.

Worked case 5: Different units

Scenario. Length A:B=2:3 in metres, while a second ratio compares B’s numerical centimetre value with C.

The tempting error. Combining because the label B repeats. 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 shared quantity must use compatible units before ratio units are matched. Convert or clarify the definition first. 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 equalise abstract ratio numbers across inconsistent measurement units. 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. Rewrite both relationships in centimetres. Do this variation after a gap, not immediately, so retrieval rather than imitation drives the solution.

Worked case 6: Different time states

Scenario. Before transfer A:B=2:3; after transfer B:C=4:5.

The tempting error. Combining the two B terms automatically. 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 before and B after may be different quantities. The shared name is not enough; time state must match. 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. Use separate labels such as B_before and B_after and connect them through the transfer. 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. Create a no-change condition under which combination becomes legal. Do this variation after a gap, not immediately, so retrieval rather than imitation drives the solution.

Worked case 7: Ratio with a fraction

Scenario. A:B=1:1.5 and B:C=3:4.

The tempting error. Avoiding the problem by rounding 1.5. 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. Clear the fractional ratio first: A:B=2:3. The shared B already has 3 units in B:C=3:4. 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. A:B:C=2:3:4. 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 actual B=12 to confirm both relationships. Do this variation after a gap, not immediately, so retrieval rather than imitation drives the solution.

Worked case 8: Smallest whole form

Scenario. A:B:C is found as 12:18:24.

The tempting error. Keeping a correct but unreduced ratio when simplest form is requested. 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. All terms share factor 6, so reduce together without changing relationships. 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. The simplest whole-number ratio is 2:3:4. 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 dividing only A and C would break the ratio. 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 the shared term?

It is the quantity label that appears in both ratios.

Why must it be equalised?

The same actual quantity cannot represent different numbers of combined ratio units.

How do I equalise it?

Use a common multiple and scale every term in each ratio consistently.

Can I reverse a ratio?

Yes, if you reverse both labels and numbers together.

Can before and after quantities be shared?

Not automatically; distinguish their time states first.

What if units differ?

Convert to compatible units before combining.

What is the best final check?

Recover each original two-part ratio from the combined ratio.

What should I study next?

Practise ratios, models and multi-step word problems in the Mathematics Learning Hub.

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

The sequence is designed to move from reference control in English, through contextual interpretation in Mathematics and fair rate comparison in Science, to a cross-subject test for exceptions. Each guide owns a different examination decision. Use the subject hubs for full topic teaching and these learner guides for performance control.

How to Perform in PSLE | Learner’s Guide Vol 0165 | Mathematics: Equalise the Shared Term Before Combining Two Ratios