The problem this guide solves
A ratio compares multiplicatively. Learners often remember that both terms must be treated alike, then wrongly assume that adding the same number to both terms will preserve the ratio. Equal treatment is not enough: the operation must preserve the multiplicative relationship. For a:b, multiplying or dividing both non-zero terms by the same non-zero factor preserves the quotient a/b. Adding the same non-zero amount k gives (a+k):(b+k), which usually changes the quotient when a and b are unequal. Cross-products provide a decisive check: a:b and c:d are equivalent exactly when a×d=b×c, with meaningful non-zero comparison terms.
This learner-first guide is part of the complete PSLE Learner’s Guide volume directory. The historical foundation remains available through the original PSLE Learning Guide, while current subject routes continue through the Mathematics hub. When you need official paper and assessment information, consult the SEAB PSLE formats examined in 2026 rather than relying on memory or hearsay.
The governing mechanism
For a:b, multiplying or dividing both non-zero terms by the same non-zero factor preserves the quotient a/b. Adding the same non-zero amount k gives (a+k):(b+k), which usually changes the quotient when a and b are unequal. Cross-products provide a decisive check: a:b and c:d are equivalent exactly when a×d=b×c, with meaningful non-zero comparison terms. The important habit is to explain why a choice follows from the structure of the problem. A remembered slogan is useful only when it points to a relationship that can be tested.
A four-step decision routine
- Step 1. Write the ratio in a fixed order and attach the quantities to the terms.
- Step 2. Ask whether the change is a common multiplication or division. If so, identify one scale factor that maps every term.
- Step 3. If the change is an addition or subtraction, do not call it equivalent merely because both terms changed equally.
- Step 4. Verify uncertain claims with simplified form, equal fractions or cross-products, then interpret the result in the story.
Use the routine slowly during learning, then compress it as fluency develops. The aim is not to perform a ritual on every easy item. The aim is to have a dependable recovery path when two answers seem plausible.
Worked examples: see the mechanism under pressure
Worked example 1
Case. 2:3 becomes 6:9. Reasoning. Both terms are multiplied by 3, so the comparison remains two parts for every three parts. The decisive evidence is the governing relationship, not how familiar the wording looks. Common trap. Seeing larger numbers and assuming the ratio must be larger. Repair. Return to the four-step routine, name the relevant quantity or grammatical/scientific relationship, and verify the final statement in context.
Worked example 2
Case. 2:3 becomes 5:6. Reasoning. Adding 3 to each term produces 5:6, but 2×6=12 while 3×5=15; the ratios are not equivalent. The decisive evidence is the governing relationship, not how familiar the wording looks. Common trap. Treating equal addition as if it were a scale factor. Repair. Return to the four-step routine, name the relevant quantity or grammatical/scientific relationship, and verify the final statement in context.
Worked example 3
Case. 4:10 simplifies to 2:5. Reasoning. Both terms are divided by 2. The common division preserves the quotient. The decisive evidence is the governing relationship, not how familiar the wording looks. Common trap. Dividing only the larger term until the numbers look small. Repair. Return to the four-step routine, name the relevant quantity or grammatical/scientific relationship, and verify the final statement in context.
Worked example 4
Case. 7:7 becomes 10:10. Reasoning. Both ratios simplify to 1:1. Equal addition happens to preserve equality because the original terms are equal. The decisive evidence is the governing relationship, not how familiar the wording looks. Common trap. Turning this special case into a general rule for unequal ratios. Repair. Return to the four-step routine, name the relevant quantity or grammatical/scientific relationship, and verify the final statement in context.
Worked example 5
Case. 3:5 becomes 12:20. Reasoning. One common factor, 4, maps both terms. The fractions 3/5 and 12/20 are equal. The decisive evidence is the governing relationship, not how familiar the wording looks. Common trap. Finding two different factors that happen to reach the targets. Repair. Return to the four-step routine, name the relevant quantity or grammatical/scientific relationship, and verify the final statement in context.
Worked example 6
Case. 3:5 becomes 6:8. Reasoning. The first term doubled but the second did not; cross-products give 24 and 30. The decisive evidence is the governing relationship, not how familiar the wording looks. Common trap. Noticing that both terms increased and overlooking different relative changes. Repair. Return to the four-step routine, name the relevant quantity or grammatical/scientific relationship, and verify the final statement in context.
Worked example 7
Case. A red-to-blue ratio of 2:3 receives 4 red and 6 blue counters. Reasoning. The added batch itself has ratio 2:3, so the combined totals keep 2:3. This is not equal addition; it is adding a proportional batch. The decisive evidence is the governing relationship, not how familiar the wording looks. Common trap. Saying any addition changes a ratio. Repair. Return to the four-step routine, name the relevant quantity or grammatical/scientific relationship, and verify the final statement in context.
Worked example 8
Case. A 2:3 mixture receives 5 units of each ingredient. Reasoning. Equal addition changes the totals to 7:8, so the original unequal ratio is not preserved. The decisive evidence is the governing relationship, not how familiar the wording looks. Common trap. Confusing equal amounts with proportional amounts. Repair. Return to the four-step routine, name the relevant quantity or grammatical/scientific relationship, and verify the final statement in context.
Worked example 9
Case. 15:25 and 3:5. Reasoning. Dividing both original terms by 5 gives the second ratio. Cross-products also give 75 on both sides. The decisive evidence is the governing relationship, not how familiar the wording looks. Common trap. Comparing term differences, 10 and 2, instead of multiplicative structure. Repair. Return to the four-step routine, name the relevant quantity or grammatical/scientific relationship, and verify the final statement in context.
Worked example 10
Case. 8:12 and 10:15. Reasoning. Both simplify to 2:3. There need not be an integer factor directly from 8 to 10 for the ratios to be equivalent. The decisive evidence is the governing relationship, not how familiar the wording looks. Common trap. Rejecting equivalent ratios because the direct scale factor is fractional. Repair. Return to the four-step routine, name the relevant quantity or grammatical/scientific relationship, and verify the final statement in context.
Worked example 11
Case. 6 boys to 9 girls becomes 10 boys to 15 girls. Reasoning. Both ratios simplify to 2:3; the group has grown by a proportional 4:6 batch. The decisive evidence is the governing relationship, not how familiar the wording looks. Common trap. Looking only at the unequal additions and missing the proportional final comparison. Repair. Return to the four-step routine, name the relevant quantity or grammatical/scientific relationship, and verify the final statement in context.
Worked example 12
Case. 6:9 becomes 9:12. Reasoning. Adding 3 to both changes 2:3 into 3:4. Cross-products 72 and 81 are unequal. The decisive evidence is the governing relationship, not how familiar the wording looks. Common trap. Using a constant difference of 3 as evidence of equivalence. Repair. Return to the four-step routine, name the relevant quantity or grammatical/scientific relationship, and verify the final statement in context.
Worked example 13
Case. A recipe for 4 people uses 6 cups of rice; for 10 people it uses 15 cups. Reasoning. People and rice are each multiplied by 2.5, so the ratio is preserved. The decisive evidence is the governing relationship, not how familiar the wording looks. Common trap. Believing scale factors must be whole numbers. Repair. Return to the four-step routine, name the relevant quantity or grammatical/scientific relationship, and verify the final statement in context.
Worked example 14
Case. A map scale of 1:50 000 is enlarged by doubling only the printed distance. Reasoning. The representation changes, so the old written scale cannot simply remain unchanged. Ratios require both quantities and units to be interpreted. The decisive evidence is the governing relationship, not how familiar the wording looks. Common trap. Applying the common-factor rule to a physical change without updating the compared quantities. Repair. Return to the four-step routine, name the relevant quantity or grammatical/scientific relationship, and verify the final statement in context.
Worked example 15
Case. 0:5 and 0:10. Reasoning. Both show zero of the first quantity, and multiplying by 2 maps the terms. The comparison is meaningful in context, though division by zero must still be avoided. The decisive evidence is the governing relationship, not how familiar the wording looks. Common trap. Using a ratio with a zero term as permission to divide by zero. Repair. Return to the four-step routine, name the relevant quantity or grammatical/scientific relationship, and verify the final statement in context.
Worked example 16
Case. 5:0 and 10:0. Reasoning. A common factor maps the ordered pairs, but the fraction 5/0 is undefined. Use the pair interpretation carefully and never claim a numerical quotient. The decisive evidence is the governing relationship, not how familiar the wording looks. Common trap. Forcing every ratio check through division by zero. Repair. Return to the four-step routine, name the relevant quantity or grammatical/scientific relationship, and verify the final statement in context.
Worked example 17
Case. 18:24 becomes 3:4. Reasoning. Divide by the greatest common factor 6. Using the greatest factor is efficient but not required for correctness. The decisive evidence is the governing relationship, not how familiar the wording looks. Common trap. Thinking only one simplification step is allowed. Repair. Return to the four-step routine, name the relevant quantity or grammatical/scientific relationship, and verify the final statement in context.
Worked example 18
Case. 18:24 becomes 9:12 then 3:4. Reasoning. Each step divides both terms by the same factor, first 2 then 3. The chain remains equivalent. The decisive evidence is the governing relationship, not how familiar the wording looks. Common trap. Assuming multi-step simplification changes the ratio. Repair. Return to the four-step routine, name the relevant quantity or grammatical/scientific relationship, and verify the final statement in context.
Worked example 19
Case. A class ratio 12:18 gains 2 boys and 3 girls. Reasoning. The added group has ratio 2:3, matching the simplified original 2:3, so the new ratio 14:21 remains equivalent. The decisive evidence is the governing relationship, not how familiar the wording looks. Common trap. Thinking additions must use the original large terms 12 and 18. Repair. Return to the four-step routine, name the relevant quantity or grammatical/scientific relationship, and verify the final statement in context.
Worked example 20
Case. A class ratio 12:18 gains 2 boys and 2 girls. Reasoning. The new ratio 14:20 simplifies to 7:10, not 2:3. The decisive evidence is the governing relationship, not how familiar the wording looks. Common trap. Calling the change fair because the same number joined each group. Repair. Return to the four-step routine, name the relevant quantity or grammatical/scientific relationship, and verify the final statement in context.
Worked example 21
Case. Which is larger, 3:4 or 4:5? Reasoning. Compare 3/4 and 4/5: cross-products are 15 and 16, so 4:5 is larger as a first-to-second quotient. The decisive evidence is the governing relationship, not how familiar the wording looks. Common trap. Subtracting terms and declaring a tie because both differences are 1. Repair. Return to the four-step routine, name the relevant quantity or grammatical/scientific relationship, and verify the final statement in context.
Worked example 22
Case. A ratio 5:8 has first term 20 in an equivalent ratio. Reasoning. The scale factor is 4, so the second term must be 32. The decisive evidence is the governing relationship, not how familiar the wording looks. Common trap. Adding 15 to the second term as well and obtaining 23. Repair. Return to the four-step routine, name the relevant quantity or grammatical/scientific relationship, and verify the final statement in context.
Worked example 23
Case. A ratio 21:28 has second term 12 in an equivalent ratio. Reasoning. Since 28 is multiplied by 3/7 to get 12, multiply 21 by 3/7 to get 9. The decisive evidence is the governing relationship, not how familiar the wording looks. Common trap. Avoiding fractional scale factors and guessing from differences. Repair. Return to the four-step routine, name the relevant quantity or grammatical/scientific relationship, and verify the final statement in context.
Worked example 24
Case. Two ratios have equal cross-products. Reasoning. For non-zero comparison terms, equal cross-products establish equal fractions and hence equivalent ordered ratios. The decisive evidence is the governing relationship, not how familiar the wording looks. Common trap. Computing cross-products but forgetting to preserve order. Repair. Return to the four-step routine, name the relevant quantity or grammatical/scientific relationship, and verify the final statement in context.
Worked example 25
Case. 3 cm:2 m. Reasoning. Convert to one unit first: 3 cm:200 cm. Only then simplify or compare. The decisive evidence is the governing relationship, not how familiar the wording looks. Common trap. Cancelling numbers before making units common. Repair. Return to the four-step routine, name the relevant quantity or grammatical/scientific relationship, and verify the final statement in context.
Worked example 26
Case. $4:$6 and 40 cents:60 cents. Reasoning. Each pair simplifies to 2:3 once its own units are common; the ratios are equivalent. The decisive evidence is the governing relationship, not how familiar the wording looks. Common trap. Comparing dollar numerals directly with cent numerals without conversion. Repair. Return to the four-step routine, name the relevant quantity or grammatical/scientific relationship, and verify the final statement in context.
Worked example 27
Case. A 1:2 paint mixture is doubled. Reasoning. Use 2:4, 3:6 or any common multiple depending on total amount. Doubling total mixture multiplies both components. The decisive evidence is the governing relationship, not how familiar the wording looks. Common trap. Adding 2 to each term and calling 3:4 a doubled mixture. Repair. Return to the four-step routine, name the relevant quantity or grammatical/scientific relationship, and verify the final statement in context.
Worked example 28
Case. The ratio of wins to games is 3:5; two more wins occur in two more games. Reasoning. The new ratio is 5:7, not equivalent to 3:5. Events added to both counts do not necessarily match the original rate. The decisive evidence is the governing relationship, not how familiar the wording looks. Common trap. Believing identical additions preserve every rate. Repair. Return to the four-step routine, name the relevant quantity or grammatical/scientific relationship, and verify the final statement in context.
Misconceptions and targeted repairs
Misconceptions persist because a wrong shortcut sometimes succeeds on easy examples. Repair requires a contrast pair: one case in which the shortcut appears to work and one nearly identical case in which it fails. Explain the difference in a complete sentence. Then solve a delayed example without seeing the explanation.
- Do not decide from appearance alone. Mark the word, quantity, variable, heading or relationship that controls the answer.
- Do not confuse an observation with an explanation. State what is given or seen before stating why it happens.
- Do not let one familiar example become a universal rule. Test a boundary case and an exception.
- Do not stop at a number or option letter. Restore the unit, grammatical role, scientific process or contextual meaning.
- Do not treat checking as repeating the same thought. Use a different representation, removal test, control, conversion or counterexample.
Independent practice with explained answers
Practice 1
Task. Is 4:7 equivalent to 12:21? Answer. Yes. Explanation. Both terms are multiplied by 3. Before reading this explanation, identify the single piece of information that controls the decision. After reading it, close the guide and restate the reasoning in your own words.
Practice 2
Task. Is 4:7 equivalent to 7:10? Answer. No. Explanation. Adding 3 to both terms changes the ratio; cross-products are 40 and 49. Before reading this explanation, identify the single piece of information that controls the decision. After reading it, close the guide and restate the reasoning in your own words.
Practice 3
Task. Complete 5:8 = 20:___. Answer. 32 Explanation. The common scale factor is 4. Before reading this explanation, identify the single piece of information that controls the decision. After reading it, close the guide and restate the reasoning in your own words.
Practice 4
Task. Simplify 24:36. Answer. 2:3 Explanation. Divide both terms by 12. Before reading this explanation, identify the single piece of information that controls the decision. After reading it, close the guide and restate the reasoning in your own words.
Practice 5
Task. Which is equivalent to 6:10: 9:15 or 9:13? Answer. 9:15 Explanation. Both 6:10 and 9:15 simplify to 3:5. Before reading this explanation, identify the single piece of information that controls the decision. After reading it, close the guide and restate the reasoning in your own words.
Practice 6
Task. A ratio 2:5 receives 6 and 15 more respectively. Is it preserved? Answer. Yes. Explanation. The added batch is three copies of 2:5, so totals 8:20 remain 2:5. Before reading this explanation, identify the single piece of information that controls the decision. After reading it, close the guide and restate the reasoning in your own words.
Practice 7
Task. A ratio 2:5 receives 6 more of each. Is it preserved? Answer. No. Explanation. The totals 8:11 are not equivalent to 2:5. Before reading this explanation, identify the single piece of information that controls the decision. After reading it, close the guide and restate the reasoning in your own words.
Practice 8
Task. Find x: 7:9 = x:36. Answer. 28 Explanation. Multiply both terms by 4. Before reading this explanation, identify the single piece of information that controls the decision. After reading it, close the guide and restate the reasoning in your own words.
Practice 9
Task. Find x: 18:30 = 3:x. Answer. 5 Explanation. Divide both original terms by 6. Before reading this explanation, identify the single piece of information that controls the decision. After reading it, close the guide and restate the reasoning in your own words.
Practice 10
Task. Are 8:12 and 14:21 equivalent? Answer. Yes. Explanation. Both simplify to 2:3. Before reading this explanation, identify the single piece of information that controls the decision. After reading it, close the guide and restate the reasoning in your own words.
Practice 11
Task. Compare 5:6 and 6:7 as fractions. Answer. 6:7 is larger. Explanation. Cross-products are 35 and 36. Before reading this explanation, identify the single piece of information that controls the decision. After reading it, close the guide and restate the reasoning in your own words.
Practice 12
Task. A 3:4 mix has 21 units of the first part. Find the second. Answer. 28 units. Explanation. The scale factor from 3 to 21 is 7. Before reading this explanation, identify the single piece of information that controls the decision. After reading it, close the guide and restate the reasoning in your own words.
Practice 13
Task. Convert and simplify 25 cm:1 m. Answer. 1:4 Explanation. One metre is 100 centimetres, so 25:100 simplifies to 1:4. Before reading this explanation, identify the single piece of information that controls the decision. After reading it, close the guide and restate the reasoning in your own words.
Practice 14
Task. True or false: if both terms rise by 5, the ratio is unchanged. Answer. False in general. Explanation. Common addition does not preserve an unequal ratio. Before reading this explanation, identify the single piece of information that controls the decision. After reading it, close the guide and restate the reasoning in your own words.
Practice 15
Task. Explain the exception 9:9 to 14:14. Answer. Both are 1:1. Explanation. Equal terms remain equal after the same addition, but this special case does not justify the rule for unequal terms. Before reading this explanation, identify the single piece of information that controls the decision. After reading it, close the guide and restate the reasoning in your own words.
Practice 16
Task. A class 10:15 gains 4:6 pupils. New ratio? Answer. 14:21 = 2:3. Explanation. The added group matches the original simplified ratio. Before reading this explanation, identify the single piece of information that controls the decision. After reading it, close the guide and restate the reasoning in your own words.
Practice 17
Task. A class 10:15 gains 4:4 pupils. New ratio? Answer. 14:19. Explanation. Equal additions change the unequal comparison. Before reading this explanation, identify the single piece of information that controls the decision. After reading it, close the guide and restate the reasoning in your own words.
Practice 18
Task. Check 16:28 and 20:35 by cross-products. Answer. Equivalent. Explanation. 16×35 and 28×20 both equal 560. Before reading this explanation, identify the single piece of information that controls the decision. After reading it, close the guide and restate the reasoning in your own words.
Delayed transfer: prove that the learning can travel
Do not finish by repeating the same surface form. Wait at least a day, then solve three mixed items: one that looks familiar, one that hides the same mechanism in a different representation, and one that contains irrelevant information. For each response, write a one-line justification. A correct answer with no transferable reason is fragile; an explained answer can be checked, repaired and reused.
On the following day, retrieve the four-step routine from memory before opening notes. Create one correct example and one tempting wrong example. Exchange them with a study partner or ask a parent to read only the question. If the intended distinction is not visible, improve the item. Designing a clean contrast forces you to understand the mechanism more deeply than copying another completed answer.
Examination control
- First pass: answer direct items efficiently, but mark any item in which two choices survive.
- Second pass: apply the governing mechanism to the marked items and write a compact check beside the working.
- Final pass: verify order, scope, units, labels, pronoun case, variables and requested form rather than merely rereading the same words.
- If time is short, preserve evidence. A short, precise chain of reasoning is more useful than a long vague paragraph.
- Never import a rule from another question without checking that the same conditions hold.
Accuracy under time pressure comes from reducing avoidable decisions. Use consistent notation and complete final statements. When an answer changes during checking, identify the exact reason for the change; do not switch merely because an alternative feels more sophisticated.
A two-week practice sequence
- Days 1–2: learn the mechanism and complete worked examples with notes open.
- Days 3–4: solve contrast pairs and explain why the tempting option fails.
- Days 5–6: mix this skill with earlier topics so the cue is not announced.
- Day 7: complete a short timed set and classify errors as reading, concept, execution or checking errors.
- Days 8–10: repair only the weak error class using fresh examples.
- Days 11–12: complete delayed transfer without notes and teach one example aloud.
- Days 13–14: perform a timed mixed review, then schedule the next retrieval rather than massing more identical questions.
For parents and tutors
Ask for the reason before offering a correction. Useful prompts include: “Which word or heading controls that choice?”, “What stayed the same?”, “What changed?”, “What alternative explanation have you ruled out?”, and “Can you make a near-example where your rule fails?” These prompts reveal whether the learner owns the mechanism.
Record errors by cause, not merely by topic. A learner may know the concept but lose marks through scope, unit conversion, case identification, uncontrolled variables or an incomplete final statement. One accurately diagnosed error is more valuable than a large undifferentiated worksheet.
Frequently asked questions
Why is adding the same number not ‘doing the same thing’?
It is the same additive action, but a ratio records multiplication or division. Additive equality does not preserve a multiplicative comparison.
Can a fractional scale factor be used?
Yes. Equivalent ratios require one non-zero common multiplier, which may be a fraction or decimal.
Must I always simplify first?
No. A direct scale factor or cross-products may be faster. Simplifying is especially useful when comparing several ratios.
What happens when both original terms are equal?
Equal addition keeps them equal, so 1:1 remains 1:1. Treat that as a special case, not the general rule.
Why must units match?
A ratio compares quantities. Different units can make bare numerals misleading; convert before simplifying.
Can I add another batch and keep the ratio?
Yes, if the added batch has the same ordered ratio as the existing quantities. Proportional addition preserves the ratio.
Next routes
Return to the complete volume directory to choose the next precise learning job, revisit the original PSLE Learning Guide for the series foundation, or continue through the Mathematics hub. Use the official SEAB examination-format page whenever a current paper detail matters.
The goal is independent control: notice the relationship, choose a method, explain the result, and check it using a genuinely different route.
Retrieval extension 1
Revisit this case without looking at the earlier explanation: 2:3 becomes 6:9. First name the controlling relationship. Then state the correct interpretation and one reason: Both terms are multiplied by 3, so the comparison remains two parts for every three parts. Finally, reject the tempting shortcut explicitly: Seeing larger numbers and assuming the ratio must be larger. Change one detail so that the answer would change, and explain why the changed condition matters. This variation step prevents memorisation of surface wording and builds a rule that can survive an unfamiliar examination context.
Retrieval extension 2
Revisit this case without looking at the earlier explanation: 2:3 becomes 5:6. First name the controlling relationship. Then state the correct interpretation and one reason: Adding 3 to each term produces 5:6, but 2×6=12 while 3×5=15; the ratios are not equivalent. Finally, reject the tempting shortcut explicitly: Treating equal addition as if it were a scale factor. Change one detail so that the answer would change, and explain why the changed condition matters. This variation step prevents memorisation of surface wording and builds a rule that can survive an unfamiliar examination context.
Retrieval extension 3
Revisit this case without looking at the earlier explanation: 4:10 simplifies to 2:5. First name the controlling relationship. Then state the correct interpretation and one reason: Both terms are divided by 2. The common division preserves the quotient. Finally, reject the tempting shortcut explicitly: Dividing only the larger term until the numbers look small. Change one detail so that the answer would change, and explain why the changed condition matters. This variation step prevents memorisation of surface wording and builds a rule that can survive an unfamiliar examination context.
Retrieval extension 4
Revisit this case without looking at the earlier explanation: 7:7 becomes 10:10. First name the controlling relationship. Then state the correct interpretation and one reason: Both ratios simplify to 1:1. Equal addition happens to preserve equality because the original terms are equal. Finally, reject the tempting shortcut explicitly: Turning this special case into a general rule for unequal ratios. Change one detail so that the answer would change, and explain why the changed condition matters. This variation step prevents memorisation of surface wording and builds a rule that can survive an unfamiliar examination context.
Retrieval extension 5
Revisit this case without looking at the earlier explanation: 3:5 becomes 12:20. First name the controlling relationship. Then state the correct interpretation and one reason: One common factor, 4, maps both terms. The fractions 3/5 and 12/20 are equal. Finally, reject the tempting shortcut explicitly: Finding two different factors that happen to reach the targets. Change one detail so that the answer would change, and explain why the changed condition matters. This variation step prevents memorisation of surface wording and builds a rule that can survive an unfamiliar examination context.
Retrieval extension 6
Revisit this case without looking at the earlier explanation: 3:5 becomes 6:8. First name the controlling relationship. Then state the correct interpretation and one reason: The first term doubled but the second did not; cross-products give 24 and 30. Finally, reject the tempting shortcut explicitly: Noticing that both terms increased and overlooking different relative changes. Change one detail so that the answer would change, and explain why the changed condition matters. This variation step prevents memorisation of surface wording and builds a rule that can survive an unfamiliar examination context.
Retrieval extension 7
Revisit this case without looking at the earlier explanation: A red-to-blue ratio of 2:3 receives 4 red and 6 blue counters. First name the controlling relationship. Then state the correct interpretation and one reason: The added batch itself has ratio 2:3, so the combined totals keep 2:3. This is not equal addition; it is adding a proportional batch. Finally, reject the tempting shortcut explicitly: Saying any addition changes a ratio. Change one detail so that the answer would change, and explain why the changed condition matters. This variation step prevents memorisation of surface wording and builds a rule that can survive an unfamiliar examination context.
Retrieval extension 8
Revisit this case without looking at the earlier explanation: A 2:3 mixture receives 5 units of each ingredient. First name the controlling relationship. Then state the correct interpretation and one reason: Equal addition changes the totals to 7:8, so the original unequal ratio is not preserved. Finally, reject the tempting shortcut explicitly: Confusing equal amounts with proportional amounts. Change one detail so that the answer would change, and explain why the changed condition matters. This variation step prevents memorisation of surface wording and builds a rule that can survive an unfamiliar examination context.
Retrieval extension 9
Revisit this case without looking at the earlier explanation: 15:25 and 3:5. First name the controlling relationship. Then state the correct interpretation and one reason: Dividing both original terms by 5 gives the second ratio. Cross-products also give 75 on both sides. Finally, reject the tempting shortcut explicitly: Comparing term differences, 10 and 2, instead of multiplicative structure. Change one detail so that the answer would change, and explain why the changed condition matters. This variation step prevents memorisation of surface wording and builds a rule that can survive an unfamiliar examination context.