PSLE ratio word problems become much easier when a Primary 6 student can answer one question before calculating: what stays the same while the quantities change? In before-and-after ratio questions, money may be spent, counters may be transferred, children may join a group or both quantities may increase. A good model records the original state, the change and the final state without assuming that one ratio unit has the same value at both times.
This is an original practice guide for the Primary 6 MOE/SEAB Mathematics examinations. It teaches equivalent ratios, three-part ratios, dividing quantities in a given ratio and connecting ratios to fractions, alongside richer change-of-state word problems. These examples are teaching exercises, not future PSLE predictions. The aim is to help pupils select the right invariant before they begin long arithmetic.
- Transfer between two groups: their combined total stays constant.
- Both groups increase or decrease by the same amount: their difference stays constant.
- Only one group changes: the unchanged group is the anchor for matching ratios.
- A quantity enters or leaves the combined system: do not assume the total remains constant.
- Ratios before and after: their units are generally different until an invariant justifies scaling.
- Check: reconstruct the original amounts, apply the stated change, then verify the final ratio.
The PSLE ratio syllabus: exactly what P6 pupils should understand
The MOE Primary Mathematics syllabus updated October 2025 lists ratio under Primary Six Standard Mathematics. Pupils learn the notation, representation and interpretation of two- and three-term whole-number ratios; equivalent and simplest-form ratios; dividing a quantity in a given ratio; missing terms in equivalent ratios; and links between fractions and ratios. The stated ratio scope excludes ratios involving fractions and decimals. This guide therefore uses whole-number ratio terms, although one unit can sometimes have a dollar or measurement value that is not an integer in a real-world interpretation.
The published SEAB Standard Mathematics examination format for examination from 2026 contains Paper 1, which excludes calculators, and Paper 2, which allows them. The entire examination has 45 questions worth 100 marks, but neither paper’s format reveals the exact topic of a future question. Ratio is syllabus knowledge; a particular before-and-after construction is not a guaranteed PSLE question.
Check the MOE Primary Mathematics syllabus and SEAB Mathematics 0008 examination document. If preparing for 2027, reconfirm year-specific administrative information on SEAB’s PSLE website. Do not rely on an old paper’s timing without checking the revised format.
First master a ratio that does not change
A basket contains apples and oranges in the ratio 3:5. There are 64 fruits altogether. Total units = 3 + 5 = 8, so one unit represents 64 ÷ 8 = 8 fruits. Apples = 3 × 8 = 24; oranges = 5 × 8 = 40. Check 24 + 40 = 64 and 24:40 simplifies to 3:5.
The ratio 3:5 means apples are 3/5 of the number of oranges. It does not mean apples are 3/5 of all the fruits. Apples are 3/8 of the total and oranges are 5/8 of the total. A child who cannot distinguish these two fractions is not ready to solve a changing-ratio question.
Ask pupils to make three statements: ‘Three units are apples’; ‘five units are oranges’; ‘eight units make the whole’. Every later question depends on accurately identifying the total number of units or the relevant part.
Now change the given information. If there are 40 oranges, that is five units, so each unit equals eight and apples equal 24. If apples exceed another group? First identify which difference of ratio units represents the difference. Ratio questions do not all begin with the combined total.
A reliable before-change-after table
A useful working layout contains three rows: before, change and after. Across the top, write the two named groups and their combined total. A line such as ‘A +12; B −12; total unchanged’ often reveals more than an entire page of unlabelled units.
Take the ratio A:B = 2:5. If ten items move from B to A, the changes are A +10 and B −10. The combined count does not change. If ten new items arrive for A instead, the changes are A +10 and B unchanged, so the combined count increases by ten. These two stories sound similar but require different invariants.
Before drawing a bar model, ask ‘Where did the items come from?’ If the items leave one group and enter the other, total is constant. If they arrive from outside, total grows. If they are removed from both groups, total falls even though the difference between groups might stay the same.
One further rule protects marks: avoid inventing a one-to-one relationship between the initial and final unit sizes. The ratio 2:5 and the ratio 3:4 both have seven total parts, but that fact alone does not make their individual units equal unless their total quantity is unchanged.
Worked example 1: direct transfer with constant total
Red and blue counters are in the ratio 2:7. Ten blue counters are transferred to the red group. Afterwards, red:blue = 4:5. How many counters were there altogether?
Before the transfer the total is nine units. Afterwards the total is also nine units, because the counters stayed within the two groups. The initial red count is two units and the final red count is four units. The increase of two units corresponds to the ten transferred counters, so one unit represents five counters.
The original total is 9 × 5 = 45 counters. Before transfer there were 10 red and 35 blue. After moving ten, there are 20 red and 25 blue; 20:25 simplifies to 4:5. Both the total and the final ratio check out.
Notice how little calculation was needed once the invariant was identified. The same nine-unit scale is valid precisely because the same counters remained inside the combined system. Without that fact, equating the units would be an unjustified shortcut.
Worked example 2: a transfer when ratio sums differ
Before a transfer, red:blue = 3:5. Eighteen blue counters are moved to the red group. Afterwards, red:blue = 3:2. Find the initial number of blue counters.
The combined total is unchanged, but the original ratio has eight units while the final ratio has five. Scale the first ratio to 15:25 and the second to 24:16, so both have 40 aligned total units. Now the red group increases from 15 to 24 aligned units, an increase of nine units.
Nine aligned units correspond to eighteen transferred counters. One unit therefore equals two counters. Initially blue had 25 × 2 = 50 counters and red had 15 × 2 = 30.
Check the actual change: red increases to 48 and blue falls to 32. The after ratio is 48:32 = 3:2. The initial blue count of 50 is therefore consistent with both states.
This example shows why the least-common-multiple technique is helpful but secondary. The first decision was that the total is constant; only then did aligning eight and five total units make sense.
Worked example 3: only one part is unchanged
Aisha and Ben have money in the ratio 3:7. Aisha keeps the same amount while Ben spends $26. Afterwards, Aisha:Ben = 3:5. How much money did Ben have initially?
Aisha is unchanged. Her three units before and three units after refer to the same amount, so the two unit sizes can be matched. Ben decreases from seven to five units, a fall of two units equal to $26. Each unit equals $13.
Ben originally held 7 × $13 = $91. Aisha held 3 × $13 = $39. Ben’s remaining money is $91 − $26 = $65; the after ratio $39:$65 simplifies to 3:5.
The combined total falls when Ben spends the money. Calling total constant here would be wrong. The unchanged Aisha amount is the valid bridge between states.
Worked example 4: the unchanged part needs scaling
The ratio of boys to girls in a club is 3:4. Six boys join, while the number of girls remains unchanged. The new ratio is 9:10. Find the original number of girls.
Girls are the constant group. Align their number of units: the original 3:4 becomes 15:20, and the final 9:10 becomes 18:20. Boys increase by three aligned units, and that increase represents six boys. Each aligned unit equals two people.
Girls = 20 × 2 = 40. Originally there were 30 boys, then 36 after six joined. The final ratio 36:40 simplifies to 9:10.
This example reveals an important trap. The original three boy units do not equal the final nine boy units one-for-one. The quantity that stays constant is the girls’ actual count, and equivalent-ratio scaling makes that fact visible.
Worked example 5: constant difference when equal amounts are added
Two children have stickers in the ratio 2:5. Each buys eighteen additional stickers. Their new ratio is 4:7. Find how many stickers the first child had before buying more.
If both children gain eighteen, the difference between their sticker counts stays constant. Initially the difference is three ratio units; afterwards it is also three ratio units. Here the same unit size can be used, so the first child’s increase from two units to four units is two units equal to eighteen.
One unit equals nine stickers, making the first child’s original amount 2 × 9 = 18. The second child originally had 45. After both gain eighteen, they have 36 and 63; 36:63 simplifies to 4:7.
Many students incorrectly call this a constant-total question. The combined total rises by thirty-six. What stays unchanged is the gap between the two quantities, not their sum.
Worked example 6: when one decreases and the other increases
John has two parts of a quantity and Ken has three parts. John gains ten items while Ken loses five items. They then have the same count. How many items did each have initially?
Their initial difference was one part. John’s gain reduces the gap by ten, and Ken’s loss reduces it by another five. The full initial gap was therefore fifteen items, so one part represents fifteen.
John initially had 2 × 15 = 30 items and Ken had 3 × 15 = 45 items. Afterwards John has 40 and Ken has 40, as required. This example also checks the direction of every change.
The lesson is to think in terms of distance between quantities. When the smaller quantity grows and the larger falls, the gap closes through both movements. Forgetting one movement creates a wrong answer despite correct individual arithmetic.
Worked example 7: three-part ratios and sharing
Three siblings divide $360 in the ratio 2:3:4. Find the amount received by the sibling with the largest share.
Total units = 2 + 3 + 4 = 9. One unit = $360 ÷ 9 = $40. The largest share is four units, or $160. Check the other shares, $80 and $120; together they sum to $360.
Now suppose the smallest share is known to be $80, but the total is not given. Two units equal $80, so one unit equals $40 and the same total of $360 follows. The method does not change; only the quantity that provides the unit value changes.
For three-part ratios, pupils must never drop the third term while constructing the total. In a:b:c, a/(a+b+c) is the fraction of the whole represented by the first group, not a/(a+b).
Worked example 8: missing term in an equivalent ratio
The ratio 5:8 is equivalent to 20:n. Find n. Five becomes twenty through multiplication by four, so eight also becomes thirty-two. Thus n = 32.
An alternative check is 20:32 = 5:8 after dividing both terms by four. Keep multiplication consistent across all terms; changing only one term alters the relationship.
If three quantities are in the ratio 2:3:5 and the first quantity represents twelve objects, multiply all three terms by six to get 12:18:30. Two units equal twelve, so one unit represents six.
Equivalent ratios are foundational for before-and-after problems because scaling lets us align the same invariant across two situations. Pupils should practise this skill without lengthy stories before applying it to a transfer.
A decision tree for any unfamiliar ratio story
Start by labelling A and B at the first moment. Next record each change with a positive or negative sign. If A increases by k and B decreases by k, the combined total remains unchanged. Align total ratio units before and after.
If both A and B increase by k or both decrease by k, the difference A − B remains unchanged. Align the difference of ratio units, taking care with which group is larger. An equal addition to both groups changes their total but not their difference.
If A is unchanged while B changes, align A. If B is unchanged while A changes, align B. If both change by unequal amounts, the difference may change by a known amount; calculate that difference carefully rather than forcing a standard invariant.
If the problem includes three groups, identify which groups exchange items and whether a fourth source or destination exists. A transfer from A to B within three groups may leave the overall total fixed even if one pair’s subtotal changes. Name the system whose total stays constant.
Only after this classification should a pupil draw a model or form equations. That small reading investment is often the most effective way to reduce repeated wrong starts.
The mini-clinic: sixteen original questions with answers
All questions below are freshly written exercises. Cover the answer and solve first. A word problem is not understood until the pupil can explain the anchor and check both states.
1. Simple two-part sharing
$135 is divided in the ratio 4:5. The larger share is $75, because nine units equal $135 and five units equal $75. Check the smaller share is $60.
2. Difference provides the unit
A:B = 3:8 and B exceeds A by 40. The five-unit difference equals 40, so one unit is eight; A is 24 and B is 64.
3. The part, not the total, is given
Blue:red = 2:7 and blue has fourteen counters. One unit is seven, so red has forty-nine and the total is sixty-three.
4. Ratio to fraction
Books:pens = 5:3. Books are 5/8 of all items, not 5/3 of the whole. Five thirds compares the books to pens.
5. Fraction to ratio
Two fifths of a group are boys and the rest are girls. Boys:girls = 2:3, because three fifths of the group are girls.
6. Simplest whole-number ratio
There are 18 green and 30 yellow marbles. Green:yellow = 3:5 after dividing both counts by six.
7. Equivalent ratio
6:11 = 24:n. Scale by four, so n = 44.
8. Three-part ratio
A:B:C = 2:4:7 and the total is 130. Thirteen units equal 130, one unit equals ten, and C has seventy.
9. Transfer and same total
A:B = 2:7. After nine units of the real item move from B to A, their ratio becomes 3:6. Original one ratio unit = nine, so original A = 18 and B = 63; after A = 27 and B = 54.
10. Transfer and equal groups
Red:blue = 4:6. Ten blue items move to red and the counts become equal. The original difference is two units; transferring ten closes the difference by twenty, so each unit is ten. Original counts: forty and sixty.
11. One group unchanged
A:B = 2:5. B loses $18 and the ratio becomes 2:3. Two units represent eighteen, so one unit equals nine and B originally had $45.
12. An equal increase
A:B = 1:3. Both receive eight items, changing the ratio to 2:4. The initial difference stays two units, and A rises by one unit equal to eight, so A initially had eight and B twenty-four. The final ratio is 16:32 = 1:2, equivalent to 2:4.
13. An equal decrease
A:B = 4:7. Both give away six items and the ratio becomes 3:6. The difference is unchanged at three units, so one unit represents six. Initial values were twenty-four and forty-two; final eighteen and thirty-six.
14. A new member joins
Boys:girls = 2:3. Five boys join, making the ratio 3:3. Girls are unchanged. One original ratio unit equals five; originally ten boys and fifteen girls.
15. Separate the required state
A has thirty counters and B has fifty. After ten counters move from B to A, the new amounts are forty and forty. The original ratio is 3:5; the final ratio is 1:1. A question about ‘initial’ must not be answered with the final ratio.
16. The total actually changes
A has twelve tokens and B has eighteen, so A:B = 2:3. When eight new tokens arrive for A, the ratio becomes 20:18 = 10:9. The combined total changes from thirty to thirty-eight. Constant-total scaling would be invalid.
Deep clinic: identical-looking ratios can come from different causes
Suppose two groups change from 2:3 to 3:4. A pupil may imagine that one item has been added to each group. That is one possible story, but not the only possible story. If the first state has twenty and thirty objects, and the final state thirty and forty, both gained ten. The difference stays ten.
If the initial groups instead have forty and sixty, the final quantities might be sixty and eighty. In that case the increments are twenty and twenty, also preserving the difference. The ratios alone cannot determine the actual quantities or the amount of change.
A third scenario may involve a transfer and addition, producing the same final ratio while changing the total. To solve such a story, we need the stated changes or another numerical fact. This is why before-and-after ratio questions must be read as relationships, not just patterns of digits.
Teach pupils to ask whether enough information exists. A responsible mathematical answer may be ‘cannot be determined from the information provided’ when a word problem lacks the scale needed for a unique count.
Five mistakes that signal a conceptual gap
The first is adding ratio terms but forgetting what the total represents. An initial red:blue ratio of 3:5 has eight combined parts, but if a third group exists those eight parts may be only part of the larger total.
The second is aligning one unit before and after merely because both expressions contain the number one. Ratio units encode relative size; they do not have a fixed physical value across changing systems.
The third is treating transfer as addition to the system. When five objects move from B to A, the total does not rise by five. It stays the same, while the individual groups change in opposite directions.
The fourth is ignoring the double change in the gap. A transfer of ten from the larger group to the smaller one narrows their difference by twenty, unless the sizes cross and the direction of comparison changes.
The fifth is answering for the wrong time. A student may compute the final number in A but the question asks for how many A had before anything changed. Always label ‘original’ and ‘new’ beside the final numbers.
Why a bar model is sometimes better than equations
Bars make the unchanged part visible. Draw the initial and final quantities in aligned rows, highlight the segment that remains fixed, and mark the exact gain or loss. This works especially well when two or three quantities are compared and the numbers are clean.
For constant total, a single complete bar divided differently before and after can reveal where units need rescaling. For constant difference, two pairs of bars can highlight that the longer bar continues to exceed the shorter by the same physical length.
Equations are also legitimate and may be efficient for pupils comfortable with them. But using a symbol does not remove the need to define what remains constant. A clear model or equation should make the same mathematical relationship transparent.
At PSLE, choose the representation the pupil can execute correctly under time pressure. Repeatedly switching methods because one tutorial claims a single magic heuristic is not always helpful.
A ten-minute warm-up and a thirty-minute practice session
Begin with three short ratio skills: simplifying 12:18, sharing 70 in 2:5, and expressing the first group of 3:7 as a fraction of the total. These checks establish basic fluency.
Next present one unchanged-part question, one transfer question and one equal-increase question without labelling their type. Ask the pupil to say the invariant before calculating. If the child chooses the wrong invariant, stop and repair that classification before continuing.
Then solve two longer mixed problems with full working and final checks. End by asking the pupil to write a one-sentence rule for each error found. The lesson is successful only when the correction changes the next independent attempt.
A pile of twenty questions answered mechanically is less useful than a few well-chosen variations that force the pupil to identify what really remained unchanged.
A four-week ratio revision ladder
Week one: strengthen whole-number ratios, simplest form, equivalent ratios and the relationship between a ratio and a fraction of the whole. Include two- and three-part questions with a variety of known totals, parts and differences.
Week two: teach constant total using only genuine internal transfers. Align total parts by scaling and verify that all original objects remain in the system. Change ratio sums so pupils cannot assume before and after units are automatically identical.
Week three: teach constant part and constant difference. Contrast one unchanged group with equal additions to both groups. Use paired stories that differ by one phrase, such as ‘gave to’ versus ‘received from outside’.
Week four: mix all cases under time limits appropriate to the child. Add questions in which both quantities change by different amounts, and ask for a short verbal justification of the invariant before any computation.
Keep an error notebook listing the first wrong decision: ‘assumed total fixed despite an outside addition’ or ‘equated units without aligning the unchanged part’. Re-test with different numbers several days later.
The examination working format that protects clarity
Label the groups clearly and use a compact table or aligned model. State the invariant, align the units, use the stated change to find one unit, then compute the requested initial or final value. The most useful working shows why a certain number of units equals the stated transfer.
Under the SEAB examination description from 2026, structured and long-answer questions require clear solution steps in the space provided. A full working method is important because an unexplained final answer conceals whether the mathematical relationship was correct.
A two-mark short-answer item with one part may award a method mark for a correct method even if the final answer is wrong, as described in the official syllabus. That is not a guarantee that unrelated workings gain credit. The safest habit is concise, mathematically meaningful working.
Do not overfill the page with repeated versions of the same bar. The purpose of the representation is to reduce uncertainty, not to consume time and attention.
Frequently asked questions about PSLE ratio
What is the fastest way to solve a before-and-after ratio question?
Identify which actual quantity stays constant, or which difference changes by a known amount. Only then align ratio units. This avoids the most serious common wrong start.
Why can I not equate initial and final units?
Because each ratio describes a different state. One unit represents a proportion of the actual total or part at that time. It becomes comparable only through an invariant and correct equivalent-ratio scaling.
What stays constant when ten beads move from B to A?
The combined number of beads, assuming none enter or leave the two-group system. A increases by ten while B decreases by ten.
What stays constant when both groups receive ten beads?
Their difference stays constant. The combined total increases by twenty.
What stays constant if only one group loses money?
The unchanged group’s amount stays constant. The combined total falls; the original and final ratio must be aligned using that unchanged group.
Are ratios involving decimal terms part of the P6 ratio description?
The MOE Primary Six Standard ratio list explicitly excludes ratios involving fractions and decimals. Whole-number ratio terms are the relevant starting scope.
Can I predict a PSLE 2027 ratio-transfer question?
No. The documented syllabus tells you what to learn, not the unreleased content of a particular future paper. Use practice to develop transferable reasoning.
Do I need to learn algebra instead of the model method?
No. Models and simple equations can express the same relationship. A clear representation the pupil understands is more important than a label for the technique.
Related learning and family guidance
A changing ratio is fundamentally a lesson about invariants: decide what changed and what did not. The same thinking appears in percentages, average questions and even geometry. When the pupil gets the relationship right, the arithmetic often becomes short.
Use eduKate Singapore’s before-and-after ratio state-change guide for a companion explanation. For terminology, see Primary 6 Mathematics vocabulary: ratio, algebra, circles and percentage.
For other Primary Six revision areas, continue to PSLE Algebra Word Problems and Paper 1 Non-Calculator Accuracy. For the transition after PSLE, the immutable Secondary 1 Mathematics tutor reference explains why mathematical structure becomes increasingly important.
At eduKateSG, the most valuable teaching intervention is the moment before the pupil chooses a method. Small-group attention allows a tutor to inspect whether the pupil really understands the words ‘transferred’, ‘remaining’, ‘each added’ and ‘unchanged’. The goal is independent reasoning, not the ability to reproduce a worked example. Families can contact eduKate Singapore to discuss work samples and revision priorities.
Official references: MOE Primary Mathematics syllabus, updated October 2025, SEAB Standard Mathematics syllabus from 2026, and SEAB PSLE information. Article prepared in November 2026; all practice examples are illustrative rather than exam predictions.