Mathematics Tutor Singapore | PSLE Maths ratio word problems often become difficult at the exact moment quantities move. A student may know that 3:5 contains eight units, yet still use the original ratio after twelve counters have transferred from one group to another. The missing skill is following the story’s changing state.
I begin with three questions: what was present before, what happened, and which relationship stays true after the action? A transfer conserves the combined total; an external addition raises that total; removing equal amounts conserves the difference, not the ratio. These distinctions help learners solve unfamiliar PSLE-style problem sums without guessing a procedure from keywords.
This teaching guide develops the unit method, bar models, worked arithmetic, a light algebraic extension, independent practice and parent guidance. Its numerical situations are original examples. The official MOE Primary Mathematics syllabus is linked below for the relevant curriculum context.
Open the ratio and transfer chapters
- Begin with the right ratio unit
- Before-and-after stories describe different mathematical actions
- Worked example: transfer from 3:5 to equal groups
- Why one transfer changes the gap by twice its size
- Addition from outside: a fixed category instead of a fixed total
- Equal removal and constant difference
- Build a reliable before-and-after bar model
- Fractions and percentages of the changing whole
- Six misconceptions to diagnose
- Twelve worked questions for independent practice
- How to revise over two weeks
- What to ask a PSLE Maths tutor
- Frequently asked questions
- Official syllabus and connected lessons
Begin with the right ratio unit
A ratio of 3:5 compares two groups using three and five equal units. The unit is not necessarily one counter. If the combined number is 96, then eight units represent 96, so one unit represents twelve. The actual counts are 36 and 60.
| Information | Meaning of known amount | Computation |
|---|---|---|
| Total 96, ratio 3:5 | Eight units | 96 ÷ 8 = 12 per unit |
| Difference 18, ratio 3:5 | Two units | 18 ÷ 2 = 9 per unit |
| First group 24, ratio 3:5 | Three units | 24 ÷ 3 = 8 per unit |
| Second group 40, ratio 3:5 | Five units | 40 ÷ 5 = 8 per unit |
I ask children to label the unit before doing arithmetic. The phrase “the difference represents two units” shows that they know why they are dividing by two. Simply writing 18 ÷ 2 with no explanation may have been copied from a similar worked solution.
There are also different meanings of a fraction. The first group is three-fifths of the second group, but three-eighths of the combined total. When a question later requests a percentage of all counters, the reference is the combined eight units. Confusing group-to-group and group-to-total is a common root of otherwise surprising errors.
Before-and-after stories describe different mathematical actions
A ratio problem may transfer, add, remove, exchange or multiply quantities. Those actions need not preserve the same mathematical relationship. In a real transfer, every counter leaving one group arrives in the other: the total is unchanged. If new counters are added from outside, the total increases. If ten are removed from both groups, the difference remains unchanged while the total decreases by twenty.
| Action | Changes | Remains fixed |
|---|---|---|
| Transfer blue to red | Red gains, blue loses | Combined count |
| Add new red counters | Red and total rise | Blue count |
| Remove ten from both | Both counts and total fall | Difference between counts |
| Multiply both groups by two | Both counts double | Ratio |
| Relabel the two groups | Category names change | Actual physical counts |
The first diagram I draw has two rows: before and after. It records two independent facts: the original quantities and the exact action that changed them. I do not let the child silently use an after ratio with a before total unless the story proves that total is unchanged.
Different ratio units can represent different absolute amounts. A bar labelled “three parts” before an addition need not contain the same unit-size as a bar labelled “three parts” after the addition. Their correspondence must be established through an unchanged category, difference or total.
Worked example: transfer from 3:5 to equal groups
Red and blue counters start in the ratio 3:5 with 96 counters altogether. Twelve blue counters are transferred into the red group. Find the new ratio.
There are eight starting units, so one equals twelve. Red has 36, blue 60. The transfer makes red 36 + 12 = 48 and blue 60 − 12 = 48. Therefore the new ratio is 1:1.
| Moment | Red | Blue | Total |
|---|---|---|---|
| Before | 36 | 60 | 96 |
| Transfer | +12 | −12 | No change |
| After | 48 | 48 | 96 |
| Ratio | 1 | 1 | Equal-sized groups |
A wrong solution such as 48 and 60 has accidentally added twelve from outside. A solution of 36 and 48 has merely removed twelve. Both errors can be detected by checking the original combined total of 96. A good tutor invites the learner to perform that check before revealing the answer.
This model is easier than many examination questions because the original total is given. If only a difference or the later ratio were known, the pupil would have to align a different invariant. That is the next layer of reasoning, not an entirely new topic.
Why one transfer changes the gap by twice its size
The original difference was 60 − 36 = 24 counters. Transferring twelve from the larger to the smaller closes the entire gap, because blue loses twelve and red gains twelve. The difference decreases by twenty-four, which is twice the amount transferred.
| Before counts | Transfer from larger to smaller | After counts | New gap |
|---|---|---|---|
| 36 and 60 | 12 | 48 and 48 | 0 |
| 20 and 50 | 5 | 25 and 45 | 20 |
| 15 and 35 | 10 | 25 and 25 | 0 |
| 40 and 70 | 8 | 48 and 62 | 14 |
If the smaller group instead gives counters to the larger, the difference grows by twice the transferred number. Direction therefore matters. A child should always mark the giving group and the receiving group with an arrow before using a “half the difference” shortcut.
For an algebra-ready learner, the starting difference is B − A. After moving t from larger B to smaller A, the difference is (B − t) − (A + t) = B − A − 2t. That equation formalises the bar model, but it is optional enrichment if the visual relationship is not yet secure.
Addition from outside: a fixed category instead of a fixed total
Consider a different problem. Red:blue starts at 2:3. Six new red counters are added from another collection. The new red:blue ratio is 3:4. What was the original total?
Represent original red as 2u and blue as 3u. Red becomes 2u + 6 while blue remains 3u, so (2u + 6):(3u) = 3:4. This means 4(2u + 6) = 9u. Expanding gives 8u + 24 = 9u, so u = 24.
Original red is 48 and original blue 72; the starting total is 120. After adding six, red is 54 while blue remains 72, giving 54:72 = 3:4. The new total is 126. The unchanged blue amount links the two ratio states.
| Quantity | Before | After adding six red |
|---|---|---|
| Red | 48 | 54 |
| Blue | 72 | 72 |
| Total | 120 | 126 |
| Ratio | 2:3 | 3:4 |
A ratio bar method gives the same result. Treat blue’s three original units and four later units as the same physical amount. On a common twelve-part scale, original red occupies eight small parts and later red occupies nine. The one extra small part represents six counters, so twenty small parts represent the original combined total of 120.
If a learner instead subtracts six from blue, they have changed an addition into a transfer. Their equation would describe a different story. It is the action, not the numbers 2:3 and 3:4, that decides which model is valid.
Equal removal and constant difference
Suppose red:blue is 3:5. Ten counters are removed from each group, and the new ratio is 1:2. Let the original amounts be 3u and 5u. The after relationship is (3u − 10):(5u − 10) = 1:2.
Cross-multiplication gives 2(3u − 10) = 5u − 10, or 6u − 20 = 5u − 10. Hence u = 10. Red originally had 30, blue 50. After removal they have 20 and 40, giving 1:2. The original total was 80.
| Measure | Before | After removing ten each |
|---|---|---|
| Red | 30 | 20 |
| Blue | 50 | 40 |
| Total | 80 | 60 |
| Difference | 20 | 20 |
| Ratio | 3:5 | 1:2 |
The ratio changes because subtracting ten is not the same as multiplying both groups by a common factor. If we multiply 30 and 50 by two, the ratio stays 3:5; if we subtract ten, it becomes 1:2. This counterexample is often a better explanation than a rule learned in isolation.
Building a reliable before-and-after bar model
A good diagram answers a question about meaning: which quantity occupies this bar? I would label four entries separately—red before, blue before, red after and blue after. Then I would mark what is transferred, added or removed, and align only the genuinely unchanged amount.
For a transfer, draw two bars before and two bars after with the combined length identical. For an addition, align the untouched category in both states. For equal removal, show how both bars shrink by the same fixed count and retain the original absolute difference.
If a student can solve a transfer only when a tutor draws the diagram, I gradually remove the scaffold. First they label a partial model, then construct their own, and eventually write a short equation or table directly. The achievement is independent selection of a representation, not loyalty to one diagram style.
Ratios, fractions and percentages of a changing whole
With a ratio 3:5, red is 3/8 of the combined collection and blue is 5/8. In the 96-counter example, red rises from 36/96 = 37.5% of the whole to 48/96 = 50% after transfer. The total is fixed while the category shares change.
After an external addition, both the numerator and denominator of a category fraction may change. Red rises from 48/120 = 40% to 54/126 = 3/7, approximately 42.86%. The new share is not 54/120, because the original total of 120 is no longer the combined number of counters.
This is why ratio, fractions and percentages are connected. All describe a relationship between a quantity and a reference. A child who keeps asking “what does the denominator represent now?” will be more reliable across all three topics.
Six misconceptions to diagnose before adding worksheets
| Observed mistake | Likely misunderstanding | Next teaching question |
|---|---|---|
| Divides a given total by five in ratio 3:5 | Does not use the full eight-unit total | What do all eight units represent? |
| Gains twelve red without reducing blue | Transfer treated as external addition | Does the total remain unchanged? |
| Red and blue both shrink but ratio assumed constant | Fixed subtraction confused with multiplication | Compare 30:50 and 20:40 |
| After ratio applied to old total following addition | States are mixed | How many items now exist? |
| Before and after bar units assumed identical | No valid invariant aligned | Which quantity genuinely stayed the same? |
| Red fraction given as 3/5 of total | Compares red-to-blue instead of red-to-whole | How many units make the whole? |
A student may have excellent multiplication facts yet still model a transfer incorrectly. That child needs work on conservation and state changes, not an unrelated drill on multiplication. Another student may draw the correct bars but mishandle subtraction; that is a different, narrower repair.
Twelve worked ratio questions
1. Find counts from a total
A:B = 2:7 with 81 objects. Nine ratio units represent 81, so A = 18 and B = 63. Check that they add to 81.
For a transfer, check the combined total; for an outside addition, check the unchanged group; for equal removal, check the difference. A solution that fails its own physical invariant must be reconsidered before simplifying the final ratio.
2. Find counts from a difference
A:B = 4:7 and B exceeds A by 21. Three units are 21, so A = 28 and B = 49.
For a transfer, check the combined total; for an outside addition, check the unchanged group; for equal removal, check the difference. A solution that fails its own physical invariant must be reconsidered before simplifying the final ratio.
3. Transfer until equal
Groups contain 25 and 45. Move ten from larger to smaller. Both contain 35, so the new ratio is 1:1.
For a transfer, check the combined total; for an outside addition, check the unchanged group; for equal removal, check the difference. A solution that fails its own physical invariant must be reconsidered before simplifying the final ratio.
4. Transfer without equality
Groups contain 28 and 52. Move eight from larger to smaller. They become 36 and 44, or 9:11.
For a transfer, check the combined total; for an outside addition, check the unchanged group; for equal removal, check the difference. A solution that fails its own physical invariant must be reconsidered before simplifying the final ratio.
5. Add from outside
Groups contain 40 and 60. Add ten to the first group. They become 50 and 60, or 5:6. The total is now 110.
For a transfer, check the combined total; for an outside addition, check the unchanged group; for equal removal, check the difference. A solution that fails its own physical invariant must be reconsidered before simplifying the final ratio.
6. Remove equally
Groups contain 30 and 50. Remove ten from each. They become 20 and 40, or 1:2. The difference remains twenty.
For a transfer, check the combined total; for an outside addition, check the unchanged group; for equal removal, check the difference. A solution that fails its own physical invariant must be reconsidered before simplifying the final ratio.
7. Find the missing group
A:B = 5:3 and A has 45. One unit is nine, so B has 27, giving a combined 72.
For a transfer, check the combined total; for an outside addition, check the unchanged group; for equal removal, check the difference. A solution that fails its own physical invariant must be reconsidered before simplifying the final ratio.
8. Reference fractions
A:B = 3:5. A is 3/8 of the combined total but 3/5 of B. Both statements can be true at once.
For a transfer, check the combined total; for an outside addition, check the unchanged group; for equal removal, check the difference. A solution that fails its own physical invariant must be reconsidered before simplifying the final ratio.
9. Use half the starting gap
Counts are 32 and 56. The difference is 24, so transfer twelve from the larger to equalise them at 44.
For a transfer, check the combined total; for an outside addition, check the unchanged group; for equal removal, check the difference. A solution that fails its own physical invariant must be reconsidered before simplifying the final ratio.
10. Reverse an addition
After ten new red are added, red:blue is 3:4 and blue = 40. Red after = 30, red before = 20; the original ratio was 1:2.
For a transfer, check the combined total; for an outside addition, check the unchanged group; for equal removal, check the difference. A solution that fails its own physical invariant must be reconsidered before simplifying the final ratio.
11. Multiply both counts
If A:B = 2:5 with 14 and 35, doubling gives 28 and 70. The ratio remains 2:5.
For a transfer, check the combined total; for an outside addition, check the unchanged group; for equal removal, check the difference. A solution that fails its own physical invariant must be reconsidered before simplifying the final ratio.
12. Compare actions
Start with 30 red and 50 blue. Add ten new red: 40:50 = 4:5, total 90. Transfer ten from blue: 40:40 = 1:1, total 80.
For a transfer, check the combined total; for an outside addition, check the unchanged group; for equal removal, check the difference. A solution that fails its own physical invariant must be reconsidered before simplifying the final ratio.
A two-week revision and delayed-retention plan
| Session | Focus | Evidence of progress |
|---|---|---|
| 1 | Identify one ratio unit from a given total | Labels all parts of the whole |
| 2 | Identify one unit from a given difference | Matches the excess units |
| 3 | Calculate a transfer | Checks fixed total |
| 4 | Predict the double effect on the gap | Explains the movement verbally |
| 5 | Add external objects to one group | Keeps the untouched group fixed |
| 6 | Remove equal numbers | Recognises constant difference |
| 7 | Align old and new bar units | Matches only an invariant |
| 8 | Convert a ratio into a fraction of the total | States the correct denominator |
| 9 | Solve an unseen mixed story | Selects method without a chapter heading |
| 10 | Return to a previously wrong problem | Works independently after a delay |
A short session can be enough when the question is carefully chosen. If the student is still unsure what a ratio unit is, more complex two-stage transfers will not repair that uncertainty. Start with clear meaning, then use progressively mixed tasks, and introduce time pressure only when the solution process is stable.
What to ask a PSLE Maths tutor
A useful tutor should be able to say exactly what the learner misunderstood. “Weak in ratio” is too broad to guide the next lesson. “The learner treats the added quantity as a transfer and forgets the changed total” is specific, testable and teachable.
- Show me how you diagnose the first incorrect step, not only the final answer.
- How do you teach a pupil to identify a fixed total, untouched group or constant difference?
- Will the child make their own before-and-after model or always fill in a prepared one?
- How do you check that the same idea works with changed wording?
- How do you protect the student’s wider school workload and wellbeing?
- Can you explain what independence should look like after a period of lessons?
Small-group lessons can make it easier to listen to each child’s explanation, but a class-size label does not guarantee effective teaching. What matters is precise feedback, sensible sequencing and whether the learner becomes less dependent on prompts. Not every student needs tuition; some need a targeted school conversation or manageable home practice.
Frequently asked questions
Does a transfer always preserve the total?
A transfer within the same closed collection preserves its total when nothing is added or removed.
Why does moving ten close a gap by twenty?
One side loses ten while the other gains ten, so the difference shrinks at both ends.
Is adding ten equivalent to transferring ten?
No. Adding from outside changes the total; transferring inside the collection preserves it.
Does removing ten from both preserve the ratio?
Not generally. It preserves the difference but commonly changes the ratio.
Must bar models be used for every ratio sum?
No. A table or sound equation can be just as valid. Models are useful when they clarify the quantities.
How should a tutor check understanding?
Use a changed story without prompts several days after the explanation, and ask the student to name the invariant.
Can ratio tuition guarantee PSLE AL1?
No. A topic can be improved through good teaching, but examinations depend on broader learning and conditions.
Official syllabus and connected Mathematics guides
The MOE Primary Mathematics syllabus updated October 2025 covers Primary 6 ratio notation, equivalent ratios, dividing quantities in a ratio and connections between fractions and ratios. The original transfer puzzles in this article illustrate applications of those concepts; they are not claims about what will appear in a particular future PSLE paper.
Use the Mathematics Learning Hub for the full education pathway. For related part-whole reasoning, read PSLE Maths Reference-Whole Problem Sums. Earlier foundations appear in Primary 3 Equivalent Fractions. For later equations, use Secondary 2 Simultaneous Equations.
Parents who want help interpreting an actual wrong ratio problem can arrange an eduKate Singapore consultation with one solved ordinary ratio question and one failed before-and-after attempt. Comparing them is more valuable than assuming the solution is another large worksheet.