Did you know that a 2:3:5 ratio has ten equal parts, not three? If three groups share200 units in that ratio, one part is twenty, so their actual shares are forty, sixty and one hundred. That is the essential part-whole idea behind multi-part ratios. The later questions may add, transfer or compare amounts, but the starting point remains the same unit.
For a Secondary 2 G2 Mathematics learner in Bukit Timah, ratio word problems often become difficult because the child divides the right number by the wrong number of parts. A statement such as ‘B has72’ refers to three units in a2:3:5 split, while ‘A+B has80’ refers to five. The correct denominator depends on precisely what the question says is known.
This guide teaches three-part allocation, mixed units, fractional ratios, shared groups, changing ratios and algebraic transfers. The worked examples are hypothetical and focus on checking totals, maintaining conservation where appropriate and explaining the part-unit reasoning before using an equation.
Quick answer: one ratio unit first
| Situation | Method |
|---|---|
| Total200, ratio2:3:5 | Ten units; each20; shares40,60,100. |
| B=72 in ratio2:3:5 | B is three units; one unit24. |
| A+B=80 in ratio2:3:5 | Five units represent80; one unit16. |
| C−A=45 in ratio2:3:5 | Three units represent45; one unit15. |
| A:B=2:3 and B:C=4:5 | Match B=12; combined8:12:15. |
| Transfer from B to A | Total stays fixed; A grows and B shrinks equally. |
| Add new objects to A | Total grows; B remains unchanged. |
Where three-part ratios fit the G2 curriculum
Ratio and proportion are established G2 Mathematics content, including comparing more than two quantities and dividing a quantity in a given ratio. A Secondary 2 school can build upon earlier number skills with complex changing-ratio word problems, but actual assignment scope varies. The 2027 SEC G2 syllabus describes the later examination framework; this article does not imply that every 2026 Secondary 2 assessment asks all the extension questions below.
Official curriculum link: SEAB’s G2 Mathematics syllabus listings and a school Secondary 2 Mathematics topic overview.
Twenty-three concepts to make ratio sharing predictable
1. A ratio compares parts, not three independent percentages
The ratio A:B:C=2:3:5 says that for every two equal units of A there are three of B and five of C. It is a relation among quantities expressed in the same units. A ratio is not automatically three fractions whose numerators should each be divided by their own part value; its full number of parts is2+3+5=10. This gives a structure for sharing a known total and checking it. The actual sizes remain unknown until a total or one part is specified.
Teaching move: Draw ten identical boxes, grouping two for A, three for B and five for C.
Mistake to avoid: Do not mistake 2:3:5 as 2%,3%,5% or as three people automatically getting the values2,3,5.
2. The total number of ratio units is the sum of the entries
For a ratio2:3:5, the whole is ten equal conceptual units. If total quantity is200, each unit is200/10=20, so the three shares are40,60 and100. The logic is divide once by the whole number of parts, then multiply by each group count. This is why an early part-total table helps students who can handle multiplication but struggle to decide their first operation.
Teaching move: Write unit value next to the ten-box model before calculating each share.
Mistake to avoid: Dividing200 by three recipients rather than ten ratio units produces a different sharing rule.
3. The part-to-whole fraction has the total parts in the denominator
A:B:C=2:3:5 gives A=2/10=1/5 of total, B=3/10, and C=5/10=1/2. These are proportions of the whole, not ratios to the other single groups. The fractions add to one, as they should when A, B and C are exclusive parts of the same total. Recognising these fractions makes conversions to percentages and probability-like sharing contexts easier.
Teaching move: Write 20%,30%,50% under the three original ratio entries.
Mistake to avoid: Writing A=2/3 of the whole would confuse A:B with A:(A+B+C).
4. A known total gives one unit by division
When a total is240 in ratio2:3:5, one unit is240/10=24. Hence A=48, B=72 and C=120. The final addition48+72+120=240 is as important as the calculations because it confirms that none of the ten parts was dropped. A child who rushes directly to three separate multiplications may confuse the base of each share; the unit value provides one common reference.
Teaching move: Ask for one unit first, before allowing any of the three share calculations.
Mistake to avoid: Rounding unit size too early can make three final shares fail to total the original.
5. A known one-group value determines the unit directly
If B represents three units and B=72, one unit is72/3=24. Then A=48 and C=120, giving total240. This is different from dividing B by the sum of all parts, which would treat a group share as though it were the entire quantity. The question often hides the total and asks for it; the one-group method reconstructs the missing whole.
Teaching move: Point to the three B boxes rather than all ten while dividing72.
Mistake to avoid: The given B value belongs to three parts, not automatically ten.
6. A known sum of two groups uses their combined ratio units
If A+B=80 in ratio2:3:5, then A+B represents five units. One unit=80/5=16, so A=32, B=48 and C=80. The full total is160. This pattern generalises to a known sum such as B+C, which would correspond to eight parts. Students should label the exact subset the number describes before dividing.
Teaching move: Highlight only A and B boxes on the ratio bar before using the number80.
Mistake to avoid: Dividing80 by all ten units incorrectly assumes it is the complete total.
7. A known difference between groups uses the difference in parts
For ratio2:3:5, the difference C−A is three parts. If the numerical difference is45, one part is45/3=15. Shares are A30, B45 and C75. The same method works for C−B, a two-part difference, or B−A, a one-part difference. It turns comparison language into arithmetic without needing a full algebraic equation.
Teaching move: Mark five C boxes above two A boxes and count the unmatched three.
Mistake to avoid: Dividing a group difference by the sum of parts is a common category error.
8. Part-to-part comparisons differ from part-to-whole shares
From A:B:C=2:3:5, A:B=2:3, while A:(B+C)=2:8=1:4. A is 2/10 of the whole but 2/5 of the combined A+B quantity. These fractions are all legitimate for their own denominators, but they answer different questions. A tutor should ask ‘compared with what?’ whenever a pupil has the correct numerator and an incorrect percentage.
Teaching move: Write three labelled comparisons for the same A value.
Mistake to avoid: The ratio A:B=2:3 does not state A is two-thirds of the whole.
9. Units must be aligned before ratios are simplified
The ratio1.5 metres to75 centimetres must be converted to one unit system:150cm:75cm=2:1. Writing1.5:75 as though the units matched would be wrong by a factor of one hundred. The same applies to hours and minutes, kilograms and grams, or dollars and cents when a ratio compares actual quantities. Ratio notation becomes dimensionless only after compatible measures are expressed on a common scale.
Teaching move: Convert both lengths into centimetres and simplify only afterward.
Mistake to avoid: A numerical simplification using different units is not a meaningful comparison.
10. Fractional and decimal ratio entries can be cleared
The ratio0.6:0.9:1.5 becomes6:9:15 after multiplying every entry by ten, then2:3:5 after dividing by three. The fraction ratio1/2:3/4:1 becomes2:3:4 by multiplying every entry by four. It is valid to multiply all entries by the same positive factor because ratio relationships remain unchanged. This avoids awkward fractional units in later sharing calculations.
Teaching move: Write the same multiplier over all three ratio terms to keep them equivalent.
Mistake to avoid: Changing only one entry rather than every entry alters the relative sizes.
11. Equivalent ratios describe the same proportions at a different scale
The ratios2:3:5 and4:6:10 are equivalent because each entry doubled. Their totals are ten and twenty units respectively, but their part-to-whole fractions remain20%,30%,50%. Simplifying to the lowest whole-number ratio makes the structure clearer and prevents large numbers from cluttering the algebra. A ratio is not tied to one absolute unit size until additional numerical information is supplied.
Teaching move: Check that corresponding fractions2/10 and4/20 agree.
Mistake to avoid: Adding the same number to all entries is not generally a valid equivalence operation.
12. Combining two pairwise ratios requires matching the shared group
Suppose A:B=2:3 and B:C=4:5. The B entries differ, so scale the first ratio by four and the second by three to make B=12: A:B=8:12 and B:C=12:15. The three-part relation is A:B:C=8:12:15. The combined ratio follows matching the same physical B quantity, not simply writing2:3:5 from unrelated denominators.
Teaching move: Make B the same height on two ratio bars before joining A and C.
Mistake to avoid: Using the shared group with different numerical scales yields inconsistent comparisons.
13. A ratio can be used to compare quantities even when the total is unknown
If A:B:C=8:12:15, A is8/35 of the whole and C is15/35. We can say C is15/8 times A without knowing the actual total. However the actual numbers of objects cannot be determined without at least one numerical size or another constraint. The distinction between relative information and absolute information prevents a student from inventing counts from a ratio alone.
Teaching move: Ask what could be calculated from8:12:15 without any numerical total.
Mistake to avoid: A ratio determines proportions but not a unique physical quantity scale.
14. A transfer between groups preserves the total
A group has A40 and B60, then twenty items move from B to A. After transfer, A60 and B40, while the total remains100. A transfer is different from adding new items: one share gains what another loses. This distinction affects the algebraic equation for a target ratio and gives an immediate total-sum check. An arrow labelled ‘B→A twenty’ can make the conservation visible.
Teaching move: Use physical counters and move ten from one pile to another without creating new ones.
Mistake to avoid: A transfer must not increase the overall total unless the story also states an addition.
15. Adding to one group changes the total
If A40 and B60 and twenty new items are added to A, the final quantities are A60, B60, total120. A transfer of twenty from B to A would instead produce A60,B40,total100. The two scenarios have identical new A but different B and total; reading the verb matters more than recognising the number twenty. A good solution should first record the before-and-after values as an expression table.
Teaching move: Write ‘add’ and ‘transfer’ side by side with separate total rows.
Mistake to avoid: Using a constant-total equation in an addition problem creates an incorrect denominator.
16. A target ratio can form a linear equation
Start with A:B=2:3 and total100, so A40 and B60. Add x new items to A until A:B=1:1. Then (40+x)/60=1, giving x=20. Check final A60 and B60. This is a proportion equation, not a special new category of algebra: first reconstruct actual groups, then represent the stated change, then apply the target ratio. It is generally easier to avoid solving directly from bare2:3 without a numerical scale.
Teaching move: Set up a three-column table ‘before’, ‘change’, ‘after’ for both groups.
Mistake to avoid: A pupil who changes both A and B when only A receives additions has changed the story.
17. A target three-part ratio may involve a transfer between two groups
Suppose A:B:C begins2:3:4 with total180, giving A40, B60, C80. Move twenty from C to A, producing60,60,60 or1:1:1. The sum stays180. The result can be understood through equalising the values: A needs twenty to reach60 and C can supply exactly twenty while B stays unchanged. This type of example connects ratio with conservation and early equation modelling.
Teaching move: Calculate each original share before adjusting the two affected groups.
Mistake to avoid: A transfer between A and C does not alter the unchanged B share.
18. Changing ratio does not always mean the total changed
If a target ratio differs from the original, check whether the problem describes adding, removing, transferring or replacing items. Each operation produces a different total relation. In a transfer the original total is conserved, while in adding or removing it changes. Some puzzles ask for an unknown contribution from outside the system; others compare rearranged parts of an unchanged whole. The word problem’s physical meaning should determine the equations rather than the other way round.
Teaching move: Underline the verbs and draw arrows showing source and destination of quantities.
Mistake to avoid: A final ratio alone does not reveal which real-world operation produced it.
19. A quantity difference can remain unchanged when both groups gain equally
If A40 and B60 each receive ten new objects, the final values50 and70 keep their difference20 but change their ratio from2:3 to5:7. This shows that adding a fixed amount to two shares does not preserve their original ratio even though their absolute difference stays the same. Similarly, multiplying both shares by the same positive factor preserves ratio but scales their difference. Students should distinguish additive and multiplicative changes.
Teaching move: Compare A:B=40:60 and50:70 on two ratio bars.
Mistake to avoid: Adding the same amount to all ratio entries is not generally a valid ratio simplification.
20. A ratio does not automatically measure fairness in a moral sense
In a Mathematics problem, a ratio describes a specified allocation rule. Whether sharing2:3:5 is fair depends on context, such as the number of people, work contributed or agreed criteria. The calculation alone does not establish that a real situation should use those shares. Hypothetical examples can use tickets, beads or funds, but should avoid asserting that any particular student deserves a larger share based only on a number in a practice question.
Teaching move: Ask which rule determines the ratio before calling an allocation fair.
Mistake to avoid: A mathematically correct proportion is not automatically a justified real-world policy.
21. Percentages provide a second check on ratio sharing
For a2:3:5 split, percentages are20%,30%,50%. If a total200 is shared as40,60,100, divide each amount by200 and obtain the matching proportions. This cross-check is useful when the question includes both ratio and percentage language. It also makes it easier to compare allocations across different totals: the absolute amounts change, but the ratio percentages remain fixed when the sharing rule does not change.
Teaching move: Convert one ratio split into percentages and verify they add to100%.
Mistake to avoid: The total percentage should not be obtained by adding the raw ratio entries as percentages.
22. An answer should include actual amounts and the total check
A pupil might write ‘A receives two parts, B three, C five’ when asked for numerical shares of240. That restates the ratio but does not answer the question. The finished calculation should name48,72 and120 in the appropriate units and show their sum240. If any item represents a whole object, the unit value must allow whole counts or the original exact division may be impossible under the story. A complete answer describes who receives what and why.
Teaching move: Require one line for unit value, one for three shares and one for verification.
Mistake to avoid: A fractional count of indivisible objects must not simply be rounded while preserving a claimed exact ratio.
23. A useful error log captures the first missing comparison
A learner can make the wrong start by dividing the total by the number of people, treating a known B value as the whole, or subtracting quantities when the story says transfer. Those are distinct errors even if all result in incorrect final shares. A tutor should ask the pupil to label the whole, identify what one ratio unit means and show the changing total. The diagnostic should lead to a new problem with different numbers, not merely another copy of the same example.
Teaching move: Have the student point to which combined ratio units each given number covers.
Mistake to avoid: A calculator cannot repair a misunderstanding of what number belongs in the denominator.
Thirty worked three-part and changing-ratio examples
Worked example 1: Share200 in 2:3:5
Question: A,B,C share200 units in ratio2:3:5. Find each.
Method: Ten parts; one part20; A40,B60,C100.
Check: 40+60+100=200.
Wrong turn: Divide by ten parts, not three recipients.
Change the problem: Share250 using same ratio.
Worked example 2: Share240
Question: Share240 in ratio2:3:5.
Method: One unit24, so shares48,72,120.
Check: 48+72+120=240.
Wrong turn: Do not divide240 by two or three separately.
Change the problem: Find C as fraction of total.
Worked example 3: Known B
Question: A:B:C=2:3:5 and B=72. Find A,C,total.
Method: One part72/3=24, A48,C120,total240.
Check: Shares ratio48:72:120 simplifies2:3:5.
Wrong turn: B is three units not ten.
Change the problem: Find values when C=150.
Worked example 4: Known C
Question: Ratio2:3:5 and C=150. Find A,B.
Method: One unit150/5=30, A60,B90.
Check: A+B+C=300.
Wrong turn: Use the known five-part group to find unit.
Change the problem: What if C=75?
Worked example 5: Known A+B
Question: A:B:C=2:3:5 and A+B=80. Find total.
Method: A+B five units, so unit16. Total10×16=160.
Check: A32,B48,C80.
Wrong turn: Do not treat80 as whole.
Change the problem: Try B+C=96.
Worked example 6: Known B+C
Question: A:B:C=2:3:5 and B+C=96. Find A and total.
Method: B+C eight parts, so unit12. A24,total120.
Check: Shares24,36,60 add120.
Wrong turn: Combine group units before dividing.
Change the problem: Try A+C=84.
Worked example 7: Known difference C−A
Question: C exceeds A by45 in ratio2:3:5. Find shares.
Method: Difference5−2=3 parts=45, so unit15. Shares30,45,75.
Check: 75−30=45, total150.
Wrong turn: Do not divide45 by ten.
Change the problem: Try difference60.
Worked example 8: Known difference C−B
Question: C exceeds B by24 in ratio2:3:5. Find all.
Method: C−B two units, unit12; shares24,36,60.
Check: 60−36=24.
Wrong turn: Use difference two parts not ratio sum.
Change the problem: What if C−B=36?
Worked example 9: Three-part 3:4:5
Question: Share144 in ratio3:4:5.
Method: Twelve parts; unit12; shares36,48,60.
Check: Sum144.
Wrong turn: Write all three categories.
Change the problem: Try total180.
Worked example 10: Three-part 4:5:6
Question: Share225 in ratio4:5:6.
Method: Fifteen parts, unit15; shares60,75,90.
Check: 60+75+90=225.
Wrong turn: Do not assume unit equals first entry four.
Change the problem: Try total150.
Worked example 11: Red to blue 3:5
Question: 64 identical items are red/blue in ratio3:5. Find red.
Method: Eight parts; each8; red24, blue40.
Check: 24/40=3/5, red share24/64=3/8.
Wrong turn: Ratio red:blue differs from red:total.
Change the problem: Find red percent37.5%.
Worked example 12: Unit conversion
Question: Simplify ratio1.5 metres:75 centimetres.
Method: Convert1.5m=150cm;150:75=2:1.
Check: Two times75=150.
Wrong turn: Same units first.
Change the problem: Try2.4m:60cm.
Worked example 13: Decimal three-part
Question: Simplify0.6:0.9:1.5.
Method: Multiply by10:6:9:15; divide3 =>2:3:5.
Check: Original fractions preserve20%,30%,50%.
Wrong turn: Scale every entry equally.
Change the problem: Try0.4:0.8:1.2.
Worked example 14: Fraction three-part
Question: Simplify1/2:3/4:1.
Method: Multiply all entries by4 =>2:3:4.
Check: Original values proportional to0.5,0.75,1.
Wrong turn: Use common multiple of denominators.
Change the problem: Try1/3:1/2:1/6.
Worked example 15: Combine pairwise ratios
Question: A:B=2:3 and B:C=4:5. Find A:B:C.
Method: Make B=12: A:B=8:12; B:C=12:15. Combined8:12:15.
Check: B consistent in both.
Wrong turn: Do not join2:3:5 blindly.
Change the problem: Use total140.
Worked example 16: Use combined ratio with total
Question: For A:B:C=8:12:15, total140. Find values.
Method: Parts35; unit4. A32,B48,C60.
Check: Sum140, and A/B=2/3, B/C=4/5.
Wrong turn: Use all35 parts.
Change the problem: Try total210.
Worked example 17: Another pairwise join
Question: A:B=5:2, B:C=3:4. Find combined ratio.
Method: Make B=6. Scale first by3=>15:6, second by2=>6:8. Ratio15:6:8.
Check: A/B=15/6=5/2; B/C=6/8=3/4.
Wrong turn: Match the shared B exactly.
Change the problem: Share58 according to this.
Worked example 18: Share58 in15:6:8
Question: Share58 in ratio15:6:8.
Method: Total29 parts; unit2; shares30,12,16.
Check: 30+12+16=58.
Wrong turn: The largest group is A.
Change the problem: Try total87.
Worked example 19: Transfer B to A
Question: A40,B60. Transfer20 from B to A. Find ratio.
Method: After transfer A60,B40, ratio60:40=3:2.
Check: Total remains100.
Wrong turn: Do not add new items to total.
Change the problem: Transfer only ten instead.
Worked example 20: Add new items to A
Question: A40,B60. Add20 new items to A. Find ratio and total.
Method: A60,B60, ratio1:1,total120.
Check: Total grows by20.
Wrong turn: Do not subtract20 from B.
Change the problem: Find amount added to make ratio1:1.
Worked example 21: Find addition to reach equal
Question: A:B=2:3 and total100. How many items add to A to make1:1?
Method: Initial A40,B60. Solve40+x=60, so x20.
Check: Final A60 B60 total120.
Wrong turn: Addition and transfer are different.
Change the problem: If target ratio3:2 by adding to A, find x50.
Worked example 22: Transfer to reach3:2
Question: A40,B60 total100. Transfer x from B to A until A:B=3:2.
Method: Target total100 split3:2 =>A60,B40, so transfer20.
Check: A60+B40=100.
Wrong turn: Transfer conserves total.
Change the problem: Try final ratio4:1.
Worked example 23: Three-part transfer
Question: A:B:C=2:3:4 total180. Transfer20 from C to A.
Method: Initial40,60,80; final60,60,60 ratio1:1:1.
Check: All three sum180.
Wrong turn: B is unchanged.
Change the problem: Transfer ten instead.
Worked example 24: B gets additional20
Question: A:B:C=1:2:3 total120. Add20 to B. Find new ratio.
Method: Initial20,40,60; after20,60,60 =>1:3:3.
Check: Total becomes140.
Wrong turn: Do not call this a transfer.
Change the problem: Remove ten from C as a second change.
Worked example 25: Solve initial unknown unit from new ratio
Question: A:B initially2:3. Add15 to A; new ratio5:6. Find original A,B.
Method: Let A=2u,B=3u. (2u+15)/(3u)=5/6 =>12u+90=15u, u=30; initial60,90.
Check: After addition75:90=5:6.
Wrong turn: Do not use 15 as the size of one ratio unit.
Change the problem: Find original total150.
Worked example 26: Add to B to reach new ratio
Question: A72,B48. Add x to B to make A:B=2:3.
Method: 72/(48+x)=2/3 gives216=96+2x, x=60.
Check: Final72:108=2:3.
Wrong turn: The denominator changes but A stays72.
Change the problem: Try target3:4.
Worked example 27: Equal additions change ratio
Question: A40,B60 each receive10. Find new ratio and compare.
Method: New A50 B70, ratio5:7, not original2:3.
Check: Difference20 unchanged, total120.
Wrong turn: Adding equal amounts doesn’t preserve ratio.
Change the problem: Try multiplying both old quantities by2 instead.
Worked example 28: Multiply all groups by same factor
Question: A:B:C=2:3:5 and original shares40,60,100. Double every group.
Method: New80,120,200, same ratio2:3:5, total400.
Check: Every original share doubled.
Wrong turn: A multiplicative scale preserves ratio.
Change the problem: What if only A doubles?
Worked example 29: Fraction of whole after sharing
Question: For2:3:5, what percent of total is C?
Method: C has5 of10 units=1/2=50%.
Check: A20%,B30%,C50% total100%.
Wrong turn: Do not write C=5% of whole.
Change the problem: Find A+C share70%.
Worked example 30: Ratio cannot fix absolute count
Question: A:B:C=2:3:5 without any other quantity. Can actual counts be found?
Method: No; possible shares include2,3,5 or20,30,50 or40,60,100. Only relative shares are fixed.
Check: All examples have same ratio.
Wrong turn: Avoid inventing a total.
Change the problem: Name one additional condition that determines the scale.
Ten ratio errors worth diagnosing separately
1. Whole divided by number of recipients
Error: An allocation of200 with2:3:5 is divided by three instead of ten units. Count the ratio parts first and calculate one shared unit.
Repair: Make a ten-box diagram with groups coloured differently.
Progress check: The next total is split and checked accurately.
2. A group value is mistaken for total
Error: B=72 in2:3:5 is divided by ten, though B represents only three parts. Divide by three to recover one unit.
Repair: Underline which ratio entries the given number covers.
Progress check: A new known-subset problem is solved without prompting.
3. Part-to-whole percentage uses wrong denominator
Error: A’s2 parts are compared against B’s3 rather than total10, giving2/3 instead of20%. State ‘of what?’ and use the correct whole.
Repair: Convert all three shares into percentages that total100%.
Progress check: The child distinguishes A:B from A:total.
4. Pairwise ratios joined without matching shared group
Error: A:B=2:3 and B:C=4:5 are incorrectly combined as2:3:5. Scale the B entries to their least common multiple12 first.
Repair: Check both original pairwise ratios against combined8:12:15.
Progress check: The next pairwise join shares a consistent B value.
5. Addition confused with transfer
Error: Adding twenty to A changes total, while transferring twenty B→A preserves total. Write before/change/after rows for every group.
Repair: Show totals100 and120 in contrasting examples.
Progress check: The word problem’s verb determines the algebra.
6. Units differ when simplifying a ratio
Error: A1.5m:75cm ratio is simplified as1.5:75. Convert to150cm:75cm=2:1 first.
Repair: Practise metres-centimetres and hours-minutes pairs.
Progress check: The next ratio has compatible quantities.
7. A change is applied to every group without reason
Error: A task says add15 to A but the child also changes B. Freeze unchanged groups in a table and only modify named groups.
Repair: Use initial A60 B90 and target5:6 to check x15.
Progress check: Independent new problem changes only the specified group.
8. A ratio value is rounded to a whole object indiscriminately
Error: A fractional unit produces non-integer counts of indivisible objects, then each share is rounded separately. This can break both exact ratio and total.
Repair: Explain why the stipulated exact allocation may be infeasible for whole objects.
Progress check: The learner states a contextual restriction rather than forcing exactness.
9. Difference problem uses the sum of parts
Error: C−A=45 in2:3:5 is divided by ten instead of the difference three units. Draw unmatched C boxes.
Repair: Divide45 by3, then reconstruct C and A.
Progress check: New difference puzzle uses correct part difference.
10. No final total or ratio check
Error: Calculated shares are reported without adding to the given total or simplifying their ratio. Reconstruct both relationships.
Repair: Use one example with a deliberate wrong share and locate it.
Progress check: The final three amounts match both ratio and total.
Forty-seven quick retrieval checks
Quick check 1: 2:3:5 total parts
Answer: 10. Reason: 2+3+5.
Quick check 2: Share200 in2:3:5 A
Answer: 40. Reason: 2×20.
Quick check 3: Same B
Answer: 60. Reason: 3×20.
Quick check 4: Same C
Answer: 100. Reason: 5×20.
Quick check 5: Share240 in2:3:5 A
Answer: 48. Reason: Unit24.
Quick check 6: Same B
Answer: 72. Reason: 3×24.
Quick check 7: Same C
Answer: 120. Reason: 5×24.
Quick check 8: B72 in2:3:5 unit
Answer: 24. Reason: 72/3.
Quick check 9: A+B80 in2:3:5 unit
Answer: 16. Reason: 80/5.
Quick check 10: A+B80 total
Answer: 160. Reason: 10×16.
Quick check 11: C−A45 in2:3:5 unit
Answer: 15. Reason: 45/(5−2).
Quick check 12: C−B24 in2:3:5 unit
Answer: 12. Reason: 24/(5−3).
Quick check 13: 3:4:5 total144 A
Answer: 36. Reason: Unit12.
Quick check 14: 3:4:5 total144 C
Answer: 60. Reason: Unit12.
Quick check 15: 4:5:6 total225 B
Answer: 75. Reason: Unit15.
Quick check 16: Red:blue3:5 total64 red
Answer: 24. Reason: 3/8×64.
Quick check 17: Red:blue3:5 red whole percent
Answer: 37.5%. Reason: 3/8.
Quick check 18: 1.5m:75cm simplest
Answer: 2:1. Reason: 150:75.
Quick check 19: 0.6:0.9:1.5 simplest
Answer: 2:3:5. Reason: Scale and divide.
Quick check 20: 1/2:3/4:1 simplest
Answer: 2:3:4. Reason: Multiply4.
Quick check 21: A:B2:3 B:C4:5 combined
Answer: 8:12:15. Reason: Match B12.
Quick check 22: Combined8:12:15 total140 A
Answer: 32. Reason: Unit4.
Quick check 23: Same B
Answer: 48. Reason: 12×4.
Quick check 24: Same C
Answer: 60. Reason: 15×4.
Quick check 25: A:B5:2 B:C3:4 combined
Answer: 15:6:8. Reason: Match B6.
Quick check 26: Share58 in15:6:8 first
Answer: 30. Reason: Unit2.
Quick check 27: A40 B60 transfer20 B→A ratio
Answer: 3:2. Reason: 60:40.
Quick check 28: A40 B60 add20 to A ratio
Answer: 1:1. Reason: 60:60.
Quick check 29: A40 B60 add20 total
Answer: 120. Reason: 100+20.
Quick check 30: A40 B60 transfer20 total
Answer: 100. Reason: Conserved.
Quick check 31: Ratio2:3 total100 original A
Answer: 40. Reason: 2/5.
Quick check 32: Same original B
Answer: 60. Reason: 3/5.
Quick check 33: Add to A to make1:1
Answer: 20. Reason: 40+x=60.
Quick check 34: Transfer B→A to make3:2
Answer: 20. Reason: Final60:40.
Quick check 35: 2:3:4 total180 original A
Answer: 40. Reason: Unit20.
Quick check 36: Same B
Answer: 60. Reason: 3×20.
Quick check 37: Same C
Answer: 80. Reason: 4×20.
Quick check 38: Transfer20 C→A new ratio
Answer: 1:1:1. Reason: 60:60:60.
Quick check 39: 1:2:3 total120 original values
Answer: 20,40,60. Reason: Unit20.
Quick check 40: Add20 to B new ratio
Answer: 1:3:3. Reason: 20:60:60.
Quick check 41: A:B2:3 add15 to A to make5:6 original A
Answer: 60. Reason: u30.
Quick check 42: Same original B
Answer: 90. Reason: 3u.
Quick check 43: A72 B48 add to B to reach2:3
Answer: 60. Reason: B final108.
Quick check 44: A40 B60 each add10 new ratio
Answer: 5:7. Reason: 50:70.
Quick check 45: A40 B60 multiply both by2 ratio
Answer: 2:3. Reason: 80:120.
Quick check 46: A share of total2:3:5
Answer: 20%. Reason: 2/10.
Quick check 47: C share of total2:3:5
Answer: 50%. Reason: 5/10.
Quick check 48: Does ratio alone determine actual total?
Answer: No. Reason: Scale unknown.
Four weeks from part-whole bars to changed ratios
Week 1: rebuild the part-total ratio bar
Use visual ten-box allocations, then solve totals and one-known-share cases. Require a final sum and ratio check rather than only three numerical answers.
Independent evidence: The learner shares a changed total among three groups without dividing by the number of people.
Week 2: known combined parts and differences
Ask for amounts when A+B or C−A is known. Label the exact group subset that the given number covers before dividing into one unit.
Independent evidence: An unseen difference problem uses difference of ratio entries rather than their sum.
Week 3: connect pairwise ratios and changing groups
Combine A:B with B:C by matching their common B, then contrast transfer and addition puzzles. Use before/after tables to keep group totals consistent.
Independent evidence: The child distinguishes a conserved-total transfer from new-item addition.
Week 4: solve mixed algebraic ratio stories
Introduce target ratios after an addition or transfer and one unit-conversion case. Require checking each new ratio by simplifying actual final amounts.
Independent evidence: Independent work identifies the correct whole, feasible units, and mathematical action.
An illustrative ninety-minute, three-pupil tutorial
The original eduKateSG Clementi Mathematics reference describes a 90-minute small group with three pupils. A lesson might start with ten minutes of ratio-bar diagnosis, fifteen minutes deriving one unit, twenty minutes of shared-total and difference examples, twenty-five minutes on independent transfers and additions, ten minutes checking equivalent ratios and units, and ten minutes recording the first wrong comparison. Current lessons and vacancies require direct confirmation.
One pupil might divide the whole by three, another might confuse addition with transfer, and a third may try to combine pairwise ratios without matching their shared quantity. These errors need different explanations. Small-group teaching is effective when each child attempts the first modelling line independently and then verifies a changed problem, rather than copying one classmate’s ratio bar.
Frequently asked questions
1. How do I share a total in a three-part ratio?
Add all three ratio entries to find the number of equal parts. Divide the total by that count to find one part, then multiply by each ratio entry. Verify all shares sum to the original total.
2. What if I know one group’s amount instead of the total?
Divide the group’s amount by its corresponding ratio entry. That gives one unit and allows you to calculate every other group and the total.
3. What if the known amount is the sum of two groups?
Add those two groups’ ratio entries and divide the known subtotal by that part count. Do not automatically use the full ratio sum unless the given is the entire total.
4. How do I use a given difference between groups?
Find the difference between their ratio entries. That number of units represents the stated numerical difference. Divide to obtain one unit and calculate all shares.
5. How can I combine A:B and B:C?
Make the value of B identical in the two equivalent ratios, using a common multiple of the B entries. Then read the combined A:B:C values from the consistent scale.
6. Why do ratios with decimals or fractions need conversion?
Multiplying all entries by one suitable common positive factor can remove fractional entries without changing the proportions. Then simplify by a common divisor when possible.
7. Is a ratio the same as a percentage?
A ratio compares parts; a percentage expresses a fraction of a chosen whole per hundred. For2:3:5, A is20% of total but A:B remains2:3.
8. What changes when items move between groups?
A transfer decreases one group and increases another by the same quantity, preserving total. Adding new items changes total. The type of action must be built into the equation.
9. Can an initial ratio and a new ratio determine how many were added?
Usually additional information such as an original total or one actual group amount is needed. Otherwise the scale may be undetermined. When enough is given, write initial groups using one unit and form the target proportion after the stated change.
10. What if the result is a fraction of a person?
An exact division of indivisible objects must produce feasible whole counts. Do not silently round and still claim the original exact ratio and total are satisfied.
11. Is this a Secondary 2 G2 Mathematics topic?
Ratio and proportion form part of the G2 syllabus. Multi-part sharing and changing-ratio applications build on lower-secondary skills; schools may differ on how involved their current word problems are.
12. Can a correct ratio show that the allocation is fair?
The arithmetic confirms the specified sharing rule, but whether the rule is fair depends on the practical agreement and context. Mathematics alone cannot decide that policy question.
13. How can parents check quickly?
Ask ‘What is one ratio unit?’, ‘Which parts does the given number refer to?’, ‘Does the total change?’ and ‘Do the final shares add to the whole and simplify to the target ratio?’
The main takeaway
Did you know? A ratio is only a relationship until a total, a subgroup or a difference supplies a scale. Find the value of one conceptual unit from the correct part of the story, then calculate the shares. If items are added or moved, reflect that action precisely and check both the new total and simplified ratio. The steps are short once the mathematical whole is clear.
Related reading: Direct and Inverse Proportion Made Clear, Algebra Word Problems and Equation Modelling and Percentage Change and Reverse Percentages.
The original Clementi three-pupil Mathematics tutorial reference remains unchanged. For current lesson arrangements and suitability, contact eduKate Singapore directly. Grades, places and subject-level pathways are not guaranteed.
Explore the other three Secondary 2 G2 Mathematics guides in this batch
This four-part collection connects foundational number skills, three-part ratio sharing, algebraic-fraction addition as a later-stage bridge, and trapezium area in composite geometry. Select the guide that matches the student’s actual school work, study a complete worked method, then attempt an independently changed example. The original Clementi tutorial reference remains unchanged.
- Secondary 2 G2 Mathematics Tutor Bukit Timah | Prime Factorisation for HCF, LCM and Perfect Squares
- Secondary 2 G2 Mathematics Tutor Bukit Timah | Adding and Subtracting Algebraic Fractions with Unlike Denominators
- Secondary 2 G2 Mathematics Tutor Bukit Timah | Trapezium Area, Perpendicular Height and Composite Regions