PSLE Mathematics Learning Guide · Guide 62 · Wintour V1.0 Companion to Guide 3
Return to the PSLE Learning Guide · Parent guide: Ratio Units and Actual Quantities
“Divide 96 in the ratio 3:5” is not asking you to divide 96 by 3 and then by 5. The ratio 3:5 describes eight equal ratio units altogether. The total quantity belongs to all eight units.
The robust method is:
add the ratio units → match the total to all units → find one unit → scale each share → verify the shares rebuild the whole.
This is a narrow companion to Guide 3. The parent guide explains ratio as a relationship and how totals, differences or one known group fix the scale. This page isolates one current Primary 6 job: dividing a quantity in a given ratio.
Public practice pages often reduce this to “add, divide, multiply”. eduKate’s missing diagnostic layer is to make clear why each step happens and how to detect wrong allocations before they become finished answers.
Two-part ratio: the complete method
Divide $96 in the ratio 3:5.
Total ratio units = 3+5 = 8.
One ratio unit = 96÷8 = 12.
First share = 3×12 = $36.
Second share = 5×12 = $60.
Check: 36+60=96 and 36:60 simplifies to 3:5.
Why the ratio terms are added
The total is being split across both groups. If A:B=3:5, A occupies 3 equal units and B occupies 5 equal units, so the whole consists of 8 equal units.
Using only 5 units for the total would pretend the total belongs only to B.
The operation “add the ratio” is not a memorised trick. It comes from matching the total to the combined ratio model.
Each ratio share is also a fraction of the total
For 3:5:
A is 3/(3+5)=3/8 of the whole.
B is 5/8 of the whole.
Therefore:
A = 3/8×96 = 36.
B = 5/8×96 = 60.
The ratio-unit method and the fraction-of-whole method are the same structure written differently.
Three-part ratio uses the same logic
Divide 180 in the ratio 2:3:4.
Total units = 2+3+4 = 9.
One unit = 180÷9 = 20.
Shares:
- 2 units = 40;
- 3 units = 60;
- 4 units = 80.
Check: 40+60+80=180 and 40:60:80 simplifies to 2:3:4.
Keep labels attached to the ratio terms
If Ali:Ben=3:5, write:
Ali → 3 units.
Ben → 5 units.
Then allocate.
A mathematically correct pair 36 and 60 can still answer the question wrongly if the names are reversed.
Money context
$245 is shared between Mei and Hana in the ratio 2:5.
Total units=7.
One unit=$35.
Mei=$70.
Hana=$175.
Check: 70+175=245.
Mass context
4.8 kg of material is divided between A and B in ratio 3:5.
Total units=8.
One unit=4.8÷8=0.6 kg.
A=1.8 kg; B=3.0 kg.
Unit consistency matters: the ratio has no measurement unit, but the allocated shares inherit kg from the total.
Length context
A 210 cm ribbon is cut into two parts in ratio 4:3.
Total units=7.
One unit=30 cm.
Lengths: 120 cm and 90 cm.
The larger ratio term must correspond to the larger length.
Discrete quantities can require divisibility checks
Divide 50 counters in the ratio 2:3.
Total units=5.
One unit=10 counters.
Shares=20 and30.
Everything works exactly.
Now suppose the total were 52 counters. One unit would be 10.4 counters, impossible if counters must remain whole and the ratio is exact. That signals a mismatch between the stated exact ratio and the discrete total.
Do not blindly round the shares; rounding can destroy both the total and the ratio.
Continuous quantities can have decimal shares
5 L divided in ratio 2:3 gives one unit=1 L and shares2 L,3 L.
7 L divided in ratio 2:3 gives one unit=1.4 L and shares2.8 L,4.2 L.
Decimals are valid because liquid can be divided continuously.
The ratio predicts the difference between shares
In ratio 3:5, the difference is 2 ratio units.
If one unit=12, the difference should be24.
Shares36 and60 differ by24.
This gives a second independent check beyond adding back to the total.
Equal ratio terms produce equal shares
Divide 84 in ratio 1:1.
Total units=2.
One unit=42.
Shares42 and42.
Ratio 1:1 is simply equal sharing.
Equivalent ratios give the same allocation
Ratio 6:10 simplifies to3:5.
Dividing96 in6:10:
Total units16; one unit6; shares36 and60.
Dividing96 in3:5:
Total units8; one unit12; shares36 and60.
The internal unit size changes, but the actual allocation is the same.
Do not simplify the actual allocated quantities as if they were ratio terms
If the shares are36 kg and60 kg, the actual masses remain36 kg and60 kg. Their relationship simplifies to3:5, but the quantities themselves do not become3 kg and5 kg.
A known share can reveal the total
A:B=2:7 and A receives18.
2 units=18, so1 unit=9.
B=63.
Total=81.
This is the reverse of allocation from a known total and is handled by the parent ratio guide.
A changed total requires a new allocation state
A total of120 is divided3:5, giving45 and75.
If the total later becomes160 but the ratio remains3:5, one unit becomes20 and the new shares are60 and100.
Do not carry the old unit value15 into the new total.
If the ratio itself changes, rebuild from the actual new conditions
Suppose45 and75 are the old shares, then15 is added to the first share. New actual values are60 and75, giving ratio4:5.
The old3:5 ratio no longer applies. Ratio is a state description, not a permanent label.
Nested allocation needs state tracking
240 is divided between A and B in ratio3:5. A then divides its share between X and Y in ratio1:2.
First allocation:
A=3/8×240=90.
B=150.
Second allocation uses A’s90 as the new whole:
X=1/3×90=30.
Y=60.
Do not divide the original240 directly by1:2 for the second stage.
Ratio allocation can be read as percentage allocation
Ratio2:3 means40% and60% of the whole.
So dividing250 in2:3 gives100 and150.
This connects ratio with percentage while keeping the same whole.
Unknown total can be represented algebraically
If total=T and ratio3:7:
first share=3/10 T.
second share=7/10 T.
The expressions add toT.
This is an extension in representation, not a requirement to use algebra for every ratio allocation.
Main worked workshop: twenty original ratio-allocation tasks
1.
Divide72 in ratio1:2.
Answer:24,48.
2.
Divide96 in ratio3:5.
Answer:36,60.
3.
Divide140 in ratio2:5.
Answer:40,100.
4.
Divide180 in ratio2:3:4.
Answer:40,60,80.
5.
Divide330 in ratio3:4:4.
Answer:90,120,120.
6.
$420 shared in ratio5:2.
Answer:$300,$120.
7.
6.4 kg divided3:5.
Answer:2.4 kg,4.0 kg.
8.
9 L divided4:5.
Answer:4 L,5 L.
9.
250 cm divided2:3.
Answer:100 cm,150 cm.
10.
A:B=7:3, total200. Find A.
Answer:140.
11.
A:B:C=1:2:7, total500. Find C.
Answer:350.
12.
A learner divides96 by3 and5 separately, obtaining32 and19.2.
Repair:96 belongs to all8 units; one unit12; shares36,60.
13.
A learner gets36 and60 but assigns60 to the3-part group.
Repair: keep labels attached;3 units=36,5 units=60.
14.
52 indivisible counters in exact ratio2:3. Is an exact allocation possible with whole counters?
Answer: no;52 is not divisible into5 equal ratio units.
15.
7.5 kg in ratio2:3.
Answer:3 kg,4.5 kg.
16.
Ratio6:9, total150.
Answer: simplify2:3; shares60,90.
17.
Ratio4:4:2, total250.
Answer:100,100,50.
18.
A:B=3:5, total96. Find difference.
Answer:24.
19.
A:B=2:3, total250. Express each as percentage.
Answer:40%,60%; shares100,150.
20.
A receives90 from a total divided3:5. Find total.
Answer:3 units=90,1 unit30,total8 units=240.
Diagnose the first weak link
Divides total by one ratio term. Total-to-all-units mapping error.
Correct shares, wrong labels. Ratio-order mapping error.
Shares do not sum to total. Arithmetic or scaling error.
Shares sum correctly but simplify to wrong ratio. Allocation error.
Rounds discrete shares to force integers. Exact-ratio feasibility error.
Four checks before accepting an allocation
- Total: do shares add back to the whole?
- Ratio: do shares simplify to the stated ratio?
- Order: is the larger term attached to the correct group?
- Feasibility: if objects are indivisible, are all exact shares whole counts?
Independent transfer check
- Divide132 in ratio5:6.
- Divide270 in ratio2:3:4.
- $560 is shared7:1. Find both shares.
- 8.4 L is divided3:4. Find both amounts.
- 240 pupils are split3:5. Find the smaller group.
- A:B:C=4:3:5, total360. Find B.
- Explain why total96 in ratio3:5 corresponds to8 ratio units, not5.
- Explain why36:60 verifies a3:5 allocation.
- An exact2:3 split is required for53 indivisible tokens. Explain the problem.
- Divide a totalT in ratio1:4 as fractions ofT.
Independent-check answers
1.60,72.
2.60,90,120.
3.$490,$70.
4.3.6 L,4.8 L.
5.90 pupils.
6.90.
7. The total contains both groups, so all3+5 units represent the whole.
8.36:60 divides by12 to3:5.
9.53 cannot be partitioned into5 equal integer ratio units, so an exact whole-token allocation is impossible.
10.T/5 and4T/5.
Wintour V1.0 return test
Change total, units and surface context but keep the same ratio. Then change the ratio while keeping the total. Finally move from direct allocation to reverse reconstruction from one known share.
A learner has portable ratio control when the method is chosen from the structure—what all ratio units represent—not from memorising “add-divide-multiply”.
The learner’s final card
What does each ratio term label? How many ratio units make the whole? What is one unit worth? What is each share? Do the shares add back to the total and simplify back to the stated ratio?
Continue through Batch 16
Use Guide 61: Part to Percentage. Continue with Guide 63: Missing Terms in Equivalent Ratios and Guide 64: Decimal Division and Rounding Accuracy.
Return to the PSLE Learning Guide Mathematics route.
Sources and boundaries
MOE Primary Mathematics syllabus, updated October 2025; SEAB 2026 PSLE formats.
All contexts and worked solutions are original eduKate teaching material. This page is a diagnostic companion to the existing ratio-unit owner and isolates direct allocation of a total in a stated ratio.