AO3 · Ratio and proportion

Ratio Heuristics

15 question types · PSLE Paper 2 · Model Method and algebra, side by side

01

Direct Part-to-Part and Part-to-Whole Allocation

heuristicUnit Value Evaluation

The ratio of the number of boys to the number of girls in an art club was 4 : 7. There were 36 more girls than boys. (a) How many members were there in the art club altogether? (b) If each girl was given 3 paintbrushes and each boy was given 2 paintbrushes, how many paintbrushes were distributed in total?

Boysuuuu4u
Boys : girls = 4:7. Draw the boys as four equal units.
Draw Boys: 4 equal unit boxes [u][u][u][u].
step 1 of 8 Practice ratio in the app

Draw comparison unit bars for boys and girls, equate the excess units to the difference, and scale the units accordingly.

  1. Draw Boys: 4 equal unit boxes [u][u][u][u].
  2. Draw Girls: 7 identical unit boxes [u][u][u][u][u][u][u].
  3. Difference = 7 − 4 = 3 units = 36.
  4. Value of 1 unit = 36 ÷ 3 = 12.
  5. (a) Total units = 4 + 7 = 11 units = 11 × 12 = 132 members.
  6. (b) Paintbrushes for 4 units of boys = (4 × 12) × 2 = 96.
  7. Paintbrushes for 7 units of girls = (7 × 12) × 3 = 252.
  8. Total paintbrushes = 96 + 252 = 348.

Common pitfalls

  • Dividing 36 by the total number of units (11) rather than the difference in units (3).
  • Multiplying the unit value (12) directly by the combined brush rate (2 + 3 = 5) instead of multiplying boys and girls by their respective rates.
02

Repeated Identity (Common Term Bridging)

heuristicEqualising the Shared Entity

The ratio of Alicia's stickers to Brenda's stickers was 3 : 5. The ratio of Brenda's stickers to Clara's stickers was 4 : 7. Alicia had 92 fewer stickers than Clara. How many stickers did the three girls have altogether?

Alicia3 unitsBrenda5 units3 : 5 row
Alicia : Brenda = 3:5. Draw Brenda as five units.
Draw Row 1: Alicia (3 units), Brenda (5 units).
step 1 of 8 Practice ratio in the app

Construct a two-row model for Brenda, subdividing the units until both rows contain identical unit counts.

  1. Draw Row 1: Alicia (3 units), Brenda (5 units).
  2. Draw Row 2: Brenda (4 units), Clara (7 units).
  3. Cut Brenda's 5 units into 4 smaller parts each (20 parts), and Brenda's 4 units into 5 parts each (20 parts).
  4. Alicia's bar becomes 3 × 4 = 12 units.
  5. Clara's bar becomes 7 × 5 = 35 units.
  6. Gap between Clara and Alicia = 35 − 12 = 23 units = 92.
  7. 1 unit = 92 ÷ 23 = 4.
  8. Total = (12 + 20 + 35) × 4 = 67 × 4 = 268 stickers.

Common pitfalls

  • Combining the ratios directly as 3 : 5 : 7 without equalizing the common term Brenda.
  • Equating the difference of 92 to (7 − 3 = 4) units from the unscaled ratios.
03

Constant Total (Internal Transfer)

heuristicTotal Invariance

Daryl and Evan had some marbles in the ratio 7 : 5. After Daryl gave 18 marbles to Evan, the ratio of Daryl's marbles to Evan's marbles became 1 : 2. How many marbles did Daryl have at first?

BeforeDaryl 7uEvan 5u12u
Daryl : Evan = 7:5, so the total is 12 units.
Draw Total bar of 12 units: Daryl gets 7 units, Evan gets 5 units.
step 1 of 6 Practice ratio in the app

Draw a constant-length total bar split into 12 parts for the initial state and 3 big parts for the final state, then compare individual partitions.

  1. Draw Total bar of 12 units: Daryl gets 7 units, Evan gets 5 units.
  2. Draw identical Total bar divided into 3 equal blocks: Daryl gets 1 block, Evan gets 2 blocks.
  3. Convert 1 block into units: 12 ÷ 3 = 4 units.
  4. Daryl's change: from 7 units down to 4 units = 3 units.
  5. 3 units = 18 ⟹ 1 unit = 6.
  6. Daryl at first = 7 × 6 = 42 marbles.

Common pitfalls

  • Comparing the 'Before' units (7) and 'After' units (1) directly (7 − 1 = 6 units) without normalizing total units.
  • Adding 18 to Daryl instead of subtracting it from Daryl.
04

Constant One Part (Single Unchanged Quantity)

heuristicEqualising the Unchanged Quantity

The ratio of the number of fiction books to non-fiction books on a shelf was 5 : 3. After the librarian added 54 non-fiction books to the shelf and no fiction books were added or removed, the ratio of fiction books to non-fiction books became 2 : 3. How many fiction books were on the shelf?

Beforefiction 5u3u5 : 3
Before: fiction : non-fiction = 5:3.
Before: Fiction has 5 units, Non-fiction has 3 units.
step 1 of 7 Practice ratio in the app

Keep the fiction bar length strictly constant while extending the non-fiction bar to reflect the addition.

  1. Before: Fiction has 5 units, Non-fiction has 3 units.
  2. After: Fiction has 2 units, Non-fiction has 3 units.
  3. Equalize Fiction bars by subdividing: 5 units × 2 = 10 units; 2 units × 5 = 10 units.
  4. Adjust Non-fiction bars accordingly: Before = 3 × 2 = 6 units; After = 3 × 5 = 15 units.
  5. Change in Non-fiction = 15 − 6 = 9 units = 54.
  6. 1 unit = 54 ÷ 9 = 6.
  7. Fiction books = 10 × 6 = 60.

Common pitfalls

  • Assuming non-fiction units did not change because both ratios show the digit '3'.
  • Adding 54 to the total of the initial ratio units (5 + 3 = 8) without equalizing the unchanged component.
05

Constant Difference (Equal Changes / Age Progression)

heuristicDifference Invariance

Five years ago, the ratio of Mr. Wong's age to his son's age was 7 : 2. In 10 years' time from now, the ratio of Mr. Wong's age to his son's age will be 2 : 1. (a) How old is Mr. Wong's son now? (b) What is the ratio of Mr. Wong's age to his son's age now in simplest form?

5 yrs agofather 7uson 2ugap 5u
Five years ago: father 7u, son 2u. The gap is 5 units.
Past Model: Father (7 units), Son (2 units). Difference = 5 units.
step 1 of 9 Practice ratio in the app

Draw comparison bars where the difference segment between Father and Son remains fixed in length, then observe the unit elongation.

  1. Past Model: Father (7 units), Son (2 units). Difference = 5 units.
  2. Future Model: Father (2 parts), Son (1 part). Difference = 1 part.
  3. To align difference, cut each part into 5 units: Father = 10 units, Son = 5 units.
  4. Each person gains: 10 − 7 = 3 units.
  5. Years passed = 5 + 10 = 15 years.
  6. 3 units = 15 ⟹ 1 unit = 5 years.
  7. Son's age 5 years ago = 2 × 5 = 10.
  8. (a) Son's age now = 10 + 5 = 15 years old.
  9. (b) Father now = 35 + 5 = 40. Ratio = 40 : 15 = 8 : 3.

Common pitfalls

  • Calculating the time interval as 10 − 5 = 5 years instead of 5 + 10 = 15 years.
  • Forgetting to add 5 years back to find the present age, reporting the age from 5 years ago as the final answer.
06

Chained Three-Party Ratios with Sub-Proportions

heuristicSub-Unit Branching

At an electronics fair, the ratio of the number of laptops to tablets to smartphones was 3 : 4 : 8. 13 of the laptops were Brand X and the rest were Brand Y. Half of the tablets were Brand X and the rest were Brand Y. None of the smartphones were Brand X. If there were 180 more Brand Y devices than Brand X devices, how many total devices were at the fair?

LaptopsXYY
Laptops are 3 blocks; 13 of them Brand X, so 1 block X and 2 Y.
Draw Laptops as 3 blocks: 1 block Brand X, 2 blocks Brand Y.
step 1 of 9 Practice ratio in the app

Represent each device category as discrete unit blocks, shade the Brand X portions, and compare total shaded versus unshaded blocks.

  1. Draw Laptops as 3 blocks: 1 block Brand X, 2 blocks Brand Y.
  2. Draw Tablets as 4 blocks: 2 blocks Brand X, 2 blocks Brand Y.
  3. Draw Smartphones as 8 blocks: all 8 blocks Brand Y.
  4. Sum Brand X blocks: 1 + 2 = 3 blocks.
  5. Sum Brand Y blocks: 2 + 2 + 8 = 12 blocks.
  6. Difference in blocks = 12 − 3 = 9 blocks = 180.
  7. 1 block = 180 ÷ 9 = 20.
  8. Grand total blocks = 3 + 4 + 8 = 15 blocks.
  9. Total devices = 15 × 20 = 300.

Common pitfalls

  • Applying fractions across the entire total of 15 units rather than to their specific individual category blocks.
  • Excluding smartphones from the Brand Y total, forgetting that having zero Brand X makes all smartphones Brand Y.
07

Number × Value in Ratio Form (Composite Grouping)

heuristicGrouping into 1 Composite Unit

In a charity box, the ratio of the number of $2 notes to $5 notes was 5 : 3, and the ratio of the number of $5 notes to $10 notes was 2 : 1. The total value of all the notes in the box was $1280. (a) How many $5 notes were in the box? (b) What was the total value of the $10 notes?

$2 : $5 = 5 : 3 and $5 : $10 = 2 : 1 → $5 notes as 6$2$2$2$2$2$2$2$2$2$210 × $2$5$5$5$5$5$56 × $5$10$10$103 × $10
Two ratios share the $5 notes: 5:3 and 2:1. Make the $5 count the same in both, 6.
Equalize $5 notes to get 1 base set:
step 1 of 6 Practice ratio in the app

Form a single representative group containing 10 two-dollar notes, 6 five-dollar notes, and 3 ten-dollar notes, then find how many such groups exist.

  1. Equalize $5 notes to get 1 base set:
  2. 1 Set contains: 10 of $2 notes, 6 of $5 notes, 3 of $10 notes.
  3. Value of 1 set = (10 × $2) + (6 × $5) + (3 × $10) = $20 + $30 + $30 = $80.
  4. Total sets = $1280 ÷ $80 = 16 sets.
  5. (a) Number of $5 notes = 16 × 6 = 96 notes.
  6. (b) Total value of $10 notes = 16 × (3 × $10) = 16 × $30 = $480.

Common pitfalls

  • Dividing $1280 by the sum of the ratio numbers (10 + 6 + 3 = 19) instead of their monetary values.
  • Multiplying the final set count by note denominations directly without scaling by the ratio quantities.
08

Simultaneous Units and Parts / Everything Changed

heuristicCross-Multiplication / Units and Parts System

The ratio of Zachary's savings to Hannah's savings was 4 : 5. After Zachary spent $45 and Hannah spent $100, the ratio of Zachary's remaining savings to Hannah's remaining savings became 3 : 2. Find Zachary's savings at first.

Zachary4uHannah5u
Before: 4u and 5u.
Before: Zachary = 4u, Hannah = 5u.
step 1 of 9 Practice ratio in the app

Represent before states with units (u) and after states with parts (p). Create simultaneous unit comparisons to eliminate parts.

  1. Before: Zachary = 4u, Hannah = 5u.
  2. After: Zachary = 3p, Hannah = 2p.
  3. Equation 1: 4u − 45 = 3p.
  4. Equation 2: 5u − 100 = 2p.
  5. Scale Eq 1 by 2: 8u − 90 = 6p.
  6. Scale Eq 2 by 3: 15u − 300 = 6p.
  7. Equate both expressions for 6p: 15u − 300 = 8u − 90.
  8. 7u = 210 ⟹ u = 30.
  9. Zachary at first = 4u = 4 × $30 = $120.

Common pitfalls

  • Assuming the unit difference (5u − 4u = u) corresponds to the difference in expenditure ($100 − $45 = $55).
  • Subtracting ratio numbers directly: (4 − 3) : (5 − 2) = 1 : 3 and attempting to equate units.
09

Three-Party Closed System Internal Transfer

heuristicMulti-Entity Total Invariance

Initially, the ratio of Alan's tokens to Bryan's tokens to Colin's tokens was 3 : 4 : 5. Alan gave 20 tokens to Bryan, and Colin gave 30 tokens to Alan. In the end, the ratio of Alan's tokens to Bryan's tokens to Colin's tokens became 2 : 3 : 1. How many tokens were in the box altogether?

BeforeAlan 3uBryan 4uColin 5u12u
Before: 3:4:5, a total of 12 units.
Before model: Alan (3u), Bryan (4u), Colin (5u) → Total = 12u.
step 1 of 7 Practice ratio in the app

Draw total invariant bars of 12 units for both states. Trace the entity that experienced only a single one-way transfer to determine the unit value.

  1. Before model: Alan (3u), Bryan (4u), Colin (5u) → Total = 12u.
  2. After model: Alan (2p), Bryan (3p), Colin (p) → Total = 6p.
  3. Standardize After bar into 12 units by multiplying each part by 2:
  4. After: Alan (4u), Bryan (6u), Colin (2u).
  5. Inspect Colin's bar: dropped from 5u to 2u (a loss of 3u).
  6. Since Colin only gave away 30 tokens: 3u = 30 ⟹ u = 10.
  7. Total tokens = 12u = 12 × 10 = 120.

Common pitfalls

  • Attempting to track Alan first, which involves two separate transactions (+30 and -20), increasing the likelihood of arithmetic errors.
  • Failing to standardize the total units before comparing individual changes.
10

Two-Way Offset Balancing (Opposite Direction Changes)

heuristicLinear Proportion Re-balancing

At a warehouse, the ratio of cartons of milk to cartons of juice was 5 : 2. After 35 cartons of milk were dispatched and 10 cartons of juice were added, the ratio of milk cartons to juice cartons became 3 : 2. How many cartons of milk were there at first?

Milk5uJuice2u
Before: milk 5u, juice 2u.
Before: Milk = 5u, Juice = 2u.
step 1 of 8 Practice ratio in the app

Draw bars showing milk decreasing and juice increasing, scale the final ratio parts to find common comparisons.

  1. Before: Milk = 5u, Juice = 2u.
  2. After: Milk = 3p = 5u − 35, Juice = 2p = 2u + 10.
  3. Multiply Juice equation by 3 and Milk equation by 2 to make parts equal to 6p:
  4. 2(5u − 35) = 6p ⟹ 10u − 70 = 6p.
  5. 3(2u + 10) = 6p ⟹ 6u + 30 = 6p.
  6. Equate: 10u − 70 = 6u + 30.
  7. 4u = 100 ⟹ u = 25.
  8. Milk at first = 5u = 5 × 25 = 125.

Common pitfalls

  • Treating both changes as reductions (writing 2u − 10 instead of 2u + 10).
  • Sign errors during transposition (e.g., writing 30 − 70 = −40 instead of 30 + 70 = 100).
11

Fractional Subsets and Residual Ratio Comparison

heuristicCommon Sub-Unit Partitioning

Lucas and Marcus had pocket money in the ratio 7 : 4. Lucas spent 35 of his money, and Marcus spent 12 of his money. Lucas had $48 more left than Marcus. (a) How much money did Lucas have at first? (b) What was the ratio of the total amount spent to the total amount left?

Lucas7 blocksMarcus4 blocks
Lucas : Marcus = 7:4, seven blocks against four.
Represent Lucas with 7 blocks and Marcus with 4 blocks.
step 1 of 10 Practice ratio in the app

Construct branching bars with subdivisions matching the fraction denominators, then compare the residual blocks.

  1. Represent Lucas with 7 blocks and Marcus with 4 blocks.
  2. Subdivide Lucas's 7 blocks into 5 parts each = 35 small units.
  3. Subdivide Marcus's 4 blocks into 5 parts each = 20 small units.
  4. Lucas leaves 25 × 35 = 14 units.
  5. Marcus leaves 12 × 20 = 10 units.
  6. Surplus of Lucas over Marcus = 14 − 10 = 4 units = $48.
  7. 1 unit = $48 ÷ 4 = $12.
  8. (a) Lucas at first = 35 × $12 = $420.
  9. (b) Spent units = (35 − 14) + (20 − 10) = 21 + 10 = 31 units.
  10. Left units = 14 + 10 = 24 units. Ratio = 31 : 24.

Common pitfalls

  • Applying Marcus's 12 to Lucas's money or to the total money.
  • Subtracting the fractions directly: 35 − 12 = 110 and equating that to $48.
12

Dual Mutual Exchange (Two-Way Sequential Pouring)

heuristicWorking Backwards / Multi-Stage Tracking

Container X and Container Y contained water in the ratio 5 : 3. First, 14 of the water in Container X was poured into Container Y. Then, 15 of the new amount of water in Container Y was poured back into Container X. In the end, Container X had 144 ml more water than Container Y. How much water was in Container X at first?

StartX 100uY 60u160u
Water only moves between X and Y, so the total bar never changes length.
Total volume is invariant throughout.
step 1 of 9 Practice ratio in the app

Work backwards from the final state using the constant total volume of the closed two-container system.

  1. Total volume is invariant throughout.
  2. End state: Container X has 144 ml more than Container Y.
  3. Before the second transfer, Container Y gave 15 of its water to X, leaving 45 in Y.
  4. Step forward systematically with scaled units: let initial X = 100u, Y = 60u.
  5. After first pour: X = 75u, Y = 85u.
  6. After second pour: Y = 85u × 45 = 68u, X = 160u − 68u = 92u.
  7. Difference = 92u − 68u = 24u = 144 ml.
  8. u = 6 ml.
  9. Container X at first = 100 × 6 = 600 ml.

Common pitfalls

  • Taking 15 of Container Y's *initial* volume (60u) instead of its *new* volume after receiving water from X (85u).
  • Assuming the final difference is simply the difference between the two transferred amounts.
13

Inverse Proportionality and Product Invariance

heuristicEqual Product Equating

Gear A, Gear B, and Gear C are meshed together in a row. The ratio of the number of teeth on Gear A to Gear B is 2 : 3, and the ratio of the number of teeth on Gear B to Gear C is 4 : 5. When Gear A makes 90 complete revolutions, how many complete revolutions does Gear C make?

A8 teethB12 teethC15 teethA: 90 rev B: 60 rev C: 48 rev
Teeth 2:3 and 4:5 share B; make B 12: A = 8, B = 12, C = 15 teeth.
Harmonize teeth count: A = 8 units, C = 15 units.
step 1 of 5 Practice ratio in the app

Set up an inverse ratio: the gear with more teeth turns fewer times in exact inverse proportion.

  1. Harmonize teeth count: A = 8 units, C = 15 units.
  2. Because teeth meshed is constant: RevA × TeethA = RevC × TeethC.
  3. Ratio of revolutions RevA : RevC is the inverse of teeth ratio: 15 : 8.
  4. 15 units of revolutions = 90 revs ⟹ 1 unit = 90 ÷ 15 = 6 revs.
  5. Revolutions for Gear C = 8 units = 8 × 6 = 48 revolutions.

Common pitfalls

  • Treating revolutions as directly proportional to teeth: setting RevC90 = 158, which would mean larger gears turn faster.
  • Multiplying revolutions across intermediate gears without accounting for common contact points.
14

Weighted Multi-Category Supposition in Ratio Form

heuristicUnit Batch Value Evaluation

A bookstore sold highlighters, pens, and markers. The ratio of the number of highlighters sold to pens sold was 3 : 4, and the ratio of pens sold to markers sold was 2 : 3. A highlighter cost $1.50, a pen cost $2.00, and a marker cost $3.50. The shopkeeper collected $65 more from the sale of markers than from the sale of pens. (a) How many pens were sold? (b) How much money was collected from all three items altogether?

1.501.501.503 highlighters2.002.002.002.004 pens3.503.503.503.503.503.506 markers
Pens appear in both ratios (3:4 and 2:3); make them 4 in both. One set: 3 highlighters, 4 pens, 6 markers.
1 Set contains: 3 highlighters, 4 pens, 6 markers.
step 1 of 8 Practice ratio in the app

Define 1 set containing 3 highlighters, 4 pens, and 6 markers. Find the revenue difference within 1 set and scale up to find the number of sets.

  1. 1 Set contains: 3 highlighters, 4 pens, 6 markers.
  2. Cost of pens in 1 set = 4 × $2.00 = $8.00.
  3. Cost of markers in 1 set = 6 × $3.50 = $21.00.
  4. Difference in 1 set = $21.00 − $8.00 = $13.00.
  5. Number of sets = $65 ÷ $13 = 5 sets.
  6. (a) Pens sold = 5 sets × 4 pens/set = 20 pens.
  7. Total money in 1 set = (3 × 1.50) + 8.00 + 21.00 = 4.50 + 8.00 + 21.00 = $33.50.
  8. (b) Grand total revenue = 5 × $33.50 = $167.50.

Common pitfalls

  • Equating the $65 difference directly to the quantity difference (6 − 4 = 2) units without factoring in item prices.
  • Omitting the price of highlighters when calculating total collection.
15

Multi-Tiered Dual-Subgroup Demographic Shift

heuristicHierarchical Units and Remainder Re-scaling

At a funfair, the ratio of the number of adults to children was 3 : 5. Among the children, the ratio of the number of boys to girls was 1 : 3. Later, 33 adults and 7 boys entered the funfair, while 6 girls left the funfair. In the end, the ratio of the number of adults to children became 1 : 1, and the ratio of the number of boys to girls became 1 : 2. How many children were at the funfair at first?

Adults3 unitsChildren5 units
Adults : children = 3:5.
Draw Adults as 3 units and Children as 5 units.
step 1 of 8 Practice ratio in the app

Construct a two-tier bar model: Tier 1 partitions the whole into adults and children; Tier 2 subdivides the children bar. Solve using the isolated subgroup change before confirming against the overall total.

  1. Draw Adults as 3 units and Children as 5 units.
  2. Subdivide each unit into 4 mini-units:
  3. Adults = 12 mini-units, Children = 20 mini-units.
  4. Children bar is split into Boys (5 mini-units) and Girls (15 mini-units).
  5. Focus on children's final ratio: End Boys : End Girls = p : 2p.
  6. Set up relationship: 2 × (5u + 7) = 15u − 6.
  7. 10u + 14 = 15u − 6 ⟹ 5u = 20 ⟹ u = 4.
  8. Calculate total initial children: 20u = 20 × 4 = 80 children.

Common pitfalls

  • Applying the initial 1 : 3 boy-to-girl ratio to the entire population of adults and children combined.
  • Ignoring the change in girls (subtracting 6) and adding all changes together.
  • Failing to scale the primary ratio (3 : 5) by 4, leading to cumbersome fractional units (1.25u and 3.75u).