Dividing a Quantity in a Given Ratio — Solutions

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1. Divide each quantity in ratio

   1. 60 in 2:3: 24 and 36
   2. $80 in 3:1: $60 and $20
   3. 150 m in 1:4: 30 m and 120 m
   4. $200 in 3:2: $120 and $80
   5. 90 kg in 5:4: 50 kg and 40 kg
   6. $720 in 5:3: $450 and $270
   7. 240 mL in 1:5: 40 mL and 200 mL
   8. $1000 in 7:3: $700 and $300

2. Three-way ratio division

   1. 240 in 1:2:3: 40, 80, 120
   2. $360 in 2:3:4: $80, $120, $160
   3. 180 cm in 1:2:3: 30 cm, 60 cm, 90 cm
   4. $500 in 1:4:5: $50, $200, $250
   5. 120 kg in 3:4:5: 30 kg, 40 kg, 50 kg
   6. $900 in 2:3:4: $200, $300, $400
   7. 1000 mL in 1:3:6: 100 mL, 300 mL, 600 mL
   8. $280 in 1:2:4: $40, $80, $160

3. Find the original total

   1. Sam $45 in 3:2: 1 part = $15, total = $75
   2. Alex $60 in 3:1: 1 part = $20, total = $80
   3. Smaller share $16 in 2:5: 1 part = $8, total = $56
   4. Larger share $80 in 4:1: 1 part = $20, total = $100
   5. Smaller share 30 g in 3:7: 1 part = 10 g, total = 100 g
   6. 40 km in 5:3: This is the larger (5-part) share; 1 part = 8 km, total = 64 km
   7. Middle share $120 in 2:3:5: 1 part = $40, total = $400
   8. Smaller share $25 in 1:4: Larger share = $100

4. Real context ratio division

   1. $500 in 3:2: $300 and $200
   2. 240 cm in 5:3: 150 cm and 90 cm
   3. $84 000 in 5:2: $60 000 and $24 000
   4. $90 000 in 2:3:4: $20 000, $30 000, $40 000
   5. 350 g in 4:2:1: 200 g flour, 100 g sugar, 50 g butter
   6. $2400 in 3:2:1: Sport $1200, Arts $800, Science $400
   7. $360 in 5:4: $200 and $160
   8. 60 L in 1:5: 10 L cordial, 50 L water

5. Compare ratio to percentage

   1. $200 split 60%:40% = ratio 3:2; shares = $120 and $80: Ratio 3:2; $120 and $80
   2. $150 in 2:3: 40% and 60%
   3. Alice 25%, Ben 75% of $600: Ratio 1:3; Alice $150, Ben $450
   4. $80 in 1:3 vs 25%/75%: Yes — ratio 1:3 means 1/4 and 3/4, which equals 25% and 75%. Both give $20 and $60.
   5. 30 students in 2:3: 12 students (40%) and 18 students (60%)
   6. $480 in 5:7: Smaller share = $200; fraction = 5/12 ≈ 41.7%

6. Multi-step ratio problems

   1. Inheritance ratio 3:4:5, eldest gets $3000 more than youngest: Difference = 2 parts = $3000, so 1 part = $1500. Total = 12 parts = $18 000. Shares: $4500, $6000, $7500
   2. Prize 5:3:2, 1st place $1200 more than 3rd: Difference = 3 parts = $1200, so 1 part = $400. Total = 10 parts = $4000
   3. Mixture 2:5, total 350 mL, costs: Solution A = 100 mL, Solution B = 250 mL; Cost = 100×$0.80 + 250×$0.50 = $80 + $125 = $205
   4. Profit in 3:2, owners get $48 000: 1 part = $24 000; Reinvested = 3 × $24 000 = $72 000; Total profit = 5 × $24 000 = $120 000

7. Share $240 in ratio 3:5 — guided bar diagram

   1. Complete working: Total parts = 3+5 = 8; Each part = $240÷8 = $30; Share A = 3×$30 = $90; Share B = 5×$30 = $150; Check: $90+$150 = $240 ✓
   2. Fraction for Share A: 3/8
   3. Percentage for Share B: 5/8 = 62.5%

8. Ratio bar 1:2:3 for 120 mL

   1. Total parts: 1 + 2 + 3 = 6 parts
   2. Value of each part: 120 ÷ 6 = 20 mL per part
   3. Volumes: Part A = 1×20 = 20 mL; Part B = 2×20 = 40 mL; Part C = 3×20 = 60 mL
   4. Check: 20 + 40 + 60 = 120 mL ✓

9. Alice, Ben, Carlos in ratio 2:3:4 — $180 budget

   1. Total parts: 2 + 3 + 4 = 9 parts
   2. Value of each part: $180 ÷ 9 = $20 per part
   3. Contributions: Alice = $40, Ben = $60, Carlos = $80
   4. Is Carlos’s share twice Alice’s?: Carlos = $80, Alice = $40; $80 = 2 × $40 — Yes, this is correct. The ratio 4:2 = 2:1, so Carlos contributes twice as much as Alice.
   5. With $270 budget: 1 part = $270÷9 = $30; Alice = $60, Ben = $90, Carlos = $120

10. Working backwards — reverse ratio

    1. Ratio 3:7, larger share $105: Larger is 7 parts → 1 part = $15; Smaller = 3×$15 = $45; Total = 10×$15 = $150
    2. $180 and $120 as simplest ratio: HCF(180,120) = 60; ratio = 3:2
    3. Ali gets $80 more than Beth, total $560: Let Beth = x; Ali = x+80; 2x+80=560 → x=240. Beth=$240, Ali=$320. Ratio = 320:240 = 4:3
    4. Red:blue = 2:5, blue = 35 tins: 1 part = 35÷5 = 7 tins; Red = 2×7 = 14 tins; Total = 14+35 = 49 tins