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Ratio · before and after with multiples

P6 · Ratio · PSLE Maths

Question

At first, the ratio of Ali's money to Raju's money was 5 : 3. Later, Ali spent $24 on school supplies, while Raju received $36 from his aunt. After that, the ratio of Ali's money to Raju's money became 2 : 3. How much money did Ali have at first?

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Hint

Draw the 5 units and 3 units at first. The final ratio is 2 : 3, so 3 of Ali's final amount equals 2 of Raju's final amount.

Show the answer and step-by-step explanation

Answer: Ali at first 80 $

  1. Draw the ratio — At first, Ali : Raju = 5 : 3. Draw Ali as 5 equal units and Raju as 3 equal units.
  2. Write final amounts — Ali spent $24, so his final amount is 5 units − $24. Raju received $36, so his final amount is 3 units + $36.
  3. Use the new ratio — Finally, Ali : Raju = 2 : 3. 3 times Ali's final amount gives 6 parts, and 2 times Raju's final amount also gives 6 parts, so they are equal.
  4. Compare multiples — Now draw both sides: 3 of Ali's final amount = 15 units − $72; 2 of Raju's final amount = 6 units + $72.
  5. Find 1 unit — These two final totals are equal. The top has 15 units but is $72 less; the bottom has 6 units but is $72 more. So the extra 9 units are worth $72 + $72 = $144.
  6. Find Ali at first — Ali had 5 units at first: 5 × $16 = $80. Check: Ali final $56, Raju final $84, and 56 : 84 = 2 : 3. Correct. 📌 Key idea: For a final ratio of 2 : 3, compare 3 of the first amount with 2 of the second amount because both make 6 parts.

Method: Use a bar model for the ratio at first. For the new ratio after changes, multiply both final amounts to make equal parts, e.g. for 2 : 3, compare 3 of the first amount with 2 of the second amount.

Practice: 3 similar questions

  1. At first, the ratio of Mei Ling's money to Siti's money was 4 : 3. Later, Mei Ling spent $18, while Siti received $24. After that, the ratio of Mei Ling's money to Siti's money became 2 : 3. How much money did Mei Ling have at first?
    Answer

    68 $ — Let 1 unit be the same amount of money. At first, Mei Ling had 4 units and Siti had 3 units. Finally: Mei Ling = 4 units − 18, Siti = 3 units + 24. Since the final ratio is 2 : 3, 3 × Mei Ling's final amount = 2 × Siti's final amount. 3 × (4 units − 18) = 12 units − 54; 2 × (3 units + 24) = 6 units + 48. 12 units − 54 = 6 units + 48, so 6 units = 102 and 1 unit = 17. Mei Ling had 4 × 17 = $68 at first.

  2. At first, the ratio of Raju's money to Siti's money was 7 : 4. Later, Raju donated $42, while Siti saved another $18. After that, the ratio of Raju's money to Siti's money became 3 : 2. How much money did Siti have at first?
    Answer

    276 $ — At first, Raju had 7 units and Siti had 4 units. Finally: Raju = 7 units − 42, Siti = 4 units + 18. The final ratio is 3 : 2, so 2 × Raju's final amount = 3 × Siti's final amount. 2 × (7 units − 42) = 14 units − 84; 3 × (4 units + 18) = 12 units + 54. 14 units − 84 = 12 units + 54, so 2 units = 138 and 1 unit = 69. Siti had 4 × 69 = $276 at first.

  3. At first, the ratio of Ali's money to Raju's money was 5 : 2. Ali gave Raju $44. After that, Raju spent $7 on lunch. In the end, the ratio of Ali's money to Raju's money was 4 : 3. How much money did Ali have at first?
    Answer

    200 $ — At first, Ali had 5 units and Raju had 2 units. After Ali gave away $44, Ali's final amount = 5 units − 44. Raju received $44 and then spent $7, so Raju's final amount = 2 units + 44 − 7 = 2 units + 37. The final ratio is 4 : 3, so 3 × Ali's final amount = 4 × Raju's final amount. 3 × (5 units − 44) = 15 units − 132; 4 × (2 units + 37) = 8 units + 148. 15 units − 132 = 8 units + 148, so 7 units = 280 and 1 unit = 40. Ali had 5 × 40 = $200 at first.

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