Multiple choice

A and B together can complete a task in 40 days. B and C together can complete the same task in 50 days. A and C together can complete the same task in 60 days. What is the respective ratio between the numbers of days taken by A and C while completing the same task alone?

  1. 7 : 13

  2. 13 : 7

  3. 7 : 12

  4. 12 : 7

  5. None of these
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A Correct answer
Explanation

1/A + 1/B = 1/40, 1/B + 1/C = 1/50, 1/A + 1/C = 1/60. Adding these: 2(1/A + 1/B + 1/C) = 1/40 + 1/50 + 1/60 = (15+12+10)/600 = 37/600. So 1/A + 1/B + 1/C = 37/1200. 1/C = 37/1200 - 1/40 = 7/1200. 1/A = 37/1200 - 1/50 = 13/1200. Ratio A:C = 1200/13 : 1200/7 = 7:13.

AI explanation

Let the total work be the least common multiple of 40, 50, and 60, which is 600 units. The combined daily work of A+B is 15 units, B+C is 12 units, and A+C is 10 units. Solving these equations, A's daily work is (15 + 10 - 12) / 2 = 6.5 units and C's daily work is (12 + 10 - 15) / 2 = 3.5 units. The ratio of the time taken by A and C is the inverse of their efficiencies, which is 1 / 6.5 : 1 / 3.5, simplifying to 3.5 : 6.5 or 7 : 13.

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