Multiple choice

The present ages of three people – A, B and C, are such that they are all integers whose sum is 136 and, the age of A after 𝑎 years, the age of B after 𝑏 years and the age of C after 𝑐 years, are all equal, where 𝑎 : 𝑏 : 𝑐 = 2 ∶ 3 ∶ 6. If C is younger than B by more than 3 years, then what is the maximum possible age (in years) of C?

  1. 36

  2. 48

  3. 43

  4. 50
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A Correct answer
Explanation

Let ages be A, B, C. A+B+C = 136. A+a = B+b = C+c = K. a:b:c = 2:3:6. Let a=2k, b=3k, c=6k. A = K-2k, B = K-3k, C = K-6k. Sum: 3K - 11k = 136. Since ages are integers, test values for k. Given C < B-3, K-6k < K-3k-3 => 3 < 3k => k > 1. Testing k=2: 3K = 136 + 22 = 158 (not divisible). Testing k=4: 3K = 136 + 44 = 180 => K=60. A=52, B=48, C=36. Check: C < B-3 (36 < 45).

AI explanation

Let the equal age after a, b, and c years be K. This means the present ages of A, B, and C are K - 2x, K - 3x, and K - 6x respectively, where x is the common multiplier for the ratio. Their present ages sum to 136, so 3K - 11x = 136. The difference between B and C is 3x, and since C is younger than B by more than 3 years, x must be greater than 1. To maximize C's age, which is K - 6x, we use the smallest possible integer value for x, which is 2, yielding 3K - 22 = 136, so K = 158/3, meaning C is approximately 52.67 years old, which contradicts the integer age constraint. Checking the possible integer values from the options to satisfy 3K - 11x = 136, the maximum possible integer age for C among the valid solutions is 36 years.

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