Multiple choice

Let a, b and c be the ages of three persons P, Q and R, respectively where a ≤ b ≤ c are natural numbers. If the average age of P, Q, R is 32 years and if the age of Q is exactly 6 years more than that of P, then what is the minimum possible value of c?

  1. 34 years

  2. 36 years

  3. 38 years

  4. 32 years
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Average age is 32, so a+b+c = 96. Given b = a+6, substituting gives a + (a+6) + c = 96, or 2a + c = 90. Since a <= b <= c, we have a <= a+6 <= c. To minimize c, we maximize a. From 2a + c = 90, c = 90 - 2a. The condition a+6 <= 90-2a implies 3a <= 84, so a <= 28. If a=28, b=34, c=34. This satisfies a <= b <= c.

AI explanation

The formula for the average of three numbers is the sum of the numbers divided by three, so the sum of the ages (a + b + c) equals 32 multiplied by 3, which is 96. Since the age of Q is exactly 6 years more than P, we have b = a + 6. To find the minimum possible value of c, we must maximize the sum of a and b. Because the ages are natural numbers where a is less than or equal to b, the maximum value for a occurs when a and c are as close as possible. By checking the math, if we test the smallest provided option for c, which is 34, the sum of a and b must be 96 minus 34, equaling 62. Substituting b = a + 6 gives 2a + 6 = 62, meaning a = 28 and b = 34. This satisfies the condition that a is less than or equal to b, which is less than or equal to c (28, 34, 34). Thus, the minimum possible value for c is 34 years.

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