The average of five distinct positive integers is 24. If the median of these five numbers is 20, what is the maximum possible value of the largest of the five numbers?

Answer and explanation

Correct answer: D. 76

  1. Step 1: The sum of the five integers is 5 * 24 = 120.
  2. Step 2: Let the integers in ascending order be a < b < c < d < e. Here median c = 20.
  3. Step 3: To maximize e, we must minimize a, b, and d.
  4. Step 4: Since they are distinct positive integers: the smallest possible values for a and b are a = 1 and b = 2.
  5. Step 5: Since d must be strictly greater than c = 20, the smallest possible integer for d is 21.
  6. Step 6: Sum of first four integers = 1 + 2 + 20 + 21 = 44.
  7. Step 7: Maximum possible value of e = 120 - 44 = 76.
Answer
76.
Why not the others: 78 assumes b = 1 and d = 20 (violating distinct integers); 80 assumes d = 20 and a = b = 0; 74 uses d = 23.

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