Question

The mean of a set of five different positive integers is $15.$ The median is $18.$ The maximum possible value of the largest of these five integers is

$19$

$24$

$32$

$35$

$40$

Solution:

The median of the set of numbers is the third largest number, which is $18.$ There are two numbers less than $18$ and two numbers greater than it.

The mean of the set is $15,$ so the sum of all the numbers is $5 \cdot 15 = 75.$ In order to maximize the largest number with this sum, the other numbers must be as small as possible.

The two numbers less than $18$ must be positive and distinct, so they must be $1$ and $2.$

The number immediately after $18$ must also be as small as possible, so it must be $19.$

Therefore, the remaining number, the maximum possible value in the set, is $75 - 1 - 2 - 18 - 19 = 35.$

Thus, D is the correct answer.

2001 AMC 8 Problem 21

Solution:

Problem 21 in Other Years