Question

Suppose that $S$ is a finite set of positive integers. If the greatest integer in $S$ is removed from $S,$ then the average value (arithmetic mean) of the integers remaining is $32.$ If the least integer in $S$ is also removed, then the average value of the integers remaining is $35.$ If the greatest integer is then returned to the set, the average value of the integers rises to $40.$ The greatest integer in the original set $S$ is $72$ greater than the least integer in $S.$ What is the average value of all the integers in the set $S?$

$36.2$

$36.4$

$36.6$

$36.8$

$37$

Solution:

Let $n=|S|,$ let $T$ be the total, $M$ the greatest, and $L$ the least. Then $\dfrac{T-M}{n-1}=32,$ $\dfrac{T-M-L}{n-2}=35,$ and $\dfrac{T-L}{n-1}=40.$

Subtracting the first from the third: $\dfrac{M-L}{n-1}=8.$ Since $M-L=72,$ we get $n-1=9,$ so $n=10.$

Then $T-M=288$ and $T-L=360.$ The middle equation gives $T-M-L=35\cdot 8=280,$ so $L=288-280=8$ and $M=80.$

Thus $T=288+80=368,$ and the average is $\dfrac{368}{10}=36.8.$

Thus, the correct answer is D.

2021 AMC 12B Spring Problem 12

Solution:

Problem 12 in Other Years