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 $S?$

$36.2$

$36.4$

$36.6$

$36.8$

$37$

Answer: D

Concepts:mean system of equations

Difficulty rating: 1540

Video solution:

Click to load, then click again to play

Written solution:

Let the sum of all the integers be $s,$ the greatest number be $g,$ the least number be $l,$ and the size of $S$ be $n.$

The two averages of sets with $n-1$ elements give $\begin{aligned} s-l&=40(n-1),\\ s-g&=32(n-1). \end{aligned}$ Subtracting and using $g-l=72$ gives $72=8(n-1),$ so $n=10.$

Removing both extremes leaves $8$ numbers with sum $8\cdot35=280.$ On the other hand, removing only the least number leaves a sum of $9\cdot40=360,$ so $g=360-280=80.$ Hence $l=80-72=8,$ and the original sum is $s=280+80+8=368.$

The required average is $\frac{s}{n}=\frac{368}{10}=36.8.$

Thus, the answer is D .

2021 AMC 10B Spring Problem 19

Answer: D

Difficulty rating: 1540

Video solution:

Written solution:

Problem 19 in Other Years