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 $S?$

$36.2$

$36.4$

$36.6$

$36.8$

$37$

Video solution:

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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 $n.$

From the info given, we know $\dfrac{s-l}{n-1} = 40$ $\dfrac{s-g}{n-1} = 32.$ Subtracting these yields $\dfrac{g-l}{n-1} = 8.$ Since we know $g-l=72,$ we know $\frac{72}{n-1} =8,$ so $n=10.$

We also know $\dfrac{s-g-l}{n-2} = 35,$ so $s-g-l = 8\cdot 35$ $=280.$ Since $\dfrac{s-l}{n-1} = 40,$ we know $s-l = 360.$ This makes $g=80.$ Using $g=l+72,$ we get $l=8.$ Thus, $s=368.$

The average is $\dfrac sn = \dfrac{368}{10}$ $= 36.8.$

Thus, the answer is D .

2021 AMC 10B Spring Problem 19

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Problem 19 in Other Years