Question

Four positive integers are given. Select any three of these integers, find their arithmetic average, and add this result to the fourth integer. Thus the numbers $29,$ $23,$ $21$ and $17$ are obtained. One of the original integers is:

$19$

$21$

$23$

$29$

$17$

Answer: B

Concepts:linear system averages sum invariant

Difficulty rating: 1840

Small Hint:

If the original sum is $S$ and the singled-out integer is $x,$ the result is $\frac{S+2x}{3}$

Big Hint:

Sum all four reported results to determine $S$

Solution:

If the original integers sum to $S,$ the result associated with singled-out integer $x$ is $x+\frac{S-x}{3}=\frac{S+2x}{3}.$ Summing all four reported results counts the total as $2S,$ so $S=\frac{29+23+21+17}{2}=45.$ The result $29$ therefore comes from $x=\dfrac{3(29)-45}{2}=21.$

Thus, one original integer is $21,$ and the correct answer is B.

1955 AMC 12 Problem 38

Answer: B

Difficulty rating: 1840

Small Hint:

Big Hint:

Solution:

Problem 38 in Other Years