1999 CEMC Gauss (Grade 8) Problems/Problem 16

Problem

In a set of five numbers, the average of two of the numbers is $12$ and the average of the other three numbers is $7$ . The average of all five numbers is

$\text{(A)}\ 8 \frac{1}{3} \qquad \text{(B)}\ 8 \frac{1}{2} \qquad \text{(C)}\ 9 \qquad \text{(D)}\ 8 \frac{3}{4} \qquad \text{(E)}\ 9 \frac{1}{2}$

Solution 1

To find the average of all numbers, we can first find the sum of all of the numbers using the averages of the two groups, and then divide them by the number of numbers ( $5$ in this case).

Let $s_1$ represent the sum of the numbers of the first group of numbers (the one with $2$ elements), and $s_2$ represent the sum of the numbers in the other group. Setting up equations, we have: