# 2005 CEMC Gauss (Grade 7) Problems/Problem 17

## Problem

The symbol $\begin{array}{|c|c|}\hline 3 & 4 \\ \hline 5 & 6 \\ \hline \end{array}$ is evaluated as $3 \times 6 + 4 \times 5 = 38$. If $\begin{array}{|c|c|}\hline 2 & 6 \\ \hline 1 & \\ \hline \end{array}$ is evaluated as $16$, what is the number that should be placed in the empty space?

$\text{(A)}\ 1 \qquad \text{(B)}\ 2 \qquad \text{(C)}\ 3 \qquad \text{(D)}\ 4 \qquad \text{(E)}\ 5$

## Solution 1

When we calculate the value of the symbol, we add the product of the numbers on each of the two diagonals. The product of the entries on the diagonal with the $1$ and the $6$ is $6$. Since the symbol is evaluated as $16$, then the product of the entries on the other diagonal is $10$. Since one of the entries on the other diagonal is $2$, then the missing entry must be $5$. The answer is therefore $E$.

## Solution 2

Let the missing number be $x$. Using the definition for the evaluation of the symbol, we know that $2x + 1\times 6 = 16$ or $2x + 6 = 16$ or $2x = 10$ or $x = 5$. The answer is $E$.