CEMC Fermat, 2016

Algebra Difficulty 2.5 Junior Find the answer Canada

Suppose that $a$ and $b$ are integers with $\text{4 lt;a lt;b lt;22}$ . If the average (mean) of the numbers $4,a,b,22$ is 13, then the number of possible pairs $(a,b)$ is

Pick one

Solution

Since the average of the four numbers $4, a, b, 22$ is 13, then $\dfrac{4+a+b+22}{4}=13$ and so $4+a+b+22=52$ or $a+b=26$ .

Since $\text{a gt; 4}$ and $a$ is an integer, then $a \geq 5$ .

Since $a+b=26$ and $\text{a lt;b}$ , then $a$ is less than half of 26, or $\text{a lt;13}$ .

Since $a$ is an integer, then $a \leq 12$ .

Therefore, we have $5 \leq a \leq 12$ .

There are 8 choices for $a$ in this range: 5, 6, 7, 8, 9, 10, 11, 12. (Note that $12-5 + 1 = 8$ .)

These give the pairs $(a,b)=(5,21),(6,20),(7,19),(8,18),(9,17),(10,16),(11,15),(12,14)$ .

Thus, there are 8 possible pairs.

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