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Problem 537

Algebra Difficulty 2.5 Find the answer CEMC Fermat

Emilia writes down the numbers $5, x$ , and 9. Valentin calculates the mean (average) of each pair of these numbers and obtains 7, 10, and 12. What is the value of $x$ ?

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Official solution

Since the average of 5 and 9 is $\frac{5+9}{2}=7$ , then the averages of 5 and $x$ and of $x$ and 9 must be 10 and 12. In other words, $\frac{5+x}{2}$ and $\frac{x+9}{2}$ are equal to 10 and 12 in some order. Adding these, we obtain $\frac{5+x}{2}+\frac{x+9}{2}=10+12$ or $\frac{14+2x}{2}=22$ and so $7+x=22$ or $x=15$ .

Source: Omni-MATH, licensed Apache-2.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.