Competition problem — Algebra — Maths Olympiad Prep

Library / /69 of 173

Algebra Difficulty 2.5 Junior Find the answer

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. Spacing and $ signs are ignored.

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 $x=15$ .

Want a route through all this instead of an archive? The track puts 2,000 problems in a working order, from AMC 10 level to the IMO shortlist.

Source: Omni-MATH, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.