If a \ast b = (a× b)/(a+b) for a, b positive integers, then what is 5 \ast 10?

2010 AMC 8 Problem 2 — Solution

Attempt Problem 2 of the 2010 AMC 8 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2010 AMC 8 solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

If $a \ast b = \dfrac{a\times b}{a+b}$ for $a,$ $b$ positive integers, then what is $5 \ast 10?$

$\ \dfrac{3}{10}$

$\ 1$

$\ 2$

$\ \dfrac{10}{3}$

$\ 50$

Reduce $\frac{50}{15}$ after adding the denominator.

Substitute the two given numbers into the definition of $a\ast b$ .

Given our definition of $a\ast b,$ we have $5 \ast 10 = \dfrac{5 \times 10}{5 + 10} = \dfrac{50}{15} = \dfrac{10}{3}.$

Thus, the answer is D .