Question

Call a fraction $\frac{a}{b},$ not necessarily in simplest form, special if $a$ and $b$ are positive integers whose sum is $15.$ How many distinct integers can be written as the sum of two, not necessarily different, special fractions?

$\ 9$

$\ 10$

$\ 11$

$\ 12$

$\ 13$

Answer: C

Concepts:fraction divisibility systematic listing

Difficulty rating: 1660

Solution:

A special fraction with denominator $b$ equals $\frac{15-b}{b}=\frac{15}{b}-1,$ where $1\le b\le14.$ We need integer values of $\frac{15}{x}+\frac{15}{y}-2.$

Taking $x\le y,$ a check of the fourteen possible denominators gives the following pairs that produce integers: $\begin{gathered} (1,1),(1,3),(1,5),(2,2),(2,6),\\ (2,10),(3,3),(3,5),(4,12),\\ (5,5),(6,6),(6,10),(10,10). \end{gathered}$ Their distinct sums are $1,2,3,4,6,7,8,13,16,18,28.$

There are $11$ such integers.

Thus, the answer is C .

2021 AMC 10B Fall Problem 7

Answer: C

Difficulty rating: 1660

Solution:

Problem 7 in Other Years