2021 Fall AMC 12B Problems/Problem 5

Problem 5

Call a fraction $\frac{a}{b}$ , not necessarily in the 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?

$\textbf{(A)}\ 9 \qquad\textbf{(B)}\ 10 \qquad\textbf{(C)}\ 11 \qquad\textbf{(D)}\ 12 \qquad\textbf{(E)}\ 13$

Solution

$a=15-b$ so the fraction is $\frac{15-b}{b}$ which is $\frac{15}{b}-1$ . We can just ignore the $-1$ part and only care about $\frac{15}{b}$ . Now we just group $\frac{15}{1}, \frac{15}{3}, \frac{15}{5}$ as the integers and $\frac{15}{2}, \frac{15}{6}, \frac{15}{10}$ as the halves. We get $30, 20, 18, 10, 8, 6$ from the integers group and $15, 10, 9, 5, 4, 3$ from the halves group (easily calculated by halving the values from the integers group). These are both $6$ integers and we see that $10$ overlaps, so the answer is $\boxed{(D)12}$ .

~lopkiloinm

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2021 Fall AMC 12B Problems/Problem 5

Problem 5

Solution