Question

Call a fraction $\dfrac{a}{b},$ 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?

$9$

$10$

$11$

$12$

$13$

Solution:

The special fractions in lowest terms include the integers $2 = \tfrac{10}{5},$ $4 = \tfrac{12}{3},$ $14 = \tfrac{14}{1};$ the half-integers $\tfrac12,$ $\tfrac32,$ $\tfrac{13}{2};$ the quarter-integers $\tfrac14$ and $\tfrac{11}{4};$ and others.

Two specials add to an integer only when their fractional parts cancel:

Integer pairs give $4, 6, 8, 16, 18, 28.$ Half-integer pairs $\left(\tfrac12, \tfrac32, \tfrac{13}{2}\right)$ give $1, 2, 3, 7, 8, 13.$ The quarter pair $\tfrac14 + \tfrac{11}{4}$ gives $3.$

The distinct integers are $1, 2, 3, 4, 6, 7, 8, 13, 16, 18, 28,$ a total of $11.$

Thus, the correct answer is C.

2021 AMC 12B Fall Problem 5

Solution:

Problem 5 in Other Years