Pat gives half of his marbles to his best friend and then a third of those remaining to his sister. If his sister receives 9 marbles, then the number Pat keeps is
(A) 27
(B) 54
(C) 18
(D) 36
(E) 9
Pat gives half of his marbles to his best friend and then a third of those remaining to his sister. If his sister receives 9 marbles, then the number Pat keeps is
(A) 27
(B) 54
(C) 18
(D) 36
(E) 9

## Solution

Let $x$ be the total number of marbles that Pat has initially.

Then he gives $\frac{1}{2} x$ to his best friend, and $\frac{1}{3} \times \frac{1}{2} x=\frac{1}{6} x$ marbles to his sister.

So $\frac{1}{6} x=9$

$$
x=54
$$

Pat keeps $x-\frac{1}{2} x-\frac{1}{6} x=\frac{1}{3} x=18$ marbles.

ANSWER: (C)