Krystyna has some raisins. She gives one-third of her raisins to Mike. She then eats 4 raisins, after which she gives one-half of her remaining raisins to Anna. If Krystyna then has 16 raisins left, how many raisins did she have to begin?
(A) 42
(B) 54
(C) 60
(D) 84
(E) 108
Solution 1

We work backwards from the last piece of information given.

Krystyna has 16 raisins left after giving one-half of her remaining raisins to Anna.

This means that she had $2 \cdot 16=32$ raisins immediately before giving raisins to Anna.

Immediately before giving raisins to Anna, she ate 4 raisins, which means that she had $32+4=36$ raisins immediately before eating 4 raisins.

Immediately before eating these raisins, she gave one-third of her raisins to Mike, which would have left her with two-thirds of her original amount.

Since two-thirds of her original amount equals 36 raisins, then one-third equals $\frac{36}{2}=18$ raisins. Thus, she gave 18 raisins to Mike and so started with $36+18=54$ raisins.

## Solution 2

Suppose Krystyna starts with $x$ raisins.

She gives $\frac{1}{3} x$ raisins to Mike, leaving her with $x-\frac{1}{3} x=\frac{2}{3} x$ raisins.

She then eats 4 raisins, leaving her with $\frac{2}{3} x-4$ raisins.

Finally, she gives away one-half of what she has left to Anna, which means that she keeps one-half of what she has left, and so she keeps $\frac{1}{2}\left(\frac{2}{3} x-4\right)$ raisins.

Simplifying this expression, we obtain $\frac{2}{6} x-\frac{4}{2}=\frac{1}{3} x-2$ raisins.

Since she has 16 raisins left, then $\frac{1}{3} x-2=16$ and so $\frac{1}{3} x=18$ or $x=54$.

Therefore, Krystyna began with 54 raisins.

ANSWER: (B)