, 2015

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.