 # Problem of the Week Problem C and Solution Sharing Grapes

## Problem

Jessica has some grapes. She gives one-third of her grapes to Callista. She then gives $$4$$ grapes to Monica. Finally, she gives one-half of her remaining grapes to Peter. If Jessica then has $$16$$ grapes left, how many grapes did Jessica begin with? ## Solution

Solution 1:

We work backwards from the last piece of information given.
Jessica has $$16$$ grapes left after giving one-half of her remaining grapes to Peter.
This means that she had $$2 \times 16 = 32$$ grapes immediately before giving grapes to Peter.
Immediately before giving grapes to Peter, she gave $$4$$ grapes to Monica, which means that she had $$32 + 4 = 36$$ grapes immediately before giving $$4$$ grapes to Monica.
Immediately before giving the $$4$$ grapes to Monica, she gave one-third of her grapes to Callista, which would have left her with two-thirds of her original amount.
Since two-thirds of her original amount equals $$36$$ grapes, then one-third equals one half of $$36$$ or $$\frac{36}{2} = 18$$ grapes.
Thus, she gave $$18$$ grapes to Callista, and so Jessica began with $$36 + 18 = 54$$ grapes.

Solution 2:

Suppose Jessica started with $$x$$ grapes.
She gives $$\frac{1}{3} x$$ grapes to Callista, leaving her with $$1 - \frac{1}{3}x = \frac{2}{3}x$$ grapes.
She then gives $$4$$ grapes to Monica, leaving her with $$\frac{2}{3} x - 4$$ grapes.
Finally, she gives away one-half of what she has left to Peter, which means that she keeps one-half of what she has left, and so she keeps $$\frac{1}{2} (\frac{2}{3} x - 4)$$ grapes.
Simplifying this expression, we obtain $$\frac{2}{6} x -\frac{4}{2} = \frac{1}{3}x - 2$$ grapes.
Since she has $$16$$ grapes left, then $$\frac{1}{3}x - 2 = 16$$ and so $$\frac{1}{3}x = 18$$ or $$x = 54$$. Therefore, Jessica began with $$54$$ grapes.