#
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.