# Packing Boxes

Look at the times that Harry, Christine and Betty take to pack boxes when working in pairs, to find how fast Christine can pack boxes by herself.

Harry, Christine and Betty are packing $36$ boxes of chocolates.

- Harry and Christine would take $2$ hours to pack the boxes
- Harry and Betty would take $3$ hours
- Christine and Betty would take $4$ hours

How many boxes does Christine pack in one hour?

If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.

**Answer**: $7\frac12$

**Comparing Harry and Christine with and without Betty**

Harry and Betty together pack $12$ boxes per hour.

Christine and Betty together pack $9$ boxes per hour.

Therefore, in one hour, Harry packs $3$ more boxes than Christine.

In one hour, Harry and Christine together pack $18$ boxes.

Since Harry packs 3 more than Christine, Harry packs $10\frac{1}{2}$ boxes and Christine packs $7\frac{1}{2}$ boxes in an hour.

**Using algebra**

Say that Harry packs $H$ boxes per hour, Christine packs $C$ boxes per hour and Betty packs $B$ boxes per hour.

$2H+2C=36$ so $H+C=18$.

$3H+3B=36$ so $H+B=12$.

$4C+4B=36$ so $C+B=9$.

Adding all three equations together gives $2H+2C+2B=39$,

so $H+C+B=19 \frac12$.

Since we know that $H+B=12$,

$C=19\frac12 - 12 = 7\frac12$.