# Problem of the Week Problem A and Solution What’s in the Pouch?

## Problem

Zoha’s class is raising money for a local charity. The class puts any money raised into a pouch, and each Thursday their teacher creates a math problem about the money in the pouch.

The following note was attached to the pouch today.

This pouch contains a total of $$\20.30$$ in Canadian money consisting of $$4$$ coins and $$3$$ bills.

What are the specific bills and coins in the pouch?

Note: The coins available in Canada are nickels that are worth $$5$$ cents, dimes that are worth $$10$$ cents, quarters that are worth $$25$$ cents, loonies that are worth $$\1$$, and toonies that are worth $$\2$$. Also, $$\1$$ is equal to $$100$$ cents. The lowest denominations of bills are worth $$\5$$, $$\10$$, and $$\20$$.

## Solution

The pouch cannot include a $$\20$$ bill since there is only $$30$$ cents more than $$\20$$, and that would mean the pouch only contained $$1$$ bill. Similarly, it cannot include two $$\10$$ bills since this would mean the pouch only contained $$2$$ bills.

If it has one $$\10$$ bill and two $$\5$$ bills, then that would be a total of $$\20$$. This is three bills.
In this case, there are $$30$$ cents remaining, which can be formed by:

• $$1$$ quarter and $$1$$ nickel for a total of $$2$$ coins

• $$3$$ dimes for a total of $$3$$ coins

• $$2$$ dimes and $$2$$ nickels for a total of $$4$$ coins

• $$1$$ dime and $$4$$ nickels for a total of $$5$$ coins

• $$6$$ nickels for a total of $$6$$ coins

So one possibility is that the pouch contains one $$\10$$ bill, two $$\5$$ bills, two dimes, and two nickels. However, we should check to see if this is the only possibility.

Could it have three $$\5$$ bills which is $$\15$$? This means there would be $$\5.30$$ remaining. The fewest number of coins you need to make $$\5$$ is two toonies and one loonie, which is a total of $$3$$ coins. But you need at least $$2$$ coins to make up $$30$$ cents. So you need at least $$5$$ coins to make $$\5.30$$, which is too many coins.

Any more attempts to come up with $$\20$$ would take more bills and coins. So the only possibility that meets the requirements of the problem is one $$\10$$ bill, two $$\5$$ bills, two dimes, and two nickels.