# Dropouts

What percentage of students who graduate have never been to France?

## Problem

A language school reports that:

• 75% of students starting a French course have never been to France.

• Of the students who have never been to France, 60% graduate.

• Of the students who have visited France before, 70% graduate.

What percentage of the students who graduate had never been to France?

*This problem is adapted from the World Mathematics Championships*

## Student Solutions

**Using a tree diagram**

The students at the language school can be represented using a tree diagram, as shown:

Image

Image

So a total of 45% + 17.5% = 62.5% of students graduate.

45% have never been to France and graduated, so we need to find 45% as a percentage of 62.5%: $$\frac{45}{62.5}=\frac{?}{100}\\

\Rightarrow \frac {90}{125}=\frac{?}{100}\\

\Rightarrow \frac {90\times100}{125}=?\\

\Rightarrow \frac{18\times5\times25\times4}{5\times25}=?\\

\Rightarrow 18\times4=?$$

So 72% of students who graduate have never been to France.

**Using a two-way table**

The students at the language school can be represented using a two-way table, as shown:

Image

60% of 75% can be found as half of 75% plus 10% of 75% = 37.5% + 7.5% = 45%,

or as 75% of 60% = 3 $\times$ 15% = 45%.

70% of 25% can be found as half of 25% plus a fifth of 25% = 12.5% + 5% = 17.5%,

or as 25% of 70% = 70% $\div$ 4 = 17.5%.

Then we can find the total percentage of students who graduate:

or as 75% of 60% = 3 $\times$ 15% = 45%.

70% of 25% can be found as half of 25% plus a fifth of 25% = 12.5% + 5% = 17.5%,

or as 25% of 70% = 70% $\div$ 4 = 17.5%.

Then we can find the total percentage of students who graduate:

Image

\Rightarrow \frac {90}{125}=\frac{?}{100}\\

\Rightarrow \frac {90\times100}{125}=?\\

\Rightarrow \frac{18\times5\times25\times4}{5\times25}=?\\

\Rightarrow 18\times4=?$$

So 72% of students who graduate have never been to France.

**Using a Venn diagram**

The students at the language school can be represented using a Venn diagram, as shown:

Image

Image

So a total of 45% + 17.5% = 62.5% of students graduate.

45% have never been to school and graduated, so we need to find 45% as a percentage of 62.5%: $$\frac{45}{62.5}=\frac{?}{100}\\

\Rightarrow \frac {90}{125}=\frac{?}{100}\\

\Rightarrow \frac {90\times100}{125}=?\\

\Rightarrow \frac{18\times5\times25\times4}{5\times25}=?\\

\Rightarrow 18\times4=?$$

So 72% of students who graduate have never been to France.