Dropouts
What percentage of students who graduate have never been to France?
Age
14 to 16
Challenge level
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.