Review question

# How many students took French, Latin and German? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R5848

## Solution

In an examination, candidates could take any one, any two or all of Latin, French and German.

Out of $100$ candidates $4$ took Latin only, $13$ took French only and $12$ took German only. There were $5$ who took all three languages and $20$ who did not take French.

The average mark in French of all of the candidates who took this subject was $50$%. Within this group there were the following average marks in French:

$\qquad$ Those taking French only, $57$%.

$\qquad$ Those taking French and Latin, $45$%.

$\qquad$ Those taking French and German, $52$%.

$\qquad$ Those taking all three languages, $35$%.

Calculate the number who took Latin and German, the number who took French and Latin and the number who took French and German.

Let’s summarise with a Venn diagram.

Note that the question doesn’t allow for someone not taking any of the three languages. Everyone here takes at least one language.

The question tells us that $20$ candidates didn’t take French. Therefore 20 candidates took only Latin, only German, or Latin and German.

From our diagram, we see that this means $4+12+x=20,$ and so $x=4$.

Let’s use that the total number of candidates is $100$. We find that $4+13+12+4+y+z+5=100,$ which means that
$\begin{equation}\label{eq:eqn1} y+z=62. \end{equation}$

What about the averages? We have that $80$ candidates took French, and their average mark was $50$, so the total number of marks earned was $4000$.

Note now that when the question says ‘French and German’ it means ‘French and German but not Latin’.

Since the average mark of those taking only French was $57$, and $13$ candidates only took French, this accounts for $57\times 13=741$ marks.

The total marks earned by those taking French and Latin is $45y$.

The total number of marks earned by those taking French and German is $52z$.

Finally, the number of marks earned by those taking all three languages is $35\times 5=175$.

This tells us that $4000=741+45y+52z+175.$

Rearranging $\eqref{eq:eqn1}$, we find that $z=62-y$, and on substituting this in, we have $4000=741+45y+52(62-y)+175,$ which yields $7y=140,$ and so $y$ is $20$, and $z$ is $62-20=42$.

Our completed Venn diagram looks like this:

So the number who took Latin and German is $4$, the number who took French and Latin is $20$, and the number who took French and German is $42$.