Draw a Venn diagram to illustrate all this information.

We are told that \(\text{16}\) learners take Maths and History. Out of these 16 learners some take Geography as well and some do not.

We are also told that \(\text{6}\) learners take Geography and History. Out of these 6 learners some take Maths as well and some do not.

Let the number of learners who take Maths, History and Geography \(= x\). Then we can draw the Venn diagram as follows:

How many learners take Maths and Geography but not History?

In the above Venn diagram the number of learners who take Maths and Geography but not History is indicated by \(y\). To find \(y\) we first need to determine \(x\).

To find \(x\) we note that the total number of learners who take History is equal to the sum of each of the following:

- The number of learners who take History only: 16
- The number of learners who take History and Maths but not Geography: \(16 - x\)
- The number of learners who take History and Geography but not Maths: \(6 - x\)
- The number of learners who take all three subjects: \(x\)

\begin{align*}
36 & = 16 + (16 - x) + (6 - x) + x \\
& = 16 + 16 - x + 6 - x + x \\
& = 38 - x \\
\therefore x & = 2
\end{align*}

Now we can find \(y\) using the same method as to find \(x\). This time we will use the total number of learners who take Maths.

\begin{align*}
30 & = 8 + (16 - x) + x + y \\
& = 8 + 14 + 2 + y \\
& = 24 + y \\
\therefore y & = 6
\end{align*}

Therefore \(\text{6}\) learners take Maths and Geography but not History.

How many learners take Geography only?

Now we need to find \(z\). We will use the total number of learners who take Geography to find \(z\).

\begin{align*}
41 & = (6 - x) + x + y + z \\
& = 4 + 2 + 6 + z \\
& = 12 + z \\
\therefore z & = 29
\end{align*}

Therefore \(\text{29}\) learners take Geography only.

How many learners take all three subjects?

When we drew the Venn diagram we let \(x\) be the number of learners that take all three subjects. We calculated \(x\) in the first question. Therefore \(\text{2}\) learners take all three subjects.