# Drawing Venn Diagrams Effectively

Consider the following situation to illustrate through Venn diagrams.

## Two Circles

There are $50$ students in a certain high school. $16$ study Physics, $13$ study Chemistry, and $15$ study both Physics and Chemistry. Illustrate this information on a Venn diagram. \begin{align} a+b+c+d &= 50 \text{ total students} \\ a+b &= 31 \text{ Physics} \\ b+c &= 28 \text{ Chemistry} \\ b &= 15 \text{ Physics and Chemistry} \\ \end{align} \begin{align} \text{Substitute } b &=15 \text{ into } a+b=31 \\ a+15 &= 31 \\ a &= 16 \\ \end{align} \begin{align} \text{Substitute } b &=15 \text{ into } b+c=28 \\ 15+c &= 28 \\ c &= 13 \\ \end{align} \begin{align} \text{Substitute } a =16, b=15, c &=13 \text{ into } a+b+c+d=50 \\ 16 +15+13+d &= 50 \\ d &= 6 \\ \end{align} ## Three Circles

Now, let’s take a look at a situation with three circles.

A school has three subjects offered: Arts, Biology and Chemistry.

$36$ students chose Arts, $39$ chose Biology, and $37$ chose Chemistry. Of those, $9$ chose both Arts and Biology, $12$ chose both Biology and Chemistry, and $11$ chose both Arts and Chemistry. $5$ chose all three subjects. $$a=5$$ \begin{align} a+d &= 9 \\ 5+d &= 9 \\ d &= 4 \\ \end{align} \begin{align} a+b &= 12 \\ 5+b &= 12 \\ b &= 7 \\ \end{align} \begin{align} a+c &= 11 \\ 5+c &= 11 \\ c &= 6 \\ \end{align} \begin{align} g+4+5+6 &= 36 \\ g &= 21 \\ \end{align} \begin{align} e + 4+5+7 &= 39 \\ e &= 23 \\ \end{align} \begin{align} f + 6+5+7 &= 37 \\ f &= 19 \\ \end{align} \begin{align} h &= 100 -(21+4+5+6+7+23+19) \\ &= 15 \\ \end{align}  