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.
$$a=5$$
$$ \begin{align}
a+d &= 9 \\
5+d &= 9 \\
d &= 4
\end{align} $$
$$ \begin{align}
a+b &= 12 \\
5+b &= 12 \\
b &= 7
\end{align} $$
$$ \begin{align}
a+c &= 11 \\
5+c &= 11 \\
c &= 6
\end{align} $$
$$ \begin{align}
g+4+5+6 &= 36 \\
g &= 21
\end{align} $$
$$ \begin{align}
e + 4+5+7 &= 39 \\
e &= 23
\end{align} $$
$$ \begin{align}
f + 6+5+7 &= 37 \\
f &= 19
\end{align} $$
$$ \begin{align}
h &= 100-(21+4+5+6+7+23+19) \\
&= 15
\end{align} $$
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