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$$\text{There are 360 people in my school. 15 take calculus, physics, and chemistry, and 15 don't take any of them. 180 take calculus. Twice as many students take chemistry as take physics. 75 take both calculus and chemistry, and 75 take both physics and chemistry. Only 30 take both physics and calculus. How many students take physics? }$$

Sep 8, 2019

#1
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$$\text{There are 360 people in my school. 15 take calculus, physics, and chemistry, and 15 don't take any of them.~\\ 180 take calculus. Twice as many students take chemistry as take physics. 75 take both calculus and chemistry, and ~\\ 75 take both physics and chemistry. Only 30 take both physics and calculus. How many students take physics? }$$

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Sep 9, 2019
#2
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There are 360 people in my school.

15 take calculus, physics, and chemistry, and

15 don't take any of them.

180 take calculus.

Twice as many students take chemistry as take physics.

75 take both calculus and chemistry, and

75 take both physics and chemistry.

Only 30 take both physics and calculus.

How many students take physics?

$$\begin{array}{|c|c|c|rl|} \hline \text{calculus} & \text{physics} & \text{chemistry} & \\ \hline x & x & x & 15 & \text{ take calculus, physics, and chemistry} \\ \hline x & & x & 75 & \text{ take both calculus and chemistry} \\ \hline x & x & & 30 & \text{ take both physics and calculus} \\ \hline x & & & 180-15-75-30 = \mathbf{60} & \text{ take only calculus} \\ \hline & & & 180 & \text{ take calculus} \\ \hline \end{array}$$

$$\begin{array}{|c|c|c|rl|} \hline \text{chemistry} & \text{physics} & \text{calculus} & \\ \hline x & x & x & 15 & \text{ take calculus, physics, and chemistry} \\ \hline x & & x & 75 & \text{ take both calculus and chemistry} \\ \hline x & x & & 75 & \text{ take both physics and chemistry} \\ \hline x & & & v & \text{ take only chemistry} \\ \hline & & & 15+75+75+v \\ & & & =165+v & \text{ take chemistry} \\ \hline \end{array}$$

$$\begin{array}{|c|c|c|rl|} \hline \text{physics} & \text{chemistry} & \text{calculus} & \\ \hline x & x & x & 15 & \text{ take calculus, physics, and chemistry} \\ \hline x & & x & 30 & \text{ take both physics and calculus} \\ \hline x & x & & 75 & \text{ take both physics and chemistry} \\ \hline x & & & u & \text{ take only physics} \\ \hline & & & 15+30+75+u \\ & & & =120+u & \text{ take physics} \\ \hline \end{array}$$

Twice as many students take chemistry as take physics:

$$\begin{array}{|rcll|} \hline 165+v &=& 2(120+u) \\ &=& 240 +2u \\ \mathbf{v} &=& \mathbf{2u+75} \\ \hline \end{array}$$

There are 360 people in my school:

$$\begin{array}{|rcll|} \hline \overbrace{180}^{\text{take calculus}} + \overbrace{u+v+ \underbrace{75}_{\text{take both physics and chemistry} } + \underbrace{15}_{\text{don't take any of them}} }^{\text{take no calculus}} &=& 360 \\ 180 + u+v+ 75 +15 &=& 360 \\ u+v+ 90 &=& 180 \\ u+v &=& 90 \\ \mathbf{u} &=& \mathbf{90-v} \quad | \quad v=2u+75 \\ u &=& 90-(2u+75) \\ u &=& 90-2u-75 \\ 3u &=& 15 \\ \mathbf{u} &=& \mathbf{5} \\\\ v &=& 2u+75 \\ v &=& 2*5 + 75 \\ \mathbf{v} &=& \mathbf{85} \\ \hline \end{array}$$

How many students take physics?

$$\begin{array}{|rcll|} \hline && 120+u \text{ take physics} \\ &=& 120 +5 \\ &=& 125 \text{ take physics} \\ \hline \end{array}$$ Sep 9, 2019