Library and gym

In a town, 7/8 of the population has a library card, and 3/4 has a gym membership. What fraction of the population has either a library card or a gym membership?

Correct answer:

p =  0.9688

Step-by-step explanation:

$$\begin{gathered} a=\frac{7}{8}=0.875 \\ b=\frac{3}{4}=0.75 \\ P(A\cup B)=P(A)+P(B)-P(A\cap B) \\ P(A\cup B)=P(A)+P(B)-P(A)\cdot P(B) \\ c=a\cdot b=\frac{7}{8}\cdot \frac{3}{4}=\frac{7\cdot 3}{8\cdot 4}=\frac{21}{32}\doteq 0.6563 \\ p=a+b-c=\frac{7}{8}+\frac{3}{4}-\frac{21}{32}=\frac{28}{32}+\frac{24}{32}-\frac{21}{32}=\frac{28+24-21}{32}=\frac{31}{32}=0.9688 \end{gathered}$$

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