Ruben

Ruben owns a restaurant. He likes to keep track of everything customers are buying. His top 3 sellers are sandwiches, salads, and pizza. He knows that 1/3 of his customers buy a sandwich, 1/2 buy a salad, and 1/4 buy a pizza.
What fraction of customers buy one of his top 3 sellers?

Correct answer:

p =  3/4

Step-by-step explanation:

$$\begin{gathered} a=\frac{1}{3}\doteq 0.3333 \\ b=\frac{1}{2}=0.5 \\ c=\frac{1}{4}=0.25 \\ P(A\cup B\cup C)=P(A)+P(B)+P(C)-P(A\cap B)-P(A\cap C)-P(B\cap C)+P(A\cap B\cap C) \\ {p}_1=a+b+c=\frac{1}{3}+\frac{1}{2}+\frac{1}{4}=\frac{4}{12}+\frac{6}{12}+\frac{3}{12}=\frac{4+6+3}{12}=\frac{13}{12}=1\frac{1}{12}\doteq 1.0833 \\ {p}_2=a\cdot b+b\cdot c+c\cdot a=\frac{1}{3}\cdot \frac{1}{2}+\frac{1}{2}\cdot \frac{1}{4}+\frac{1}{4}\cdot \frac{1}{3}=\frac{3}{8}=0.375 \\ {p}_3=a\cdot b\cdot c=\frac{1}{3}\cdot \frac{1}{2}\cdot \frac{1}{4}=\frac{1}{24}\doteq 0.0417 \\ p=p_1-p_2+p_3=\frac{13}{12}-\frac{3}{8}+\frac{1}{24}=\frac{26}{24}-\frac{9}{24}+\frac{1}{24}=\frac{26-9+1}{24}=\frac{18}{24}=\frac{3}{4}=0.75 \end{gathered}$$

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