8. a survey of students in a school cafeteria had these results:\n• 50% of the students drink bottled…

8. a survey of students in a school cafeteria had these results:\n• 50% of the students drink bottled water.\n• 56% eat fruit.\n• 43% follow a low - fat diet.\n• 22% drink bottled water and follow a low - fat diet.\n• 23% follow a low - fat diet and eat fruit.\n• 27% drink bottled water and eat fruit.\n• 15% drink bottled water, follow a low - fat diet, and eat fruit.\nwhat percent of the students do not drink bottled water, do not follow a low - fat diet, and do not eat fruit?

8. a survey of students in a school cafeteria had these results:\n• 50% of the students drink bottled water.\n• 56% eat fruit.\n• 43% follow a low - fat diet.\n• 22% drink bottled water and follow a low - fat diet.\n• 23% follow a low - fat diet and eat fruit.\n• 27% drink bottled water and eat fruit.\n• 15% drink bottled water, follow a low - fat diet, and eat fruit.\nwhat percent of the students do not drink bottled water, do not follow a low - fat diet, and do not eat fruit?

Answer

Explanation:

Step1: Use the principle of inclusion - exclusion

Let (A) be the set of students who drink bottled water, (B) be the set of students who follow a low - fat diet, and (C) be the set of students who eat fruit. The formula for (|A\cup B\cup C|) is (|A|+|B|+|C|-|A\cap B|-|B\cap C|-|A\cap C|+|A\cap B\cap C|). Given (|A| = 50%), (|B|=43%), (|C| = 56%), (|A\cap B|=22%), (|B\cap C| = 23%), (|A\cap C|=27%), (|A\cap B\cap C| = 15%).

Step2: Calculate (|A\cup B\cup C|)

[ \begin{align*} |A\cup B\cup C|&=50 + 43+56-22 - 23-27+15\ &=(50 + 43+56)+15-(22 + 23+27)\ &=149 + 15-72\ &=92% \end{align*} ]

Step3: Find the percentage of students who do none of these

The percentage of students who do not drink bottled water, do not follow a low - fat diet, and do not eat fruit is (100%-|A\cup B\cup C|). [100 - 92=8%]

Answer:

8%