Question

1. a survey of students in a school cafeteria had these results:

• 50% of the students drink bottled water.
• 56% eat fruit.
• 43% follow a low - fat diet.
• 22% drink bottled water and follow a low - fat diet.
• 23% follow a low - fat diet and eat fruit.
• 27% drink bottled water and eat fruit.
• 15% drink bottled water, follow a low - fat diet, and eat fruit.

what percent of the students do not drink bottled water, do not follow a low - fat diet, and do not eat fruit?

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%