Questions: Jeff Friedman was a section chief for an electric utility company. After the numbers in one of his reports to the management of the utility didn't add up, he was reassigned to the home economics department. He interviewed 126 people in a suburban shopping center to discover some of their cooking habits. He obtained the results shown to the right. 57 use microwave ovens; 62 use electric ranges; 55 use gas ranges; 25 use microwave ovens and electric ranges; 17 use microwave ovens and gas ranges; 7 use both gas and electric ranges 2 use all three; 3 uses none of the three. Should he be reassigned one more time? Why or why not?

Solution Steps

To determine if Jeff Friedman's numbers add up correctly, we need to verify the total number of people interviewed based on the given data. We will use the principle of inclusion-exclusion to calculate the total number of people who use at least one of the three cooking methods and compare it to the total number of people interviewed.

1. Use the principle of inclusion-exclusion to calculate the total number of people who use at least one of the three cooking methods.
2. Subtract the number of people who use none of the three methods from the total number of people interviewed.
3. Compare the calculated total with the given total number of people interviewed to check for consistency.

Using the principle of inclusion-exclusion, we calculate the number of people who use at least one of the three cooking methods:

\[ \text{at\_least\_one} = (57 + 62 + 55 - 25 - 17 - 7 + 2) = 127 \]

We add the number of people who use none of the three methods to the number of people who use at least one method:

\[ \text{total\_with\_none} = 127 + 3 = 130 \]

We compare the calculated total number of people with the given total number of people interviewed:

\[ \text{total\_with\_none} = 130 \quad \text{vs} \quad \text{total\_interviewed} = 126 \]

Since \(130 \neq 126\), the numbers in Jeff Friedman's report do not add up correctly.

Final Answer