In a survey of a city, it was found that 90 percent of the people in the city own a refrigerator and 15 percent own a washing machine. If everybody owns at least one appliance, what percentage owns both?

### Concept & Formula This is a classic Set Theory problem that can be solved using the Principle of Inclusion-Exclusion for two sets. $$ n(A \cup B) = n(A) + n(B) - n(A \cap B) $$ ### Step-by-Step Solution * **Given:** * Percentage owning a refrigerator, $n(A) = 90\%$ * Percentage owning a washing machine, $n(B) = 15\%$ * Everybody owns at least one appliance, meaning the union of both sets covers the entire population: $n(A \cup B) = 100\%$ * **Calculation:** Let $x$ be the percentage of people who own both appliances, $n(A \cap B)$. Apply the formula: $100 = 90 + 15 - x$ $100 = 105 - x$ $x = 105 - 100$ $x = 5\%$ ### Exam Strategy & Shortcut Simply add the percentages of the two individual groups together and subtract $100$ (since there are no people outside the sets). $90 + 15 = 105$. The "extra" $5\%$ over $100$ must be the overlap (the people counted twice because they own both). ### Common Pitfall Assuming that the percentages must add up to exactly $100\%$ without overlap. Some students mistakenly subtract $15\%$ from $100\%$ and get confused when it doesn't equal $90\%$. Always remember that individual set percentages can exceed $100\%$ when summed if there is an intersection. ### Final Answer Therefore, the correct answer is **5 percent**.