Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 4.4 AIME Find the answer Italy

Problem:

In the S. Maria neighborhood there are $9897$ televisions. Only three families in the neighborhood do not own televisions, while $4\%$ have two, $2,5\%$ have $3$ , and $0,5\%$ have as many as $8$ . All the other families own exactly one television. How many families live in the S. Maria neighborhood?

Pick one

Solution

Solution:

The answer is (D). If we give a television as a gift to the families that don't have one, the percentage of families that have only one television becomes $93\%$ , and the total number of televisions $9900$ . So, letting $N$ be the number of families in the neighborhood, we have the following relation:
$N \frac{93}{100} + 2N \frac{4}{100} + 3N \frac{2,5}{100} + 8N \frac{0,5}{100} = 9900.$
From which
$N\left(\frac{93}{100} + \frac{8}{100} + \frac{7,5}{100} + \frac{4}{100}\right) = N\left(\frac{186 + 16 + 15 + 8}{200}\right) = N \frac{225}{200} = 9900,$
so $N = 8800$ .

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Source: MathNet, licensed CC-BY-4.0. Statement translated into English from it; metadata (topic, difficulty) added by this project.