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Problem 556

AMC 10/12, early questions

Algebra Difficulty 3.6 Multiple choice CEMC Fermat · Canada · 2014

Sergio recently opened a store. One day, he determined that the average number of items sold per employee to date was 75. The next day, one employee sold 6 items, one employee sold 5 items, and one employee sold 4 items. The remaining employees each sold 3 items. This made the new average number of items sold per employee to date equal to 78.3. How many employees are there at the store?

Pick one

Official solution

Suppose that there are $n$ employees at Sergio’s store.

After his first average calculation, his $n$ employees had sold an average of 75 items each, which means that a total of $75n$ items had been sold.

The next day, one employee sold 6 items, one sold 5, one sold 4, and the remaining $(n-3)$ employees each sold 3 items.

After this day, the total number of items sold to date was $75n+(6+5+4+(n-3)3)$ or $75n+15+3n-9$ or $78n+6$ .

Since the new average number of items sold per employee was 78.3, then $\dfrac{78n+6}{n}=78.3$ or $78n+6=78.3n$ .

Therefore, $0.3n = 6$ or $n = 20$ .

Thus, there are 20 employees in the store.

Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.