# Discrete math: Is the survey accurate?

A library has conducted a survey of its readers. The survey asked its $10,000$ readers about their reading habits and the number of books that they have borrowed from the library in $2012$. It has found that its readers claimed to have borrowed $75,000$ books in $2012$. The library has also reviewed the borrowing records of its books and found that on average a book was borrowed $5$ times. The library has $20,000$ books.

Is this survey accurate?

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why the downvote? – Carry on Smiling Mar 9 '13 at 23:28

HINT: If you add up the numbers of times each of the $20,000$ books in the library was borrowed to get the total number of borrowings, and divide that by $20,000$, you get the average of $5$ borrowings per book. That tells you that the total number of borrowings must have been ... what? Does this agree with the number of borrowings reported by the library’s readers?

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the average number of rentals per books is the same as the total number of rentals divided by the number of books.

in other words the library's data says $\frac{n}{20000}=5$ where $n$ is the total number of records.

multiplying both sides by $20,000$ one can see $n=100,000$

therefore the survey is off by $25,000$

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