In 2013, a State Library received a three-year grant to add 69,678 books to their collection. During a used book sale, the library sold 430 books. If the collection now is five times more than the original number of books, how many books did the library hold before it received the grant?

Answer 1

Let \( x \) represent the number of books the library held before it received the grant. The library received a grant to add 69,678 books to their collection.

According to the problem, after receiving the grant, the library also sold 430 books. Therefore, the total number of books after the grant and after the sale is:

\[ x + 69678 - 430 \]

This can be simplified to:

\[ x + 69248 \]

We are told that the current total number of books is five times the original number of books:

\[ x + 69248 = 5x \]

To solve for \( x \), we rearrange the equation:

\[ 69248 = 5x - x \]

This simplifies to:

\[ 69248 = 4x \]

Now, we divide both sides by 4:

\[ x = \frac{69248}{4} \] \[ x = 17312 \]

Thus, the library held 17,312 books before it received the grant. We can verify this:

1. Original number of books: \( 17312 \)
2. Books added by the grant: \( 69678 \)
3. Total books after the grant: \( 17312 + 69678 = 86990 \)
4. Books sold: \( 430 \)
5. Final number of books: \( 86990 - 430 = 86560 \)
6. Check if this final number is five times the original number: \( 5 \times 17312 = 86560 \)

The calculations confirm that the final number of books, 86560, is indeed five times the original number. Therefore, the answer is:

\[ \boxed{17312} \]