3/4 of the books in the library are English books and the rest are Chinese books. There are 2849 more English books than Chinese books. What is the total number of books in the library?

Question

Roman Alvarez · Accepted Answer

#2xx2849 = 5698 # books

Working with fractions, we know the following: #3/4# are English books. #1/4# are Chinese books. The difference between the fractions is: #3/4 -1/4 = 2/4 =1/2# The difference between the number of books is 2849. So if half of the number of books is 2849, the total number of books is; #2849 xx 2 = 5698# OR: #1/2 xx x = 2849# #x = 2xx2849 = 5698#

Gabriel Agustin · Answer

undefined Let $ E $ represent the number of English books and $ C $ represent the number of Chinese books. Given that $ \frac{3}{4} $ of the books are English and the rest are Chinese, we have:

$$ E = \frac{3}{4}(E + C) $$
$$ E = \frac{3}{4}(E + \frac{1}{4}(E)) $$
$$ E = \frac{3}{4}E + \frac{3}{16}E $$
$$ \frac{16}{16}E = \frac{12}{16}E + \frac{3}{16}E $$
$$ \frac{4}{16}E = \frac{3}{16}E $$
$$ \frac{1}{4}E = \frac{3}{16}E $$
$$ 16E = 12E + 3E $$
$$ 16E = 15E $$
$$ E = \frac{16}{15}E $$

So, we find that there are $ \frac{16}{15} $ times as many English books as the difference between English and Chinese books. Let's denote $ D $ as this difference. Thus,

$$ E = \frac{16}{15}D $$

Given that there are 2849 more English books than Chinese books, we have:

$$ E - C = 2849 $$
$$ \frac{16}{15}D - \frac{1}{4}D = 2849 $$
$$ \frac{64}{60}D - \frac{15}{60}D = 2849 $$
$$ \frac{49}{60}D = 2849 $$
$$ D = 2849 \times \frac{60}{49} $$
$$ D = 3500 $$

Thus, there are 3500 Chinese books.

Now, to find the total number of books in the library:

$$ E + C = \frac{16}{15}D + D $$
$$ E + C = \frac{16}{15} \times 3500 + 3500 $$
$$ E + C = 5600 + 3500 $$
$$ E + C = 9100 $$

So, the total number of books in the library is 9100.