# Ratio of the number of English and Chinese books

In a library, the ratio of the number of English books to the number of Chinese books was 8:5. After the librarians bought 50 more Chinese books, the ratio became 4:3.

a) How many English books were there in the library? b) Find the total number of English and Chinese books in the library in the end.

My work.

Initial Ratio (before)
English: Chinese
8: 5

Initial Ratio (after)
English: Chinese
4:3

*Since English books is the fixed number. I'm guessing we have to times 2 in the Initial Ratio (after), so....

8: 5 (before)
8: 6 (after)

since 6units - 5units = 1unit

*We know the difference is 50. So 1unit = 50.

a) Since we have 8 units for English. 8 * 50 = 400.
b) Since we have 8 units for English and 6 units for Chinese. 50 * 14 = 700

Is my work mathematically correct?

Yup, it is correct.

Verification:

Initially we have $$8 \times 50= 400$$ English books and $$5 \times 50= 250$$ Chinese books. The ratio was $$8:5$$.

After which, we have $$8 \times 50=400$$ English books and $$(5+1) \times 50 = 300=250+50$$ Chinese books. The ratio is now $$8:6=4:3$$.

Slower method:

$$\frac{E}{C}=\frac{8}{5}$$, $$\frac{E}{C+50}=\frac{4}{3}=\frac{8}{6}$$

$$8C=5E, 6E=8C+400$$

$$E=400$$, $$C=250$$.

• Thank you for the slower method. – RukaiPlusPlus Oct 5 '18 at 2:51