Question

A shelf at a bookstore displays 27 books. Of these 27 books, 9 of the books are nonfiction books. The store owner adds 6 new fiction books to the shelf and wants the ratio of fiction to nonfiction books to remain the same. What is the total number of nonfiction books that the store should display, after the new fiction books are added?

Answer 1

I have a feeling that it is 18.

Answer 2

I think you'll need to use variables.

Answer 3

Well I thought that nine goes to twenty seven three times, and so six will have to be multiplied by 3 to get 18.

Answer 4

n for nonfiction books.
f for fiction books.

Answer 5

okay so first of all, is 18 the wrong answer?

Answer 6

Hold on one second, I am going to solve it on my own and then try to explain.

Answer 7

The initial ratio is: \[\frac{9}{27}
\]

Answer 8

When we add the \(6\) fiction books, then we have: \[
\frac{9}{27+6}
\]

Answer 9

We need to add a certain number of non-fiction books, we'll suppose that we need to add \(n\) of them. Then we want: \[
\frac{9}{27} = \frac{9+n}{27+6+n}
\]

Answer 10

How would I solve for n?

Answer 11

One thing you can do is cross multiply.

Answer 12

Basically, \[
9\times (27+6+n) = 27\times (9+n)
\]

Answer 13

So 9(27+6+n) and 27(9+n)?

Answer 14

And I need to use the distributive property right?

Answer 15

yes

Answer 16

I got 243 + 54 + 9n and 243 + 27n. Am I Correct?

Answer 17

You still need to solve for \(n\).

Answer 18

Okay. 243 + 54 + 9n = 243 + 27n

Answer 19

I subtracted 9n from both sides. Got:
243 + 54 = 243 + 18n
Subtracted 243 from both sides, got
54= 18n
Divided n from both sides, and n equals to 3

Answer 20

So is it 3?

Answer 21

Well, does \[
\frac{9}{27} = \frac{9+3}{27+6+3}
\]?