A bookshelf holds both paperback and hardcover books

[BTGmoderatorDC]

A bookshelf holds both paperback and hardcover books. The ratio of paperback books to hardcover books is 22 to 3. How many paperback books are on the shelf?

(1) The number of books on the shelf is between 202 and 247, inclusive.

(2) If 18 paperback books were removed from the shelf and replaced with 18 hardcover books, the resulting ratio of paperback books to hardcover books on the shelf would be 4 to 1

I am confused in the statements, can some experts help me identify if the statements are sufficient?

OA D

[ErikaPrepScholar]

We know that the ratio of paperback to hardcover is 22 : 3 - this means that for every 25 books on the shelf, 22 will be paperback and 3 will be hardcover. It isn't possible to have part of a book, so we know that the total number of books must be divisible by 25. For instance, we could have 50 books, with 44 paperback and 6 hardcover, but we couldn't have 40 books, because then we would have 35.2 paperback and 4.8 hardcover.

Statement 1

There is only one multiple of 25 between 202 and 247: 225. This means that there must be (225 / 25) * 22 = 198 paperback books on the shelf. Sufficient.

Statement 2

If 18 paperback books were removed from the shelf and replaced with 18 hardcover books, the resulting ratio of paperback books to hardcover books on the shelf would be 4 to 1

Let's build an equation. If we have x groups of 25 books on the shelf:

22x - 18 : 3x + 18 = 4 : 1

Remember, we can always write ratios as fractions to more easily manipulate them:

$$\frac{22x\ -\ 18}{3x\ +\ 18}\ =\ \frac{4}{1}$$

22x - 18 = 12 x + 72
10 x = 90
x = 9

For every group of 9 books, there are 22 paperbacks, so there are 9 * 22 = 198 paperback books on the shelf. Sufficient.

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Hi lheiannie07,

We're told that the ratio of paperback books to hardcover books is 22 to 3. We're asked for the number of paperback books on the shelf. To start, we know that the number of paperback books MUST be a multiple of 22 and the number of hardcover books MUST be an equivalent multiple of 3. By extension, the TOTAL number of books MUST be a multiple of 25.

1) The number of books on the shelf is between 202 and 247, inclusive.

Fact 1 gives us a 'range' of possible totals. However, since the total number of books MUST be a multiple of 25, there's ONLY ONE option here: 225. We know that 22 out of every 25 books are paperbacks, thus the number of paperback books is (9)(22) = 198
Fact 1 is SUFFICIENT

2) If 18 paperback books were removed from the shelf and replaced with 18 hardcover books, the resulting ratio of paperback books to hardcover books on the shelf would be 4 to 1

With Fact 2, we can set up an equation based on the 'starting' number of books and the 'ending ratio of books:

Paperback/Hardcover = 22X/3X
After replacing 18 paperbacks with 18 hardcovers, we have...

(22X - 18)/(3X + 18) = 4/1

We can now solve for X...
22X - 18 = 12X + 72
10X = 90
X = 9

Therefore, the number of paperback books (22X) is (22)(9) = 198
Fact 2 is SUFFICIENT

Final Answer: D

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