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# A bookshelf holds both paperback and hardcover books. The

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Joined: 06 Sep 2013
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Concentration: Finance
A bookshelf holds both paperback and hardcover books. The [#permalink]

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11 Oct 2013, 17:05
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Question Stats:

72% (01:28) correct 28% (01:30) wrong based on 201 sessions

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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
[Reveal] Spoiler: OA
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Posts: 44298
Re: A bookshelf holds both paperback and hardcover books. The [#permalink]

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11 Oct 2013, 17:11
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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. The # of books is 22x+3x=25x, so it's a multiple of 25. There is only one multiple of 25 between 202 and 247, inclusive: 225 --> 25x=225 --> x=9 --> the # of paperback books 22x=22*9. 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. (22x-18)/(3x+18)=4/1 --> x=9 --> the # of paperback books 22x=22*9. Sufficient.

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Re: A bookshelf holds both paperback and hardcover books. The [#permalink]

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16 Oct 2013, 01:36
Bunuel wrote:
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. The # of books is 22x+3x=25x, so it's a multiple of 25. There is only one multiple of 25 between 202 and 247, inclusive: 225 --> 25x=225 --> x=9 --> the # of paperback books 22x=22*9. 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. (22x-18)/(3x+18)=4/1 --> x=9 --> the # of paperback books 22x=22*9. Sufficient.

I solved from (1) this way-
202< P+H < 247 We know 22H=3P

Finally we will get

177.76<P<280.68

we wil get som many solutions....
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Re: A bookshelf holds both paperback and hardcover books. The [#permalink]

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16 Oct 2013, 02:22
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Expert's post
honchos wrote:
Bunuel wrote:
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. The # of books is 22x+3x=25x, so it's a multiple of 25. There is only one multiple of 25 between 202 and 247, inclusive: 225 --> 25x=225 --> x=9 --> the # of paperback books 22x=22*9. 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. (22x-18)/(3x+18)=4/1 --> x=9 --> the # of paperback books 22x=22*9. Sufficient.

I solved from (1) this way-
202< P+H < 247 We know 22H=3P

Finally we will get

177.76<P<280.68

we wil get som many solutions....

First of all it should be $$202\leq{P+H}\leq{247}$$.

Next, we'll get $$177.76\leq{P}\leq{217.36}$$. From 22H=3P we know that P must be a multiple of 22. There is only one multiple of 22 in that range: 198.

Not a good way to solve.
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Re: A bookshelf holds both paperback and hardcover books. The [#permalink]

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23 Nov 2017, 15:38
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Expert's post
Hi All,

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

[Reveal] Spoiler:
D

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Re: A bookshelf holds both paperback and hardcover books. The   [#permalink] 23 Nov 2017, 15:38
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