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In a home library consisting of 108 books, some hardcover an [#permalink]
16 Nov 2013, 05:23

7

This post was BOOKMARKED

00:00

A

B

C

D

E

Difficulty:

65% (hard)

Question Stats:

57% (03:26) correct
43% (02:37) wrong based on 155 sessions

In a home library consisting of 108 books, some hardcover and some softcover, exactly 2/3 of the hard cover and exactly 1/4 of the softcover books are nonfiction. What is the greatest possible number of nonfiction books in this home library?

Re: In a home library consisting of 108 books, some hardcover an [#permalink]
16 Nov 2013, 05:46

6

This post received KUDOS

Expert's post

registerincog wrote:

In a home library consisting of 108 books, some hardcover and some softcover, exactly \(2/3\) of the hard cover and exactly \(1/4\) of the softcover books are nonfiction. What is the greatest possible number of nonfiction books in this home library?

A) 18 B) 40 C) 67 D) 72 E) 96

Method I: # of hardcover books : 12x ; # of softcover : 108-12x

Thus, 8x+(27-3x) = # of non-fiction books(n)

or 5x = n-27

\(12x = \frac{n-27}{5}*12\) As No of hardcover books is an integral quantity, thus, (n-27) has to be a multiple of 5.

Only option C makes that.

Method II: As a greater fraction of hardcover books is NF, we must maximize that in no. Also, the remaining no of books(which are softcover) must be a multiple of 4.

As lcm of both 3 and 4 is 12, thus the maximum # of hardcover books = 108-12 = 96 and # of softcover = 12.

Thus, # of non-fiction :\(\frac{2}{3}*96+\frac{1}{4}*12 = 64+3 = 67.\)

Re: In a home library consisting of 108 books, some hardcover an [#permalink]
02 Apr 2014, 10:18

4

This post received KUDOS

The ratio of hardcove and softcover is not given. To maximize the # of books I choose 105xhardcover and 3xsoftcover. 2/3 of 105 = 70 and 1/3 of 3 = 1 70+1 = 71 F) should be the answer

registerincog wrote:

What is the greatest possible number of nonfiction books in this home library? A) 18 B) 40 C) 67 D) 72 E) 96

Re: In a home library consisting of 108 books, some hardcover an [#permalink]
06 Apr 2014, 06:24

Expert's post

boonoobo wrote:

The ratio of hardcove and softcover is not given. To maximize the # of books I choose 105xhardcover and 3xsoftcover. 2/3 of 105 = 70 and 1/3 of 3 = 1 70+1 = 71 F) should be the answer

registerincog wrote:

What is the greatest possible number of nonfiction books in this home library? A) 18 B) 40 C) 67 D) 72 E) 96

F) 71

It is \(\frac{1}{4}\) not \(\frac{1}{3}\) _________________

Re: In a home library consisting of 108 books, some hardcover an [#permalink]
04 Sep 2014, 20:21

2

This post received KUDOS

Expert's post

2

This post was BOOKMARKED

registerincog wrote:

In a home library consisting of 108 books, some hardcover and some softcover, exactly 2/3 of the hard cover and exactly 1/4 of the softcover books are nonfiction. What is the greatest possible number of nonfiction books in this home library?

A) 18 B) 40 C) 67 D) 72 E) 96

Hardcover + Softcover = 108

(2/3)*Hardcover + (1/4)*Softcover = Non fiction

We want to maximize the Non fiction books. Note that if more books are Hardcover, more books will be Non Fiction since a higher proportion (2/3 as compared with 1/4 of softcover) of Hardcover books are Non Fiction. Since there must be Softcover books too, we must have at least 4 Softcover books (so that (1/4)*4 = 1 - an integer) So the rest 104 can be Hardcover but note that (2/3)*104 is not an integer since 104 is not divisible by 3.

If there are 8 softcover books, 100 is again not divisible by 3. If there are 12 softcover books, 96 is divisible by 3. So number of hardcover books much be 96.

(2/3)*96 + (1/4)*12 = Non fiction 64 + 3 = 67 = No of non fiction books

Re: In a home library consisting of 108 books, some hardcover an [#permalink]
24 Sep 2014, 21:43

VeritasPrepKarishma wrote:

registerincog wrote:

In a home library consisting of 108 books, some hardcover and some softcover, exactly 2/3 of the hard cover and exactly 1/4 of the softcover books are nonfiction. What is the greatest possible number of nonfiction books in this home library?

A) 18 B) 40 C) 67 D) 72 E) 96

Hardcover + Softcover = 108

(2/3)*Hardcover + (1/4)*Softcover = Non fiction

We want to maximize the Non fiction books. Note that if more books are Hardcover, more books will be Non Fiction since a higher proportion (2/3 as compared with 1/4 of softcover) of Hardcover books are Non Fiction. Since there must be Softcover books too, we must have at least 4 Softcover books (so that (1/4)*4 = 1 - an integer) So the rest 104 can be Hardcover but note that (2/3)*104 is not an integer since 104 is not divisible by 3.

If there are 8 softcover books, 100 is again not divisible by 3. If there are 12 softcover books, 96 is divisible by 3. So number of hardcover books much be 96.

(2/3)*96 + (1/4)*12 = Non fiction 64 + 3 = 67 = No of non fiction books

Re: In a home library consisting of 108 books, some hardcover an [#permalink]
12 Oct 2014, 05:24

mau5 wrote:

registerincog wrote:

In a home library consisting of 108 books, some hardcover and some softcover, exactly \(2/3\) of the hard cover and exactly \(1/4\) of the softcover books are nonfiction. What is the greatest possible number of nonfiction books in this home library?

A) 18 B) 40 C) 67 D) 72 E) 96

Method I: # of hardcover books : 12x ; # of softcover : 108-12x

Thus, 8x+(27-3x) = # of non-fiction books(n)

or 5x = n-27

\(12x = \frac{n-27}{5}*12\) As No of hardcover books is an integral quantity, thus, (n-27) has to be a multiple of 5.

Only option C makes that.

Method II: As a greater fraction of hardcover books is NF, we must maximize that in no. Also, the remaining no of books(which are softcover) must be a multiple of 4.

As lcm of both 3 and 4 is 12, thus the maximum # of hardcover books = 108-12 = 96 and # of softcover = 12.

Thus, # of non-fiction :\(\frac{2}{3}*96+\frac{1}{4}*12 = 64+3 = 67.\)

C.

hey, i wanna ask you about the method 1, how about 72 which is also a multiple of 5. (72-27)=45 i can't get it...thanks

In a home library consisting of 108 books, some hardcover an [#permalink]
09 Nov 2014, 12:46

1

This post received KUDOS

I tried applying the concept I learnt here somewhere: So I got 2/3 HC + 1/4 SC. and we have to maximize this.

Let's take a common LCM o12.

8/12 HC + 3/12SC.

3/12 (HC+SC)+ 5/12 HC

3/12 (108) + 5/12 * HC

27 + 5/12 * HC...But I don't understand what to do now...

I do understand the plug in way, but would like to understand this one as well please.

EDIT: I figured it out. So now plug in the greatest possible value of HC under 108 which is divisible by 12. that would be 96. That gives us 5/12 * 96 = 40.

Therefore non fiction books are 40+27=67

gmatclubot

In a home library consisting of 108 books, some hardcover an
[#permalink]
09 Nov 2014, 12:46

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