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# In a home library consisting of 108 books, some hardcover an

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In a home library consisting of 108 books, some hardcover an [#permalink]  16 Nov 2013, 05:23
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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
[Reveal] Spoiler: OA
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Re: In a home library consisting of 108 books, some hardcover an [#permalink]  16 Nov 2013, 05:46
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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 :$$\frac{2}{3}*96+\frac{1}{4}*12 = 64+3 = 67.$$

C.
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Re: In a home library consisting of 108 books, some hardcover an [#permalink]  29 Mar 2014, 06:25
OK this is going to be quick

Take LCM of 2/3 and 1/4 = 12

Now we need to maximize hardcover

Then Softcover LCM = 12 * 1/4 = 3

So Hardcover the remainder = 96 * 2/3 = 64

Total # = 3+64=67

C stands
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Re: In a home library consisting of 108 books, some hardcover an [#permalink]  02 Apr 2014, 10:18
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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
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
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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
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 $$\frac{1}{3}$$
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Re: In a home library consisting of 108 books, some hardcover an [#permalink]  04 Sep 2014, 15:55
jlgdr wrote:
OK this is going to be quick

Take LCM of 2/3 and 1/4 = 12

Now we need to maximize hardcover

Then Softcover LCM = 12 * 1/4 = 3

So Hardcover the remainder = 96 * 2/3 = 64

Total # = 3+64=67

C stands

Hi jlgdr

Your method seems crisp and nice. However can u please elaborate it lil more...
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Re: In a home library consisting of 108 books, some hardcover an [#permalink]  04 Sep 2014, 20:21
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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

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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

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