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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]

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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]

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16 Nov 2013, 05:46
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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

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]

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29 Mar 2014, 06:25
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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]

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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]

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06 Apr 2014, 06:24
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]

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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]

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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]

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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]

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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]

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

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03 Apr 2016, 10:09
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

to get the greatest number of non-fiction books, hardcover needs to be maximized while softcover minimized.
S=12
H=96
1/4 * S = 3
2/3 * H = 64
3+64=67

C
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In a home library consisting of 108 books, some hardcover an [#permalink]

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03 Apr 2016, 11:39
1. note higher proportion of hardcover nonfiction to softcover nonfiction--2/3:1/4
2. working downward from 108, identify the highest possible ratio of hardcovers to softcovers,
where hardcovers are a multiple of 3 and softcovers are a multiple of 4, with a lcm of 12
3. highest possible ratio is 108-12=96 hardcover:12 softcover
4. (2/3)(96)+(1/4)(12)=64+3=67 maximum nonfiction books in library
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22 Sep 2016, 06:36
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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

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