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

Question

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

Attachment:

1001.png [ 1.03 MiB | Viewed 12101 times ]

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

KarishmaB · Accepted Answer

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

Answer (C)

mau5 · Answer

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.

jlgdr · Answer

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

boonoobo · Answer

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 :roll:

F) 71

mau5 · Answer

It is  \frac{1}{4}  not  \frac{1}{3}

maggie27 · Answer

Hi jlgdr

Your method seems crisp and nice. However can u please elaborate it lil more...

maggie27 · Answer

Thank you so much Karishma

bachquocanh90 · Answer

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

usre123 · Answer

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

mvictor · Answer

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

gracie · Answer

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

Nevernevergiveup · Answer

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

Attachment:

1001.png [ 1.03 MiB | Viewed 12101 times ]

ScottTargetTestPrep · Answer

Since 2/3 of the hardcover are nonfiction as opposed to only 1/4 of the softcover, we want as many hardcover books as possible in the library if we want to determine the greatest possible number of nonfiction books.

However, we can we can’t have all 108 books as hardcover books since it’s stated that there is a least one softcover book. In fact, the number of softcover books must be a multiple of 4 since exactly 1/4 of them are nonfiction. Similarly, the number of hardcover books must be a multiple of 3 since exactly 2/3 of the hardcover books are nonfiction.

Let’s say there are 4 softcover books in the library; then there would be 104 hardcover books. However, 104 is not a multiple of 3.

Now let’s try 8 softcover books; then there would be 100 hardcover books. However, 100 is not a multiple of 3, either.

Finally, let’s try 12 softcover books; then there would be 96 hardcover books, and 96 is a multiple of 3. Thus, the greatest possible number of nonfiction books in the library is:

⅔(96) + ¼(12) = 64 + 3 = 67

Answer: C

jk11 · Answer

I like your method. But could you please show how you would have solved the question had you put s in terms of h (instead of h in terms of s)? Thanks

Kinshook · Answer

Since 2/3>1/4
We have to maximise hardcover books such that

H= 108-S
H max = 96
S min = 12

NF max = 2/3*96+1/4*12=64+3=67

IMO C

Posted from my mobile device

TheMBmonster · Answer

We are adding numbers two numbers to get to a new number

This new number is the sum of 2/3 x and ¼y . and that X+Y = 108. So if you have more X, you have less why.

From this we can deduce that we want to maximized X and minimize Y

X is a multiple of 3 and Y an multiple of 4.

Therefore, we can reformulate the question to ask. What is the smallest multiple of 4 that we can subtract from 108 in order to get a multiple of 3.

This would be 12, since 108 is already a multiple of 3. 108-12 =96

bumpbot · Answer

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