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# The ratio of textbooks to novels to atlases in a certain classroom is

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Math Expert
Joined: 02 Sep 2009
Posts: 44290
The ratio of textbooks to novels to atlases in a certain classroom is [#permalink]

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21 Apr 2017, 07:50
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95% (hard)

Question Stats:

43% (02:46) correct 57% (02:50) wrong based on 120 sessions

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The ratio of textbooks to novels to atlases in a certain classroom is 2 to 4 to 3. If there are no other books in the classroom, and if the teacher adds textbooks until novels make up 32% of the books present, which of the following could be the number of atlases in the room?

A. 9
B. 16
C. 34
D. 51
E. 78
[Reveal] Spoiler: OA

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Posts: 250
Re: The ratio of textbooks to novels to atlases in a certain classroom is [#permalink]

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21 Apr 2017, 09:07
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The ratio of textbooks to novels to atlases in a certain classroom is 2 to 4 to 3. If there are no other books in the classroom, and if the teacher adds textbooks until novels make up 32% of the books present, which of the following could be the number of atlases in the room?

A. 9
B. 16
C. 34
D. 51
E. 78

Textbook = T, Novel = N, Atlas = A.

T:N:A = 2:4:3

Currently percentage of N = 4/(4+2+3) = 4/9 = 44.44%

T is added until N becomes 32% or we can say

if 100 is the new total number of books 32 of them is N
or if 50 is the new total number of books 16 of them is N
or if 25 is the new total number of books 8 of them is N

So, we see that N should be a multiple of 8.

There is no change in the number of Atlas, hence as per its ratio, Atlas should be a multiple of 3.

B and C are out.

A. 9

If Atlas is 9, N will be 12 (not multiple of 8). Out. [as ratio of T:N:A is 2:4:3]

D. 51

If Atlas is 51, N will be 68 (not multiple of 8). Out.

E. 78

If Atlas is 78, N will be 104 (multiple of 8). Hence, the answer.

We can also see that if N = 104, T=52 and A=78, T:N:A = 2:4:3
If 91 more T are added, % of N = 104/325=32%
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Re: The ratio of textbooks to novels to atlases in a certain classroom is [#permalink]

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21 Apr 2017, 09:16
Bunuel

As per my experience, this question should belong to 700 level category.
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Re: The ratio of textbooks to novels to atlases in a certain classroom is [#permalink]

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21 Apr 2017, 20:05
14101992 wrote:
Bunuel

As per my experience, this question should belong to 700 level category.

It can be done more easily than your experience. However, in any even, Bunuel generally posts these all initially as 600-700 and the questions are then adjusted up or down in level of difficulty as responses come in.
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The ratio of textbooks to novels to atlases in a certain classroom is [#permalink]

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21 Apr 2017, 21:12
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Bunuel wrote:
The ratio of textbooks to novels to atlases in a certain classroom is 2 to 4 to 3. If there are no other books in the classroom, and if the teacher adds textbooks until novels make up 32% of the books present, which of the following could be the number of atlases in the room?

A. 9
B. 16
C. 34
D. 51
E. 78

The ratio of $$T:N:A = 2:4:3$$, therefore let number of $$T$$, $$N$$ & $$A$$ be $$2x$$, $$4x$$ & $$3x$$ respectively.

If $$n$$ Textbooks are added then total number of books will become $$= 2x+4x+3x+n = 9x+n$$

As number of Novels are same so we have: $$4x = 32$$%$$of(9x+n)$$

Solving the equation we will get: $$n = \frac{7x}{2}$$. As $$n$$ has to be a positive integer, this implies that $$x$$ is a multiple of $$2$$

So, number of Atlases $$=3x$$ is a multiple of $$2$$ & $$3$$ i.e. 6

Only option E satisfies this criteria.

Last edited by niks18 on 15 Oct 2017, 01:22, edited 1 time in total.
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Joined: 02 Jul 2017
Posts: 1
Re: The ratio of textbooks to novels to atlases in a certain classroom is [#permalink]

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11 Jul 2017, 23:20
Let's take the unknown multiplier approach to solve this ratio problem.
The initial numbers of textbooks, novels, and atlases are 2x, 4x, and 3x, respectively. The total number of books is
2x+4x+3x=9x.

Use y to represent the number of textbooks that are added. The total number of books will then be
9x+y, and the novels will represent
4x/(9x+y)
of the books. Since we wish for the novels to be 32% of the total number of books, we can write
4x/(9x+y)=32/100=8/25

100x=72x+8y
28x=8y
7x=2y

And since books are integers we can say that x is a multiple of 2 and y is a multiple of 7. In turn, the atlases must be a multiple of 2 as well as 3. The only multiple of 6 in the answer choices is 78, so E is correct.
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Joined: 23 Dec 2013
Posts: 16
Re: The ratio of textbooks to novels to atlases in a certain classroom is [#permalink]

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12 Jul 2017, 00:11
Calculaye the number of text books added. The number is 35.
So total number of textbooks= 55.
Using the ratio, 2:4:3, calculate as follows
2/9 of X= 55. Keep the value of X in [RAISED HAND].
Now use the ratio, 2:4:3.
3 of Atlas = X.
Calculate Atlas (Number). It comes to 77.5 which is equivalent to 78 since number of books cannot be a fraction.

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Intern
Joined: 27 Sep 2017
Posts: 11
Re: The ratio of textbooks to novels to atlases in a certain classroom is [#permalink]

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05 Mar 2018, 12:29
T : N : A
2x : 4X : 3X

Total = 2X + 3X + 4X = 9X So Current % of N = 4/9
After adding some textbooks the New % of N became 32/100 = 8/25

T : N : A
... : 8X : ....
Total = 25X

Notice that no change happened to both N & A, but N which was 4x became 8x , So that means A which was 3x will be 6X . so the answer must be divisible by 6 . Answer E is the only one that is divisible by 6.

Hence E.

Another Way to approach:

Since no change happened in the real numbers of both N & A . After adding textbooks, the ratio of N : A will remain the same just as before adding textbooks.

N:A Before adding textbooks = 4/3 , after adding text box N:Total became 32/100---> 8/25
4/3 = 8/x "where x stands for Athletes books which was 3" , you will end up with x=6 . N:A new = 8x/6X
So the answer must be divisible by 6 .

Hence E.
Re: The ratio of textbooks to novels to atlases in a certain classroom is   [#permalink] 05 Mar 2018, 12:29
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