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Each book on a certain shelf is labeled by a single category

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New post 09 Feb 2014, 01:44
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Each book on a certain shelf is labeled by a single category. For every 2 history books, there are 7 fantasy books and for every 3 fantasy books, there are 5 reference books. If the proportion of history to reference books is doubled, while the proportion of fantasy to reference books is maintained, which of the following could be the number of history books if there are fewer than 60 fantasy books on the shelf after the changes?

A. 6
B. 21
C. 24
D. 35
E. 36
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New post 09 Feb 2014, 02:04
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Mountain14 wrote:
Each book on a certain shelf is labeled by a single category. For every 2 history books, there are 7 fantasy books and for every 3 fantasy books, there are 5 reference books. If the proportion of history to reference books is doubled, while the proportion of fantasy to reference books is maintained, which of the following could be the number of history books if there are fewer than 60 fantasy books on the shelf after the changes?

A. 6
B. 21
C. 24
D. 35
E. 36


For every 2 history books, there are 7 fantasy books:
H:F = 2:7 = 6:21.

For every 3 fantasy books, there are 5 reference books:
F:R = 3:5 = 21:35.

From above, we have that H:F:R = 6:21:35.

The proportion of history to reference books is doubled, while the proportion of fantasy to reference books is maintained--> H:F:R = (6*2):21:35 = 12:21:35.

There are fewer than 60 fantasy books --> there are 21 or 21*2=42 fantasy books, thus there are 12 or 24 history books.

Answer: C.
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New post 09 Feb 2014, 02:05
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Bunuel wrote:
Mountain14 wrote:
Each book on a certain shelf is labeled by a single category. For every 2 history books, there are 7 fantasy books and for every 3 fantasy books, there are 5 reference books. If the proportion of history to reference books is doubled, while the proportion of fantasy to reference books is maintained, which of the following could be the number of history books if there are fewer than 60 fantasy books on the shelf after the changes?

A. 6
B. 21
C. 24
D. 35
E. 36


For every 2 history books, there are 7 fantasy books:
H:F = 2:7 = 6:21.

For every 3 fantasy books, there are 5 reference books:
F:R = 3:5 = 21:35.

From above, we have that H:F:R = 6:21:35.

The proportion of history to reference books is doubled, while the proportion of fantasy to reference books is maintained--> H:F:R = (6*2):21:35 = 12:21:35.

There are fewer than 60 fantasy books --> there are 21 or 21*2=42 fantasy books, thus there are 12 or 24 history books.

Answer: C.


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Hope it helps.
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New post 09 Feb 2014, 02:34
Great!!... You are too fast with your solutions.... :-D
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Re: Each book on a certain shelf is labeled by a single category  [#permalink]

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New post 24 Sep 2014, 22:43
2
History ................. Fantasy ............... Reference

2 .............................. 7

..................................... 3 ...................... 5

Equalizing

6 ............................ 21 .............................. 35

Proportion of history to reference books is doubled; other kept as it is

12x ............................... 21x ................................ 35x

Answer has to be a multiple of 12; Option A, B, D can be ignored

Multiple of 21 less than 60 = 42

x = 2

History books = 24

Answer = C
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New post 02 Nov 2014, 18:09
Bunuel wrote:
Mountain14 wrote:
Each book on a certain shelf is labeled by a single category. For every 2 history books, there are 7 fantasy books and for every 3 fantasy books, there are 5 reference books. If the proportion of history to reference books is doubled, while the proportion of fantasy to reference books is maintained, which of the following could be the number of history books if there are fewer than 60 fantasy books on the shelf after the changes?

A. 6
B. 21
C. 24
D. 35
E. 36


For every 2 history books, there are 7 fantasy books:
H:F = 2:7 = 6:21.

For every 3 fantasy books, there are 5 reference books:
F:R = 3:5 = 21:35.

From above, we have that H:F:R = 6:21:35.

The proportion of history to reference books is doubled, while the proportion of fantasy to reference books is maintained--> H:F:R = (6*2):21:35 = 12:21:35.

There are fewer than 60 fantasy books --> there are 21 or 21*2=42 fantasy books, thus there are 12 or 24 history books.
Answer: C.


Hi Bunuel,

I have a question regarding the highlighted text above. I realize that we have an integer constraint(books) but how can you assume that there can only be 21 or 42 books? Should we not spend any time to try and see if we can bring this down? Meaning, what if there were 20 fantasy books, or 30 or any other number for that matter?

Thanks!
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Re: Each book on a certain shelf is labeled by a single category  [#permalink]

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New post 03 Nov 2014, 02:19
russ9 wrote:
Bunuel wrote:
Mountain14 wrote:
Each book on a certain shelf is labeled by a single category. For every 2 history books, there are 7 fantasy books and for every 3 fantasy books, there are 5 reference books. If the proportion of history to reference books is doubled, while the proportion of fantasy to reference books is maintained, which of the following could be the number of history books if there are fewer than 60 fantasy books on the shelf after the changes?

A. 6
B. 21
C. 24
D. 35
E. 36


For every 2 history books, there are 7 fantasy books:
H:F = 2:7 = 6:21.

For every 3 fantasy books, there are 5 reference books:
F:R = 3:5 = 21:35.

From above, we have that H:F:R = 6:21:35.

The proportion of history to reference books is doubled, while the proportion of fantasy to reference books is maintained--> H:F:R = (6*2):21:35 = 12:21:35.

There are fewer than 60 fantasy books --> there are 21 or 21*2=42 fantasy books, thus there are 12 or 24 history books.
Answer: C.


Hi Bunuel,

I have a question regarding the highlighted text above. I realize that we have an integer constraint(books) but how can you assume that there can only be 21 or 42 books? Should we not spend any time to try and see if we can bring this down? Meaning, what if there were 20 fantasy books, or 30 or any other number for that matter?

Thanks!


We have that H:F:R = 12:21:35 and that there are fewer than 60 fantasy books (F < 60). Thus, F is a multiple of 21 less than 60: 21 or 42.

Hope it's clear.
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New post 07 Nov 2014, 13:42
Bunuel wrote:
russ9 wrote:
Hi Bunuel,

I have a question regarding the highlighted text above. I realize that we have an integer constraint(books) but how can you assume that there can only be 21 or 42 books? Should we not spend any time to try and see if we can bring this down? Meaning, what if there were 20 fantasy books, or 30 or any other number for that matter?

Thanks!


We have that H:F:R = 12:21:35 and that there are fewer than 60 fantasy books (F < 60). Thus, F is a multiple of 21 less than 60: 21 or 42.

Hope it's clear.


Yes, I get that part, but couldn't we multiply 21 by a non integer which would result in an integer? Do we not have to try those combinations?
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Each book on a certain shelf is labeled by a single category  [#permalink]

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New post 09 Nov 2014, 05:08
russ9 wrote:
Bunuel wrote:
russ9 wrote:
Hi Bunuel,

I have a question regarding the highlighted text above. I realize that we have an integer constraint(books) but how can you assume that there can only be 21 or 42 books? Should we not spend any time to try and see if we can bring this down? Meaning, what if there were 20 fantasy books, or 30 or any other number for that matter?

Thanks!


We have that H:F:R = 12:21:35 and that there are fewer than 60 fantasy books (F < 60). Thus, F is a multiple of 21 less than 60: 21 or 42.

Hope it's clear.


Yes, I get that part, but couldn't we multiply 21 by a non integer which would result in an integer? Do we not have to try those combinations?


We are told that the number of fantasy books is a multiple of 21. Multiples of 21 are 21, 42, 63, ... For example, 30 = 21*30/21 is NOT a multiple of 21.
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Re: Each book on a certain shelf is labeled by a single category  [#permalink]

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New post 09 Nov 2014, 07:53
Bunuel wrote:
Mountain14 wrote:
Each book on a certain shelf is labeled by a single category. For every 2 history books, there are 7 fantasy books and for every 3 fantasy books, there are 5 reference books. If the proportion of history to reference books is doubled, while the proportion of fantasy to reference books is maintained, which of the following could be the number of history books if there are fewer than 60 fantasy books on the shelf after the changes?

A. 6
B. 21
C. 24
D. 35
E. 36


For every 2 history books, there are 7 fantasy books:
H:F = 2:7 = 6:21.

For every 3 fantasy books, there are 5 reference books:
F:R = 3:5 = 21:35.

From above, we have that H:F:R = 6:21:35.

The proportion of history to reference books is doubled, while the proportion of fantasy to reference books is maintained--> H:F:R = (6*2):21:35 = 12:21:35.

There are fewer than 60 fantasy books --> there are 21 or 21*2=42 fantasy books, thus there are 12 or 24 history books.

Answer: C.


Sorry to bother you,
If instead of --> the proportion of history to reference books is doubled, while the proportion of fantasy to reference books is maintained
We were given --> the proportion of history to reference books is halved, while the proportion of fantasy to reference books is maintained

Then H:F:R = 6:(21*2):(35*2) = 12:42:70.
Is this correct?
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Re: Each book on a certain shelf is labeled by a single category  [#permalink]

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New post 12 Sep 2016, 07:21
Hello,

Can someone clarify below for me please?

When the question stem says: for every 2 History books there are 7 Fantasy books.

I translated this as 2H = 7F => H/F = 7/2.

May I know why this interpretation is wrong?
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New post 15 Nov 2016, 00:28
Pretz wrote:
Hello,

Can someone clarify below for me please?

When the question stem says: for every 2 History books there are 7 Fantasy books.

I translated this as 2H = 7F => H/F = 7/2.

May I know why this interpretation is wrong?


You can't write it as 2H = 7F.

Take a simple example - For every candy given to A, B gets 3 candies.

So if A gets 1 candy B would be getting 3 candies, if A gets 2 candies B would be getting 6 candies.

So if you are asked about ratio of candies A:B it would be 1:3 or 2:6 and so on.

So in original question it would be H:F is 2:7

HTH.

Bunuel -- is the use of word proportion in the original question correct? Shouldn't it be ratio instead?

TIA.
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New post 18 Jul 2017, 14:00
Each book on a certain shelf is labeled by a single category. For every 2 history books, there are 7 fantasy books and for every 3 fantasy books, there are 5 reference books. If the proportion of history to reference books is doubled, while the proportion of fantasy to reference books is maintained, which of the following could be the number of history books if there are fewer than 60 fantasy books on the shelf after the changes?

A. 6
B. 21
C. 24
D. 35
E. 36

H:F = 2:7 & F:R = 3:5

To get the proportion of H:R we need to combine these ratios (same thing as finding a common denominator).

H : F : R
2 : 7, 3 : 5

H : F : R
6 : 21 : 35

So the ratio of H:F = 6:35. Doubling it gives us 12:35.

So now we have:

H:F:R
12:21:35

And we are told that there are fewer than 60 Fantasy books.

Remember that for the series of proportions H:F:R these are the ratios of one to the other three, not the absolute number. The absolute number is each one of these ratios multiplied by a common multiplier 'm'.

H:F:R
12m:21m:35m

Each of these terms represents an discrete number of books. So 21m (the number of Fantasy books) is less than 60. Therefore, we are considering all multiples of 21 less than 60. There are only two of those, 21 and 42.

So there are either 21 or 42 books which means the multiplier m is either 1 or 2.

This means that the options for the number of history books is either 12 or 24. Of the answer choices, only 24 is present so (C).
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Re: Each book on a certain shelf is labeled by a single category  [#permalink]

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New post 23 Nov 2018, 04:52
Pretz wrote:
Hello,

Can someone clarify below for me please?

When the question stem says: for every 2 History books there are 7 Fantasy books.

I translated this as 2H = 7F => H/F = 7/2.

May I know why this interpretation is wrong?




I was struck in the same doubt and here is my take at it .

When we say for every 2 history book you get 7 fantasy books ,this implies 2H for Every 7F .....H/F =2/7 means 7F for every 2H..thats what the ratio means.

When we write 2H=7F,it means H/F=7/2 and so this implies 2 fantasy books for every 7 history books.

This is such a subtle and good concept.



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Re: Each book on a certain shelf is labeled by a single category   [#permalink] 23 Nov 2018, 04:52
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