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

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

Say the proportion is h:f:r...
Now h:f is nothing but \(\frac{h}{f}\), so when it is doubled, it becomes \(\frac{2*h}{f}\) or 2h:f
Also f:r is the same....so 2h:f:r is new proportion..

But say it is halved
Now h:f is nothing but \(\frac{h}{f}\), so when it is halved, it becomes \(\frac{h}{2f}\) or h:2f
Also f:r is the same, so 2f:2r....so h:2f:2r is new proportion..

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).

Great!!... You are too fast with your solutions....

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.

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?

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.

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.



Press Kudos if it helps!!

Bunuel VeritasKarishma chetan2u

Just out of curiosity, when the problem says :

The proportion of history to reference books is doubled, while the proportion of fantasy to reference books is maintained

The new ratio becomes H:F:R = (6*2):21:35 = 12:21:35.

Why is there no automatic change in ratio between History to Fantasy (I am struggling to visualize, why this still stays the same - besides the obvious that the problem has not mentioned it and hence assumed to be obvious.)

I'm not sure if my method is correct.

Considering 2x:7x
and 7x < 60
I wrote: 2+7=9 (sum of history and fantasy books)
I doubled the number of history books: 2*2=4
found the greatest common number between 4, 9 less than 60= 54

and solved this: (54/9)*4=24

solution attached

Hi guys i have happened to translate it in a different way

How did you guys comprehend correctly

If the proportion of history to reference books is doubled

I correctly got the combined ration of H:F:R
6:21:35


"while the proportion of fantasy to reference books is maintained"
how did you guys know it must be done from a combined ration

I kept the same 3:5 ratio

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