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Intern  Joined: 02 Apr 2013
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Question Stats: 56% (03:02) correct 44% (02:33) wrong based on 388 sessions

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Keats Library purchases a number of new books, all in the category of biography; the library does not acquire any other books. With the addition of the new biographies, the biography collection of the library amounts to 37.5% of the new total number of books in the library. If prior to the purchase, only 20% of the books in Keats Library were biographies, by what percent has the number of biographies in the library increased?

A. 17.5%
B. 62.5%
C. 87.5%
D. 140%
E. 150%
Math Expert V
Joined: 02 Sep 2009
Posts: 64216
Re: Keats Library purchases a number of new books, all in the ca  [#permalink]

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JBGMAT wrote:
Before the purchase: the biography collection of the library amounted to 20% (1/5 th) of the total number of books. So, before the purchase, 1 out of 5 books were biographies.

After the purchase: the biography collection of the library amounted to 37.5% (3/8 th) of the NEW total number of books. So, after the purchase, 3 out of 8 books were biographies.

Hope it's clear.

I am not convinced with the MGMAT's solution presented above. Would be nice if you could present more detailed calculations. Thanks

Keats Library purchases a number of new books, all in the category of biography; the library does not acquire any other books. With the addition of the new biographies, the biography collection of the library amounts to 37.5% of the new total number of books in the library. If prior to the purchase, only 20% of the books in Keats Library were biographies, by what percent has the number of biographies in the library increased?
A. 17.5%
B. 62.5%
C. 87.5%
D. 140%
E. 150%

Say prior to the purchase, there were total of 100 books. We are told that, by that time, the biography collection of the library amounted to 20% (1/5 th) of the total number of books, thus there were 20 biographies.

After that, x biography books were purchased, thus the total number of books became 100+x, out of which 20+x were biographies. Since after the purchase the biography collection of the library amounted to 37.5% (3/8 th) of the NEW total number of book, then $$20+x=\frac{3}{8}*(100+x)$$ --> $$x=28$$.

So, we have that initially there were 20 biography books and after the purchase there were 20+28=48 biography books --> % increase = $$\frac{difference}{original}*100 = \frac{28}{20}*100=140%$$.

Hope it's clear.
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Intern  Joined: 02 Apr 2013
Posts: 13
Re: Keats Library purchases a number of new books, all in the ca  [#permalink]

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Since the the biography collection of the library amounts to 37.5%

I thought of assuming the actual number to be 75.

and thus the total number of new books would come to 200

Thus the number of books before the purchase would be 200-75 = 125 (Since only the biographies were added)

thus 20% of 125 = 25

therefore the % increase in biographies would be (75-25)/25 = 200% (which is no where in the answer choices....

Can someone plz tell me where am I going wrong.......?????
Intern  Joined: 02 Apr 2013
Posts: 13
Re: Keats Library purchases a number of new books, all in the ca  [#permalink]

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the OE is attached .....
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Math Expert V
Joined: 02 Sep 2009
Posts: 64216
Re: Keats Library purchases a number of new books, all in the ca  [#permalink]

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manugmat123 wrote:
Since the the biography collection of the library amounts to 37.5%

I thought of assuming the actual number to be 75.

and thus the total number of new books would come to 200

Thus the number of books before the purchase would be 200-75 = 125 (Since only the biographies were added)

thus 20% of 125 = 25

therefore the % increase in biographies would be (75-25)/25 = 200% (which is no where in the answer choices....

Can someone plz tell me where am I going wrong.......?????

The red part is not correct. Out of 75 biographies not all of them were recently purchased.
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Intern  Joined: 02 Apr 2013
Posts: 13
Re: Keats Library purchases a number of new books, all in the ca  [#permalink]

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ooooooh...

yes yes  thanks a lot
Intern  Joined: 31 Dec 2012
Posts: 10
WE: Analyst (Consulting)
Re: Keats Library purchases a number of new books, all in the ca  [#permalink]

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Bunuel wrote:
manugmat123 wrote:
Since the the biography collection of the library amounts to 37.5%

I thought of assuming the actual number to be 75.

and thus the total number of new books would come to 200

Thus the number of books before the purchase would be 200-75 = 125 (Since only the biographies were added)

thus 20% of 125 = 25

therefore the % increase in biographies would be (75-25)/25 = 200% (which is no where in the answer choices....

Can someone plz tell me where am I going wrong.......?????

The red part is not correct. Out of 75 biographies not all of them were recently purchased.

Hi,

Keats Library purchases a number of new books, all in the category of biography; the library does not acquire any other books. With the addition of the new biographies, the biography collection of the library amounts to 37.5% of the new total number of books in the library. If prior to the purchase, only 20% of the books in Keats Library were biographies, by what percent has the number of biographies in the library increased?

A. 17.5%
B. 62.5%
C. 87.5%
D. 140%
E. 150%

Pls throw some more light on this part. "With the addition of the new biographies, the biography collection of the library amounts to 37.5% of the new total number of books"

Regards,

Jasmine
Math Expert V
Joined: 02 Sep 2009
Posts: 64216
Re: Keats Library purchases a number of new books, all in the ca  [#permalink]

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1
JBGMAT wrote:
Bunuel wrote:
manugmat123 wrote:
Since the the biography collection of the library amounts to 37.5%

I thought of assuming the actual number to be 75.

and thus the total number of new books would come to 200

Thus the number of books before the purchase would be 200-75 = 125 (Since only the biographies were added)

thus 20% of 125 = 25

therefore the % increase in biographies would be (75-25)/25 = 200% (which is no where in the answer choices....

Can someone plz tell me where am I going wrong.......?????

The red part is not correct. Out of 75 biographies not all of them were recently purchased.

Hi,

Keats Library purchases a number of new books, all in the category of biography; the library does not acquire any other books. With the addition of the new biographies, the biography collection of the library amounts to 37.5% of the new total number of books in the library. If prior to the purchase, only 20% of the books in Keats Library were biographies, by what percent has the number of biographies in the library increased?

A. 17.5%
B. 62.5%
C. 87.5%
D. 140%
E. 150%

Pls throw some more light on this part. "With the addition of the new biographies, the biography collection of the library amounts to 37.5% of the new total number of books"

Regards,

Jasmine

Before the purchase: the biography collection of the library amounted to 20% (1/5 th) of the total number of books. So, before the purchase, 1 out of 5 books were biographies.

After the purchase: the biography collection of the library amounted to 37.5% (3/8 th) of the NEW total number of books. So, after the purchase, 3 out of 8 books were biographies.

Hope it's clear.
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Intern  Joined: 31 Dec 2012
Posts: 10
WE: Analyst (Consulting)
Re: Keats Library purchases a number of new books, all in the ca  [#permalink]

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Hi,

Keats Library purchases a number of new books, all in the category of biography; the library does not acquire any other books. With the addition of the new biographies, the biography collection of the library amounts to 37.5% of the new total number of books in the library. If prior to the purchase, only 20% of the books in Keats Library were biographies, by what percent has the number of biographies in the library increased?

A. 17.5%
B. 62.5%
C. 87.5%
D. 140%
E. 150%

Pls throw some more light on this part. "With the addition of the new biographies, the biography collection of the library amounts to 37.5% of the new total number of books"

Regards,

Jasmine[/quote]

Before the purchase: the biography collection of the library amounted to 20% (1/5 th) of the total number of books. So, before the purchase, 1 out of 5 books were biographies.

After the purchase: the biography collection of the library amounted to 37.5% (3/8 th) of the NEW total number of books. So, after the purchase, 3 out of 8 books were biographies.

Hope it's clear.[/quote]

I am not convinced with the MGMAT's solution presented above. Would be nice if you could present more detailed calculations. Thanks
Intern  Joined: 31 Dec 2012
Posts: 10
WE: Analyst (Consulting)
Re: Keats Library purchases a number of new books, all in the ca  [#permalink]

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Yes, its clear now. Many thanks Bunuel.

Regards,
Intern  Joined: 12 Dec 2013
Posts: 23
Re: Keats Library purchases a number of new books, all in the ca  [#permalink]

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The mistake I made on this problem was that I took (37.5% - 20%) / 20% = 17.5/20 = 87.5%. Can someone explain why this is not correct?
Math Expert V
Joined: 02 Sep 2009
Posts: 64216
Re: Keats Library purchases a number of new books, all in the ca  [#permalink]

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bekerman wrote:
The mistake I made on this problem was that I took (37.5% - 20%) / 20% = 17.5/20 = 87.5%. Can someone explain why this is not correct?

That's wrong because 37.5% and 20% are from different quantities.

Consider this, if biography books were 10% of 100 books and then became 50% of 1,000,000 books would it be correct to say that the increase in the number of biographies was (50-10)/10 = 400%?

Hope it's clear.
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Joined: 16 Oct 2010
Posts: 10467
Location: Pune, India
Re: Keats Library purchases a number of new books, all in the ca  [#permalink]

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2
manugmat123 wrote:
Keats Library purchases a number of new books, all in the category of biography; the library does not acquire any other books. With the addition of the new biographies, the biography collection of the library amounts to 37.5% of the new total number of books in the library. If prior to the purchase, only 20% of the books in Keats Library were biographies, by what percent has the number of biographies in the library increased?

A. 17.5%
B. 62.5%
C. 87.5%
D. 140%
E. 150%

Responding to a pm:

Yes, you can do it using weighted averages:

You are mixing a collection consisting of 20% biographies with a collection consisting of 100% biographies to give a collection consisting of 37.5% biographies. The weights of the two collections will be:

w1/w2 = (A2 - Aavg)/(Aavg - A1) = (1 - 3/8)/(3/8 - 1/5) = (5/8)/(7/40) = 25/7

So if initial collection had a total of 25 books (including 1/5th biographies i.e. 5 biographies), 7 new biographies were added.

Biographies have increased by (7/5)*100 = 140%

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Re: Keats Library purchases a number of new books, all in the ca  [#permalink]

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VeritasPrepKarishma wrote:
manugmat123 wrote:
Keats Library purchases a number of new books, all in the category of biography; the library does not acquire any other books. With the addition of the new biographies, the biography collection of the library amounts to 37.5% of the new total number of books in the library. If prior to the purchase, only 20% of the books in Keats Library were biographies, by what percent has the number of biographies in the library increased?

A. 17.5%
B. 62.5%
C. 87.5%
D. 140%
E. 150%

Responding to a pm:

Yes, you can do it using weighted averages:

You are mixing a collection consisting of 20% biographies with a collection consisting of 100% biographies to give a collection consisting of 37.5% biographies. The weights of the two collections will be:

w1/w2 = (A2 - Aavg)/(Aavg - A1) = (1 - 3/8)/(3/8 - 1/5) = (5/8)/(7/40) = 25/7

So if initial collection had a total of 25 books (including 1/5th biographies i.e. 5 biographies), 7 new biographies were added.

Biographies have increased by (7/5)*100 = 140%

Thanks a lot Karishma for the quick response. That made my life much easier.
Highlighted above in red is where I went wrong earlier.
After reaching w1/w2=25/7 I considered that number of biographies in earlier collection is 25 (vs 5 in your solution)
Is there a technique to identify if w1 refers to the whole collection or only the biographies?
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Joined: 16 Oct 2010
Posts: 10467
Location: Pune, India
Re: Keats Library purchases a number of new books, all in the ca  [#permalink]

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keyrun wrote:
VeritasPrepKarishma wrote:
manugmat123 wrote:
Keats Library purchases a number of new books, all in the category of biography; the library does not acquire any other books. With the addition of the new biographies, the biography collection of the library amounts to 37.5% of the new total number of books in the library. If prior to the purchase, only 20% of the books in Keats Library were biographies, by what percent has the number of biographies in the library increased?

A. 17.5%
B. 62.5%
C. 87.5%
D. 140%
E. 150%

Responding to a pm:

Yes, you can do it using weighted averages:

You are mixing a collection consisting of 20% biographies with a collection consisting of 100% biographies to give a collection consisting of 37.5% biographies. The weights of the two collections will be:

w1/w2 = (A2 - Aavg)/(Aavg - A1) = (1 - 3/8)/(3/8 - 1/5) = (5/8)/(7/40) = 25/7

So if initial collection had a total of 25 books (including 1/5th biographies i.e. 5 biographies), 7 new biographies were added.

Biographies have increased by (7/5)*100 = 140%

Thanks a lot Karishma for the quick response. That made my life much easier.
Highlighted above in red is where I went wrong earlier.
After reaching w1/w2=25/7 I considered that number of biographies in earlier collection is 25 (vs 5 in your solution)
Is there a technique to identify if w1 refers to the whole collection or only the biographies?

When you say the concentration (for lack of a better term) is 20%, ask yourself - of what? 20% of what? It is 20% of the original collection. It means the weight w1 is original collection. When you say A2 is 100%, ask yourself - 100% of what? 100% of the new addition (which are all biographies). So the new addition is w2.

The formula is just a rephrasing of

Cavg = (C1*w1 + C2*w2)/(w1 + w2)

If C1 is 20%, you need w1 to be the original collection.

20% of original collection.

20% * Original Collection

and so on...
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Re: Keats Library purchases a number of new books, all in the ca  [#permalink]

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Bunuel wrote:
bekerman wrote:
The mistake I made on this problem was that I took (37.5% - 20%) / 20% = 17.5/20 = 87.5%. Can someone explain why this is not correct?

That's wrong because 37.5% and 20% are from different quantities.

Consider this, if biography books were 10% of 100 books and then became 50% of 1,000,000 books would it be correct to say that the increase in the number of biographies was (50-10)/10 = 400%?

Hope it's clear.

Bunuel, is the tip-off here (re: unknown quantities) that the equation uses %s instead of absolute numbers?
- meaning, 20% can be 1/5 or 20/100 (quantities changed)
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Re: Keats Library purchases a number of new books, all in the ca  [#permalink]

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manugmat123 wrote:
Keats Library purchases a number of new books, all in the category of biography; the library does not acquire any other books. With the addition of the new biographies, the biography collection of the library amounts to 37.5% of the new total number of books in the library. If prior to the purchase, only 20% of the books in Keats Library were biographies, by what percent has the number of biographies in the library increased?

A. 17.5%
B. 62.5%
C. 87.5%
D. 140%
E. 150%

We can let the original number of books = 100. Thus, there were 20 biographies in the collection and 80 non-biographies. We can also let x = the number of new biographies and we can create the following equation:

(20 + x)/(100 + x) = 0.375

20 + x = 37.5 + 0.375x

0.625x = 17.5

x = 28

So, now we have 20 + 28 = 48 biographies. We use the formula percent increase = (new - old)/old x 100 to determine the percentage increase:

(48 - 20)/20 x 100

28/20 x 100

7/5 x 100 = 140%

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Re: Keats Library purchases a number of new books, all in the ca  [#permalink]

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When dealing with ratios, proportions, and percentages don't forget that each can be satisfied by an infinite number of combinations. Take $$\frac{1}{5}$$ for example:

There could be $$1$$ biography and $$5$$ total books $$\frac{1}{5}=20\%$$

There could be $$2$$ biography and $$10$$ total books $$\frac{2}{10}=20\%$$

There could be $$3$$ biography and $$15$$ total books $$\frac{3}{15}=20\%$$

This pattern proceeds into positive infinite. MANY of the questions related to ratios, proportions, and percentages will ask you about the nature of the relationship after some change. You can use some variable multiplier to handle the fact that infinite numbers could satisfy the relationship described by the ratio, proportion, or percentage. I'll call the multiplier $$m$$ and the discrete number of biographies added $$x$$

$$\frac{m+x}{5m+x}=\frac{3}{8}$$

$$8m + 8x=15m+3x$$

$$5x=7m$$

$$\frac{x}{m}=\frac{7}{5}$$

So the multiplier is some multiple of 5 and the number of books added is the same multiple of 7.

Test that out to be sure.

$$\frac{5}{25}=20\%$$

$$\frac{5+7}{25+7}=\frac{12}{32}=\frac{3}{8}$$

So if you have $$5$$ biographies of $$25$$ total books, and you add $$7$$ biographies, you'll then have $$12$$ biographies in $$32$$ total books. This satisfies the questions constraints.

$$\frac{12}{5}-1=140\%$$

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Re: Keats Library purchases a number of new books, all in the ca  [#permalink]

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spence11 wrote:
When dealing with ratios, proportions, and percentages don't forget that each can be satisfied by an infinite number of combinations. Take $$\frac{1}{5}$$ for example:

There could be $$1$$ biography and $$5$$ total books $$\frac{1}{5}=20\%$$

There could be $$2$$ biography and $$10$$ total books $$\frac{2}{10}=20\%$$

There could be $$3$$ biography and $$15$$ total books $$\frac{3}{15}=20\%$$

This pattern proceeds into positive infinite. MANY of the questions related to ratios, proportions, and percentages will ask you about the nature of the relationship after some change. You can use some variable multiplier to handle the fact that infinite numbers could satisfy the relationship described by the ratio, proportion, or percentage. I'll call the multiplier $$m$$ and the discrete number of biographies added $$x$$

$$\frac{m+x}{5m+x}=\frac{3}{8}$$

$$8m + 8x=15m+3x$$

$$5x=7m$$

$$\frac{x}{m}=\frac{7}{5}$$

So the multiplier is some multiple of 5 and the number of books added is the same multiple of 7.

Test that out to be sure.

$$\frac{5}{25}=20\%$$

$$\frac{5+7}{25+7}=\frac{12}{32}=\frac{3}{8}$$

So if you have $$5$$ biographies of $$25$$ total books, and you add $$7$$ biographies, you'll then have $$12$$ biographies in $$32$$ total books. This satisfies the questions constraints.

$$\frac{12}{5}-1=140\%$$

Why does 1 need to be subtracted in the last step??
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Re: Keats Library purchases a number of new books, all in the ca  [#permalink]

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Because the question is asking for the percent increase not the percent of the original number.

Percent increase is given by:

$$\frac{new-old}{old}\rightarrow\frac{new}{old}-1$$

Percent difference from the original value is given by:

$$\frac{new}{old}$$ Re: Keats Library purchases a number of new books, all in the ca   [#permalink] 30 Aug 2017, 13:16

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