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An author receives 10% of the publisher’s net receipts in royalties on the first 10,000 copies of the author’s book sold, 12% on the next 15,000 copies sold, and 15% on all copies sold thereafter. By what percent does the ratio of the royalty percentage to number of copies decrease from the first 10,000 copies to the next 15,000 copies?
(A) 2%
(B) 10%
(C) 15%
(D) 20%
(E) 25%
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(A) 2%
(B) 10%
(C) 15%
(D) 20%
(E) 25%
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Bunuel
For the first 10000 copies, the author receives 10% of the publisher's net receipts in royalties.
The author receives 12% of the publisher's receipts for the next 15000 copies.
If the publisher made 1$ per copy sold, the first 10000 copies would bring in \(1000$(\frac{10}{100} * 10000)\)
The next 15000 copies would bring in \(1800$(\frac{12}{100} * 15000)\)
The difference is \(\frac{1800}{15000} - \frac{1000}{10000} = \frac{3600 - 3000}{30000} = \frac{600}{30000} = \frac{1}{50}\)
As a percentage, the ratio decreases from the first 10000 to the next 15000 is \((\frac{1}{50})/(\frac{1000}{10000}) = \frac{1}{50} * \frac{10000}{1000} * 100 = 20\)
Therefore, the percentage by which the ratio decreases from first 10000 to next 15000 is \(20\)% (Option D)
Bunuel
Method I - Lengthy arithmetic
(1) Ratios of royalty percentage to book copies sold?
First set, 10% earned on 10,000 sold: \(\frac{0.10}{10,000}\)
Next set, 12% earned on 15,000 sold: \(\frac{0.12}{15,000}\)
(2) Rewrite ratios - LCD of 30,000
\(\frac{0.10}{10,000}=\frac{0.3}{30,000}\)
\(\frac{0.12}{15,000}=\frac{0.24}{30,000}\)
(3) "By what percent does ..." = Percent change (in this case, percent decrease)
Percent change: \((\frac{New-Old}{Old}*100)\)
Calculate the change, then multiply by 100
\(\frac{(\frac{0.24}{30,000}-\frac{0.30}{30,000})}{(\frac{0.30}{30,000})}=\)
\(\frac{(-\frac{0.06}{30,000})}{(\frac{0.30}{30,000})}=(-\frac{0.06}{30,000}*\frac{30,000}{0.30})=\)
\(-\frac{0.06}{0.30}=-\frac{1}{5}=(-0.20*100) =\)
\(20\) percent decrease
Answer D
Method II. Faster arithmetic
We can treat ratios as fractions. Make these numbers more manageable. Multiply both ratios by (100*100 = 10,000). Result:
1) Decimal in both numerators moves two places to the right
2) Number in denominators loses 2 zeros, thus we have
\(\frac{10}{100}=\frac{5}{50}\)
and
\(\frac{12}{150}=\frac{4}{50}\)
Percent decrease? \(\frac{New-Old}{Old}*100\)
\(\frac{(\frac{4}{50}-\frac{5}{50})}{(\frac{5}{50})}=\frac{(-\frac{1}{50})}{(\frac{5}{50})}=(-\frac{1}{50}*\frac{50}{5})=\)
\(-\frac{1}{5}=(-0.20*100)=20\) percent decrease
Answer D
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Bunuel
The royalty percentage for the first 10,000 copies is:
0.10/10,000 = 10/1,000,000 = 5/500,000
The royalty percentage for the next 15,000 copies is:
0.12/15,000 = 12/1,500,000 4/500,000
The percent decrease is:
(4/500,000 - 5/500,000)/(5/500,000) x 100
(-1/500,000)/(5/500,000) x 100
-1/5 x 100 = -1/5 x 100 = -20 = 20 percent decrease
Answer: D
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First 10000 copies(A)= 10/10,000
Next 15000 copies(B)= 12/15,000
(B-A)*100/A
= 20 percent
Posted from my mobile device
Next 15000 copies(B)= 12/15,000
(B-A)*100/A
= 20 percent
Posted from my mobile device
pushpitkc
Hey Pushpit,
I only understood for first case author receives = $ 1000
and for second he receives = $ 1800
Other than this I am not able to understand. Please see the highlighted part.
Can you explain to me what the question is asking for. Not able to decipher what the question is asking.
Akshdeep
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Very unlike any question on the GMAT; too much unnecessary information. To the best of my knowledge, the GMAT does not do that
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The author makes 10% on the first 10,000 copies, then 12% on the next 15,000 copies. We want to know the percent decrease going from 10/10,000 to 12/15,000, or from 5/5,000 to 4/5,000. We're just dealing with ratios, so we can multiply everything by whatever we want. So we'll have the same percent decrease going from 5 to 4, and the percent decrease is 1/5 = 20%.
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Bunuel
Given: An author receives 10% of the publisher’s net receipts in royalties on the first 10,000 copies of the author’s book sold, 12% on the next 15,000 copies sold, and 15% on all copies sold thereafter.
Asked: By what percent does the ratio of the royalty percentage to number of copies decrease from the first 10,000 copies to the next 15,000 copies?
An author receives 10% of the publisher’s net receipts in royalties on the first 10,000 copies of the author’s book sold
The ratio of the royalty percentage to number of copies = 10% / 10,000
An author receives 12% on the next 15,000 copies
The ratio of the royalty percentage to number of copies = 12% / 15,000
By what percent does the ratio of the royalty percentage to number of copies decrease from the first 10,000 copies to the next 15,000 copies \(=\frac{\frac{12\%}{15,000} - \frac{10\%}{10,000}}{\frac{10\%}{10,000}}=\frac{8- 10}{10} = -20\%\)
IMO D
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Bunuel
Ratio of royalty percentage to number of copies in first case = 10/10,000 = 1/1000 = 15/15,000 (to make it comparable)
Ratio of royalty percentage to number of copies in second case = 12/15,000
So the ratio goes from 15/15000 to 12/15000. What percentage reduction is this? Since we are looking for percentage reduction, we can ignore the common denominator since it will anyway get cancelled to get (15 - 12)/15 * 100 = 20%
Answer (D)
If you are unsure, see this \(\frac{\frac{15}{15000} - \frac{12}{15000}}{\frac{15}{15000} } * 100 = \frac{15 - 12}{15} * 100 = 20%\)
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10/(100)*(10000) = m
12/(100)*(15000) = n
n/ m = 120/150
Percentage decrease = (m-n)/ m * 100
= (1 - n/m )* 100
= (1- 120/150) * 100
= (1-4/5) * 100
= 1/5 * 100 = 20% ANs . GMATinsight
12/(100)*(15000) = n
n/ m = 120/150
Percentage decrease = (m-n)/ m * 100
= (1 - n/m )* 100
= (1- 120/150) * 100
= (1-4/5) * 100
= 1/5 * 100 = 20% ANs . GMATinsight
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i was solving this problem thinking that i needed to consider the royalties on ALL the copied sold up to 25.000 (first 10.000 and next 15.000)
in that case the royalties are
first 10.000 --> (10%*10000)/10.000 = 10%
next 15.000 --> (10%*10.000+12%*15000)/25.000 = 2800/25000 = 11.2%
so we have a absolute increase of 1.2%, with respect to the royalties of the first 10.000 we would have had --> 1.2%/10% = 12% increase
but instead the problem was simply asking for the increase from the royalties in the first 10.000 (10%) to the royalties in the next 15.000 (12%)
--> (12% - 10%)/10% = 2%/10% = 1/50 / 1/10 = 1/5 = 20%
in that case the royalties are
first 10.000 --> (10%*10000)/10.000 = 10%
next 15.000 --> (10%*10.000+12%*15000)/25.000 = 2800/25000 = 11.2%
so we have a absolute increase of 1.2%, with respect to the royalties of the first 10.000 we would have had --> 1.2%/10% = 12% increase
but instead the problem was simply asking for the increase from the royalties in the first 10.000 (10%) to the royalties in the next 15.000 (12%)
--> (12% - 10%)/10% = 2%/10% = 1/50 / 1/10 = 1/5 = 20%
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