Percentage change

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gmattoend: Junior | Next Rank: 30 Posts; Posts: 13; Joined: Sun Apr 23, 2017 6:14 am

by gmattoend » Fri Apr 20, 2018 2:00 pm

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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%

Dear Experts Sirs,

How could I solve this question?

Many appreciate

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Source: Beat The GMAT — Problem Solving | Fri Apr 20, 2018 2:00 pm

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Terry@ThePrincetonReview: Junior | Next Rank: 30 Posts; Posts: 25; Joined: Mon Mar 12, 2018 10:26 amFollowed by: 1 member; Location: Minneapolis, Minnesota, USA

by Terry@ThePrincetonReview » Fri Apr 20, 2018 10:53 pm

The logic of this question is elusive. Why would anyone want to know this? What would the answer to this question tell us? Don't worry about it! Just take the question at face value. Notice that the information about the third tier of royalties ("15% on all copies sold thereafter") isn't even necessary in order to solve. Don't be distracted by unnecessary information.

This is a percent change question--specifically a percent decrease question---so the answer is calculated according to the following formula:

$$\frac{\left(original\ value\right)-\left(end\ value\right)}{\left(original\ value\right)}\times100$$

The original value is the "ratio of the royalty percentage to number of copies ... [for] the first 10,000 copies". Take this expression at face value using the numbers given for the first tier of royalties: royalty percentage = 10, and number of copies = 10,000, so the ratio = 10/10,000 = 1/1,000 = 0.001.

The end value is the "ratio of the royalty percentage to number of copies ... [for] the next 15,000 copies". From the wording of the question, we know that this will be a lesser value. Again, take this expression at face value using the numbers given for the second tier of royalties: royalty percentage = 12, and number of copies = 15,000, so the ratio = 12/15,000 = 4/5,000 = 8/10,000 = 0.0008.

Plugging these ratios into the formula above:

$$\frac{\left(original\ value\right)-\left(end\ value\right)}{\left(original\ value\right)}\times100\ =\ \frac{0.001-0.0008}{0.001}\times100=\frac{0.0002}{0.001}\times100\ =\frac{0.2}{1}\ \times100\ =\ 20$$

The formula calculates the percent value, so the answer is 20%, choice D.

Terry Serres
The Princeton Review
Certified Instructor, Content Developer
Online Tutoring
[email protected]

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GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

by GMATGuruNY » Sat Apr 21, 2018 2:29 am

Since the question stem asks for a PERCENT CHANGE, we can make the math easier by using easier values.
The percent change will remain constant as long as the first set of copies and the second set of copies are in the following ratio:
10000:15000 = 10:15.
Thus, the prompt can be rephrased as follows:

gmattoend wrote:An author receives 10% of the publisher's net receipts in royalties on the first 10 copies of the author's book sold, 12% on the next 15 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 copies to the next 15 copies?

(A) 2%
(B) 10%
(C) 15%
(D) 20%
(E) 25%

Since a 10% royalty is earned on the first 10 copies, we get the following royalty-to-copies ratio:
10/10 = 1.
Since a 12% royalty is earned on the next 15 copies, we get the following royalty-to-copies ratio:
12/15 = 4/5.
Since the ratio decreases from 1 to 4/5, it decreases by 1/5 = 20%.

The correct answer is D.

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gmattoend: Junior | Next Rank: 30 Posts; Posts: 13; Joined: Sun Apr 23, 2017 6:14 am

by gmattoend » Sat Apr 21, 2018 12:51 pm

GMATGuruNY wrote:Since the question stem asks for a PERCENT CHANGE, we can make the math easier by using easier values.
The percent change will remain constant as long as the first set of copies and the second set of copies are in the following ratio:
10000:15000 = 10:15.
Thus, the prompt can be rephrased as follows:

gmattoend wrote:An author receives 10% of the publisher's net receipts in royalties on the first 10 copies of the author's book sold, 12% on the next 15 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 copies to the next 15 copies?

(A) 2%
(B) 10%
(C) 15%
(D) 20%
(E) 25%
Since a 10% royalty is earned on the first 10 copies, we get the following royalty-to-copies ratio:
10/10 = 1.
Since a 12% royalty is earned on the next 15 copies, we get the following royalty-to-copies ratio:
12/15 = 4/5.
Since the ratio decreases from 1 to 4/5, it decreases by 1/5 = 20%.

The correct answer is D.

Dear GMATGuru Sir,
I try to solve the question with another way but I still confused.

I assumed the copy price is $1, So
the total price 10,000 copies = $10,000,
Then, The 10% royalty of 10,000 copies = $1000, therefore, the percentage = 1000/10000 = 0.1
the total price 15,000 copies = $15,000
Then, The 12% royalty of 15,000 copies = $1800, therefore,the percentage = 1800/15000 = 0.12

Percentage change = (New value - Initial value) /Initial value

Apply in formula = (0.12 - 0.1)/ 0.1 = 0.2. This means it s 20%

Although the answer is correct, why did not my formula or answer produce negative sign to indicate decrease change? However, I applied the values in your solution, It did produce negative sign: ( 4/5 -1) /1 = -1/5 = -0.2

Many appreciate

Gmattoend

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GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

by GMATGuruNY » Sat Apr 21, 2018 2:00 pm

gmattoend wrote:Dear GMATGuru Sir,
I try to solve the question with another way but I still confused.

I assumed the copy price is $1, So
the total price 10,000 copies = $10,000,
Then, The 10% royalty of 10,000 copies = $1000, therefore, the percentage = 1000/10000 = 0.1
the total price 15,000 copies = $15,000
Then, The 12% royalty of 15,000 copies = $1800, therefore,the percentage = 1800/15000 = 0.12

Percentage change = (New value - Initial value) /Initial value

Apply in formula = (0.12 - 0.1)/ 0.1 = 0.2. This means it s 20%

Although the answer is correct, why did not my formula or answer produce negative sign to indicate decrease change? However, I applied the values in your solution, It did produce negative sign: ( 4/5 -1) /1 = -1/5 = -0.2

Many appreciate

Gmattoend

The question stem asks for the percent change in the following RATIO:
(royalty percentage) : (number of copies).'
Initial ratio = 10 : 10,000.
New ratio = 12 : 15,000.
From 10/10000 to 12/15000 = a 20% DECREASE.
In your solution, you calculate the percent change only for the values in red:
From 10 to 12 = a 20% INCREASE.
The question stem does not ask for this percent change.

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Scott@TargetTestPrep: GMAT Instructor; Posts: 8137; Joined: Sat Apr 25, 2015 10:56 amFollowed by: 29 members; Location: Los Angeles, CA; Thanked: 43 times

by Scott@TargetTestPrep » Thu Apr 26, 2018 3:03 pm

gmattoend wrote: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%

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

Scott Woodbury-Stewart
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[email protected]

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