Problem Solving

bml1105 wrote:A pharmaceutical company received $3 million in royalties on the first $20 million in sales of the generic equivalent of one of its products and then $9 million in royalties on the next $108 million in sales. By approximately what percent did the ratio of royalties to sales decrease from the first $20 million in sales to the next $108 million in sales?

(A) 8%
(B) 15%
(C) 45%
(D) 52%
(E) 56%

First $20 million: royalties/sales ratio = 3/20 = 36/240
Next $108 million: royalties/sales ratio = 9/108 = 1/12 = 20/240

Noticed that I rewrote both with the SAME DENOMINATOR.
So, now all we need to is determine the percent change from 36 to 20.
To do so, we could use some more lengthy calculations [e.g., 100(36-20)/36]
HOWEVER, notice that, if we start at 36, a 50% decrease would give us 18.
So going from 36 to 20, must be a decrease that's LESS THAN 50% (but also pretty close to 50%)
Only one answer choice works.

Answer: C

Cheers,
Brent

bml1105 wrote:A pharmaceutical company received $3 million in royalties on the first $20 million in sales of the generic equivalent of one of its products and then $9 million in royalties on the next $108 million in sales. By approximately what percent did the ratio of royalties to sales decrease from the first $20 million in sales to the next $108 million in sales?

(A) 8%
(B) 15%
(C) 45%
(D) 52%
(E) 56%


Answer: C

In the explanation, it says "the percent decrease in the royalties to sales ratios is 100 times the quotient of the difference in the ratios divided by the ratio of royalties to sales for the first $20 million in sales". What?? How did they get to this? Can anyone explain it a bit better?

Thank you in advance!

Solution:

This is a percent decrease problem. We will use the formula:
percent change = [(new - old)/old] x 100 to calculate the final answer.

We first set up the ratios of royalties to sales. The first ratio will be for the first $20 million in sales, and the second ratio will be for the next $108 million in sales. Because all of the sales are in millions, we do not have to express all the trailing zeroes in our ratios.

First $20 Million

royalties/sales = 3/20

Next $108 Million

royalties/sales = 9/108 = 1/12

We can plug 1/12 and 3/20 into our percent change formula:

[(new - old)/old] x 100

[(1/12 - 3/20)/(3/20)] x 100

Because the first term is a complex fraction, we can simplify it by multiplying each fraction within the complex fraction by the LCM of 12 and 20, which is 60:

1/12 x 60 = 5 and 3/20 x 60 = 9

Therefore, we have:

[(5 - 9)/9] x 100

(-4/9) x 100

At this point we can stop and consider the answer choices. Since we know that 4/9 is just a bit less than ½, we know that (-4/9) x 100 is about a 45% decrease.

Answer:C

bml1105 wrote: ↑

Sun Apr 20, 2014 7:19 pm

A pharmaceutical company received $3 million in royalties on the first $20 million in sales of the generic equivalent of one of its products and then $9 million in royalties on the next $108 million in sales. By approximately what percent did the ratio of royalties to sales decrease from the first $20 million in sales to the next $108 million in sales?

(A) 8%
(B) 15%
(C) 45%
(D) 52%
(E) 56%


Answer: C

Solution:

This is a percent decrease problem. We will use the formula: percent change = (new – old)/old x 100 to calculate the final answer.

We first set up the ratios of royalties to sales. The first ratio will be for the first 20 million in sales and the second ratio will be for the next 108 million in sales. Because all of the sales are in millions, we do not have to express all of the trailing zeros in our ratios.

First 20 Million

royalties/sales = 3/20

Next 108 Million

royalties/sales = 9/108 = 1/12

Because each ratio is not an easy number to use, we can simplify them by multiplying each by the LCM of the two denominators, which is 60. Keep in mind that we are able to do this only because our answer choices are expressed in percentages.

First 20 Million

royalties/sales = (3/20) x 60 = 9

Next 108 Million

royalties/sales = (1/12) x 60 = 5

We can plug 9 and 5 into our percent change formula:

(new – old)/old x 100

[(5 – 9)/9] x 100

-4/9 x 100

At this point we can stop and consider the answer choices. Since we know that 4/9 is just a bit less than ½, we know that -4/9 x 100 is about a 45% decrease.

Answer: C

Hi All,

We’re told that a pharmaceutical company received $3 million in royalties on the FIRST $20 million in sales and then $9 million in royalties on the NEXT $108 million in sales. We’re asked by what APPROXIMATE percentage did the ratio of royalties on the first $20 million DECREASE to the ratio of royalties on the next $108 million.

The question clearly refers to Percentage Change, so we’ll need to use the Percentage Change Formula to answer it.

Percent Change = (New – Old)/Old = Difference/Original

First though, we need to define the two ratios involved:

$3 million/$20 million = 3/20 = .15

$9 million/$108 million = 9/108 = 1/12 = .083333

Instead of directly placing those decimals into the Percentage Change Formula, I’m going to multiply them both by 100 (which will make the final calculation easier to look at…

Old = 15
New = 8.333

(New – Old)/Old = (8.3333 – 15)/15 = -6.6666/15

Since -6/15 would be a 40% decrease, we know that -6.6666/15 would be a bit more of a decrease (but not a 50% decrease, since that would be -7.5/15) …

Final Answer: C

GMAT Assassins aren’t born, they’re made,
Rich