Problem Solving

Can you please help with this problem?

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 milion in royalties on the next $108 million in sales. By approximately what percent did the ratio of rouyalties 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%

Correct answer is C

required ratio for first 20million in sales = 3/20--------(1)
required ratio for next 108 million in sales = 9/108-----(2)

required %ge = ((1)-(2))/(1)

ratio of royalties to sales

in sale 1: 3M/20M--> 3/20

in sale 2: 9M/108M-> 1/12


Change (with respect to original value-sale 1)

change/original value --> (3/20 - 1/12)/ 3/20--> (36-20)/240 * 20/3 --> 16/240 * 20/3--> 16/12*3-->4/9

4/9 is less than 0.5 (45 / 90 ) but more than 0.4 (4/10)

The percent change is 4/9 * 100 : between 40 and 50%

The solutions posted are good.

Because the question says "decrease", we need to use the percent change formula. (Words such as "decrease/increase", "greater/less than" combined with the word "percent" is a huge signal that you should use the percent change formula.)

The question can be reworded as:

"3/20 is what percent greater than 9/108?"

% change = (change/orginal)*100%

Work through the (change/original) part, and then multiply by 100% at the end.

(3/20 - 9/108)/9/108

"9/108" simplifies to 1/12 (we know that 108 is divisibe by 9 because 1+0+8 is divisible by 9)

So we have:

(3/20 -1/12)/(1/12)

We need to find the LCM of 20 and 12. 20 is 4*5 while 12 is 4*3. So, the LCM is 4*5*3 or 60:

(9-5)/60)/(9/60)

4/9

4/9 is just .44444 (1/9 is .1111; 2/9 is .2222, etc.)

So we have .4444*100% = 44.4444%

The question said "approximately." Among the answer choices, 45% is the best approximation.

(Note that according to rounding rules, it would b 44%).

I am little confused.

% change = (change/orginal)*100%

Work through the (change/original) part, and then multiply by 100% at the end.

(3/20 - 9/108)/9/108

Shouldn`t it be (3/20 - 9/108)/3/20 ???

cvec wrote:Can you please help with this problem?

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 milion in royalties on the next $108 million in sales. By approximately what percent did the ratio of rouyalties 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%

Correct answer is C

The question asks for an approximation, so we don't have to be exact in our calculations.

Original ratio = 3/20 = 15/100
Decreased ratio = 9/108, not too far from 9/100
Since 15/100 and 9/100 have the same denominator, we have to worry only about the change in the numerators.
Percent decrease from 15 to 9 is (15 - 9)/15 * 100 = 6/15 * 100 = 40%.
The only possible answer is C.

Ratio of the royalties went from 3/20 to 9/108

Thus, decrease in the ratio = 3/20 - 1/12 = 1/15

Now, the question asks you how much is the % decrease in the ratio. The ratio decreased by 1/15 from 3/20
So, another way to work this is "What % of 3/20 gives you 1/15"

x * (3/20) = (1/15) = > x = 20/45 = 4/9 = 0.44

Thus, ratio went down by 44.4%, hence (C) is the closest.

Hello Mitch,

I followed this approach for PS-Question 115 - OG 13th Edition.

I compared 3/20 to 9/x. So the proportion must have been 9/60 instead of 9/108.

So, found the change in value -> (108 - 60) and applied the formula --> Percent Increase or Decrease = (Change in Value / Original Value) * 100

(48 / 108) * 100 = 44.4 %

Though I arrived at the answer, I'm not sure if this method is good enough that it can be followed.

Thanks!

Regards,
Arvind.

arvysri wrote:Hello Mitch,

I followed this approach for PS-Question 115 - OG 13th Edition.

I compared 3/20 to 9/x. So the proportion must have been 9/60 instead of 9/108.

So, found the change in value -> (108 - 60) and applied the formula --> Percent Increase or Decrease = (Change in Value / Original Value) * 100

(48 / 108) * 100 = 44.4 %

Though I arrived at the answer, I'm not sure if this method is good enough that it can be followed.

Thanks!

Regards,
Arvind.

Your reasoning is sound.
To determine the percent increase from one fraction to another, we can proceed as follows:

1. Convert the fractions so that they have the same numerator
2. Determine the percent increase between the denominators

2/5 is what percent greater than 1/4?
Converting 1/4 so that it has the same numerator as 2/5, we get:
2/5 and 2/8.
Percent increase between the denominators = Difference/Original * 100 = (8-5)/5 * 100 = 3/5 * 100 = 60%.
Thus, 2/5 is 60% greater than 1/4.

We can follow the same line of reasoning to calculate a percent decrease:
1/4 is what percent less than 2/5?
Converting 1/4 so that it has the same numerator as 2/5, we get:
2/8 and 2/5.
Percent decrease between the denominators = Difference/Original * 100 = (8-5)/8 * 100 = 3/8 * 100 = 37.5%.
Thus, 1/4 is 37.5% less than 2/5.

I find this way of solving percents problem totally new.
May not use it, though.

GMATGuruNY wrote: The question asks for an approximation, so we don't have to be exact in our calculations.

Original ratio = 3/20 = 15/100
Decreased ratio = 9/108, not too far from 9/100
Since 15/100 and 9/100 have the same denominator, we have to worry only about the change in the numerators.
Percent decrease from 15 to 9 is (15 - 9)/15 * 100 = 6/15 * 100 = 40%.
The only possible answer is C.

You have changed denominator from 108 to 100 and got the approximate answer. What if the answer choices like 35%, 40%, 45%. :!: :!:

ProGMAT wrote: You have changed denominator from 108 to 100 and got the approximate answer. What if the answer choices like 35%, 40%, 45%. :!: :!:

If the answer choices are very close, the exact percent change can be calculated as follows.

Denominator approach:
Convert the ratios so that the denominators are the same.
Calculate the percent change in the numerators.

Numerator approach:
Convert the ratios so that the numerators are the same.
Calculate the percent change in the denominators.

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 milion in royalties on the next $108 million in sales. By approximately what percent did the ratio of rouyalties 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%

Denominator approach:
Old ratio: 3/20 = 9/60.
New ratio: 9/108 = 1/12 = 5/60.
From 9 to 5, the percent change in the numerators = (9-5)/9 * 100 = 4/9 * 100 ≈ 44.44.

Numerator approach:
Old ratio: 3/20 = 9/60.
New ratio = 9/108.
From 60 to 108, the percent change in the denominators = (108-60)/60 * 100 = 48/108 * 100 = 4/9 * 100 ≈ 44.44.

But we should always check the answer choices BEFORE we calculate.
Here, the answer choices are NOT very close and the question stem asks for an APPROXIMATION.
Thus, we can save time by estimating, as I did in my post above.