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Struggling with Algebra Variables

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Struggling with Algebra Variables

by avada » Tue Sep 04, 2012 1:41 pm
Problem 198 from OG13

Last Sunday a certain store sold copies of Newspaper A for $1 and Newspaper B for 1.25. If r percent of the stores revenue came from newspaper sales of Paper A and if p percent of the newspapers sold that day were A, which of the following expresses r in terms of p?
100p/125-p
150p/250-p
300p/375-p
400p/500-p
500p/625-p

This is how I approached the problem. I am not good enough to just figure it out so I make up numbers like manhattan recommends. So I used 6 copies sold of A and 4 of B. This gives us 10 total copies sold which means that p=60. Now r must be the revenue and if we work out the rev it comes to 11 of which 6 is comprised of A. 5 is roughly 54.5 percent of 11 so now I just plug it in. I know that the denominator multiplied by 54 should be roughly equal to the numerator and went from there.

I want to see if there is a faster way. This took me far too long
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by GMATGuruNY » Tue Sep 04, 2012 2:42 pm
avada wrote:Problem 198 from OG13

Last Sunday a certain store sold copies of Newspaper A for $1 and Newspaper B for 1.25. If r percent of the stores revenue came from newspaper sales of Paper A and if p percent of the newspapers sold that day were A, which of the following expresses r in terms of p?
100p/125-p
150p/250-p
300p/375-p
400p/500-p
500p/625-p
Plug in numbers that make the math easy.

If the number of copies of A = the number of copies of B, then p=50, since 1/2 of the newspapers sold are A.

If 4 copies of each type of newspaper are sold, then the total revenue will be an integer:
Revenue from 4 copies of A = 4*1 = 4.
Revenue from 4 copies of B = 4*1.25 = 5.
Total revenue = 4+5 = 9.

Since r% of the revenue comes from A:
r = revenue from A/total revenue * 100 = 4/9 * 100 = 400/9. This is our target.

Now we plug p=50 into the answers to see which yields our target of 400/9.
The denominator of the correct answer choice must yield a multiple of 9.
For an integer to be a multiple of 9, the sum of its digits must be a multiple of 9.
A quick scan of the denominators reveals that only D works:

125-p = 125-50 = 75. Not a multiple of 9.
250-p = 250-50 = 200. Not a multiple of 9.
375-p = 375-50 = 325. Not a multiple of 9.
500-p = 500-50 = 450, which is a multiple of 9.
625-p = 625-50 = 575. Not a multiple of 9.

The correct answer is D.
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by adthedaddy » Tue Sep 04, 2012 3:12 pm
You can also solve by the following algebraic approach.

A is sold for $1
B is sold for $1.25

Revenue of A = r% of the total revenue
Sale of A = p% of the total news papers sold

Let n = total newspapers sold

Thus, Sale of A = p% of n = (p/100)*n
and Sale of B = n - A = n - (p/100)*n = (100-p)*n/100

Revenue of A = Total Sales * price of each news paper
Revenue of A = (p/100)n*1 $
Revenue of B = [(100-p)n/100]*1.25 $

Now, its given that Revenue of A = r%

i.e. (Revenue of A / Total revenue) = r/100
i.e. (pn/100) / [(pn/100)+(100-p)n/100]*1.25 = r/100
i.e. (pn)/[pn + (5n/4)*(100-p)] = r/100
i.e. [400p/(500-p)] = r

Ans: Option - D
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by GMATGuruNY » Tue Sep 04, 2012 3:30 pm
adthedaddy wrote:You can also solve by the following algebraic approach.

A is sold for $1
B is sold for $1.25

Revenue of A = r% of the total revenue
Sale of A = p% of the total news papers sold

Let n = total newspapers sold

Thus, Sale of A = p% of n = (p/100)*n
and Sale of B = n - A = n - (p/100)*n = (100-p)*n/100

Revenue of A = Total Sales * price of each news paper
Revenue of A = (p/100)n*1 $
Revenue of B = [(100-p)n/100]*1.25 $

Now, its given that Revenue of A = r%

i.e. (Revenue of A / Total revenue) = r/100
i.e. (pn/100) / [(pn/100)+(100-p)n/100]*1.25 = r/100
i.e. (pn)/[pn + (5n/4)*(100-p)] = r/100
i.e. [400p/(500-p)] = r

Ans: Option - D
Good work.
If you're going to use algebra, the following might be a bit easier.

Let the number of newspapers sold = 100.
Since p% are A, the number of copies of A sold = p/100 * 100 = p.
Thus, the number of copies of B sold = 100-p.

Revenue from A = p*1 = p.
Revenue from B = (100-p)(1.25) = 125 - 1.25p.
Total revenue = p + (125 - 1.25p) = 125 - .25p.

Since r% of the total revenue comes from A:
r = revenue from A/total revenue * 100 = p/(125 - .25p) * 100 = 100p / (125 - .25p).

Since the answer choices do not include any decimals, we must clear .25 by multiplying by 4/4:
(100p)(4) / (125 - .25p)(4) = 400p/(500-p).

The correct answer is D.
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