beat the gmat PS problem：a certain store sold

[echo-ji-520]

Hi,everyone
how to solve this PS simplely and quickly ? Please, help!!!:

Last Sunday a certain store sold copies of Newspaper A for $1.00 each and copies of Newspaper B for $1.25 each, and the store sold no other newspapers that day. If r percent of the store’s revenues from newspaper sales was from Newspaper A and if p percent of the newspapers that the store sold were copies of newspaper A, which of the following expresses r in terms of p?
A. 100p / (125 - p)
B. 150p / (250 - p)
C. 300p / (375 - p)
D. 400p / (500 - p)
E. 500p / (625 - p)

OA is D. and I use this way to solve it :
Let No of papers of A type = A ,revenue from A type = A $ ;
Let No of papers of B type = B ,revenue from B type = 1.25B $

r (A + 1.25B) / 100 = A ----(1)

p (A + B) / 100 = A --------(2)

then Solve both: (1 ) rA + 1.25rB = 100A (100-r)A = 1.25rB ............

at last I got the answer D,but the whole process take me 380s .I want to know if there is a simple way to solve it quickly ? thank you !

[Ben Ku]

If you're spending a lot of time on algebraic manipulations on this problem, you might want to try plugging in numbers.

Suppose we sell 20 newspaper A and 80 newspaper B

r = (20) / (20 + 1.25(80)) = (20) / (20 + 100) = 20 / 120 = 1/6 or 16.7%
p = 20 / 100 = 1/5 or 20%

We'll plug in p = 20 to the answer choices and see which gives 16.7.
A. 100p / (125 - p) = 2000 / 115 = 400 / 23 => Incorrect
B. 150p / (250 - p) = 3000 / 230 = 300 / 23 => Incorrect
C. 300p / (375 - p) = 6000 / 355 = 1200 / 71 => Incorrect
D. 400p / (500 - p) = 8000 / 480 = 50 / 3 = 16.7 => Correct
E. 500p / (625 - p) = 10000 / 605 = 2000 / 121 => Incorrect



Here's the Algebraic Approach:

(1) r (A + 1.25B) = 100A
(2) p (A + B) = 100A

We can solve for B:
(3) B = (100A/p) - A

Substituting (3) into (1) and simplify:
r (A + 1.25 [(100A/p) - A]) = 100A
r (A + 125A/p - 1.25A) = 100A

Divide both sides by A:
r (1 + 125/p - 1.25) = 100
r (125/p - 0.25) = 100
r = 100 / (125/p - 0.25)

Multiply the right side by 4p / 4p:
r = 400p / (500 - p)

Hope that helps.

[davetzulin]

*edit, didn't realize this was such an old thread. and this is an OG 13 problem anyway

[tim]

thanks for being careful about that, Dave..