Practice GMAT Data Sufficiency Question
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Given that A = 3y + 8x, B = 3y  8x, C = 4y + 6x, and D = 4y  6x, what is the value of x*y?
 AB + CD = 275
 AD  BC = 420
Correct Answer: B
 This problem deals with polynomials and factoring, as well as simultaneous equations with two variables. Factoring or expanding where necessary will help greatly in solving this problem.

Evaluate Statement (1) alone.
 First, multiply A and B, then C and D.
AB = (3y + 8x)(3y  8x) = 9y^{2}  64x^{2}
CD = (4y + 6x)(4y  6x) = 16y^{2}  36x^{2}  Now add AB and CD.
AB + CD = 25y^{2}  100x^{2}
Factor a 25 out of the right side of the equation.
AB + CD = 25(y^{2}  4x^{2})
Notice that the polynomial on the right side can be factored.
AB + CD = 25(y + 2x)(y  2x)
 Since AB + CD = 275, substitute this value into the equation.
275 = 25(y + 2x)(y  2x)
Divide both sides by 25.
11 = (y + 2x)(y  2x)  Let P = y + 2x and Q = y  2x. There are only four ways that 11 can be the product of the two numbers P and Q: P = 1 and Q = 11, or P = 11 and Q = 1, or P = 1 and Q = 11, or P = 11 and Q = 1. Examine the first two possibilities.
 First, P = 1 and Q = 11. Write out P and Q fully.
P = y + 2x = 1
Q = y  2x = 11
Using linear combination, add both sides of the two equations together.
2y = 10
Which means that y = 5. Plug y = 5 back into either equation and get x = 3.  Secondly, P = 11 and Q = 1. Write out P and Q fully.
P = y + 2x = 11
Q = y  2x = 1
Using linear combination, add both sides of the two equations together.
2y = 10
Which means that y = 5. Plug y = 5 back into either equation and get x = 3.  In the first case, y = 5 and x = 3, which means that x*y = 15. However, in the second case, when y = 5 and x = 3, x*y = +15. Therefore it is not possible to determine the value of x*y since the sign cannot be determined.
 Statement (1) is NOT SUFFICIENT.
 First, multiply A and B, then C and D.

Evaluate Statement (2) alone.
 First, multiply A and D, then B and C.
AD = (3y + 8x)(4y  6x) = 12y^{2} + 14xy  48x^{2}
BC = (3y  8x)(4y + 6x) = 12y^{2}  14xy  48x^{2}  Now subtract BC from AD; almost all the terms cancel out.
AD  BC = 14xy  (14xy) = 28xy
Since AD  BC = 420, substitute this value into the equation.
420 = 28xy.
Divide both sides by 28.
xy = 15  Statement (2) is SUFFICIENT.
 First, multiply A and D, then B and C.
 Since Statement (1) alone is NOT SUFFICIENT and Statement (2) alone is SUFFICIENT, answer B is correct.
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