Practice GMAT Data Sufficiency Question
Return to the list of practice GMAT data sufficiency questions.
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) = 9y2 - 64x2
CD = (4y + 6x)(4y - 6x) = 16y2 - 36x2
- Now add AB and CD.
AB + CD = 25y2 - 100x2
Factor a 25 out of the right side of the equation.
AB + CD = 25(y2 - 4x2)
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.
-
Evaluate Statement (2) alone.
- First, multiply A and D, then B and C.
AD = (3y + 8x)(4y - 6x) = 12y2 + 14xy - 48x2
BC = (3y - 8x)(4y + 6x) = 12y2 - 14xy - 48x2
- 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.
- Since Statement (1) alone is NOT SUFFICIENT and Statement (2) alone is SUFFICIENT, answer B is correct.
Return to the list of practice GMAT data sufficiency questions.