Practice GMAT Data Sufficiency Question

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15a + 6b = 30, what is the value of a-b?

b = 5 – 2.5a
9b = 9a – 81

A)	Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
B)	Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
C)	BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D)	EACH statement ALONE is sufficient.
E)	Statement (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

Correct Answer: B

Be aware that simply because you have two equations with two unknowns does not mean that a solution exists. You must have two unique equations with two unknowns in order for a solution to exist.
Evaluate Statement (1) alone.
1. There are two possible ways to solve this problem:
  Method (1): Substitute b from Statement (1) into the original equation.
  15a + 6(5 – 2.5a) = 30
  15a + 30 - 15a = 30
  30 = 30
  0 = 0
  Based upon this answer, the equation in Statement (1) is the equation in the original question solved for b. Consequently, we only have one equation and two unknowns. There is not enough information to determine a-b.
  
  Method (2): Rearrange the equation in Statement (1) and subtract this equation from the original equation.
  b = 5 – 2.5a
  b + 2.5a = 5
  2.5a + b = 5
  Multiply by 6 so b's cancel:15a + 6b = 30
  This method also shows that the equation in Statement (1) is nothing more than the original equation rearranged. Consequently, we only have one equation and two unknowns. There is not enough information to determine a-b.
2. Statement (1) is NOT SUFFICIENT.
Evaluate Statement (2) alone.
1. Try to line up the two equations so that you can subtract them:
  9b = 9a – 81
  81 + 9b = 9a
  81 = 9a - 9b
  Statement (2) Equation: 9a - 9b = 81
  Original Question Equation: 15a + 6b = 30
  At this point, you can stop since you know that you have two unique equations and two unknowns. Consequently, there will be a solution for a and for b, which means there will be one unique value for a-b. Statement (2) is SUFFICIENT.
2. If you want to solve to see this (Note: Do not solve this in a test as it takes too much time and is not necessary):
  Multiply (2) by 4: 36a - 36b = 324
  Multiply Original by 6: 90a + 36b = 180
  
  6*Original + 2*Statement(2): (90a + 36a) + (36b + -36b) = 180 + 324
  126a = 204
  a = 4
  
  Solve for b:
  9b = 9(4) - 81 = -45
  b = -5
  
  a - b = 4 - (-5) = 4 + 5 = 9
Since Statement (1) alone is NOT SUFFICIENT and Statement (2) alone is SUFFICIENT, answer B is correct.

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