# Practice GMAT Problem Solving Question

The average (arithmetic mean) of a and b is 90; the average (arithmetic mean) of a and c is 150; what is the value of (b-c)/2?
 A) 60 B) 120 C) 30 D) -60 E) -40
1. Translate the question into algebra, using the formula that the average equals the sum of the terms divided by the number of terms.
90 = (a + b)/2
Multiply by 2: 180 = a + b

150 = (a + c)/2
Multiply by 2: 300 = a + c
2. At this point, there are two ways to manipulate the algebra.
3. Method 1
1. Line up the two equations so that you can subtract them.
Equation 1: a+b = 180
Equation 2: a+c = 300
2. Notice that the equations are lined up such that if you subtract them, you end up with b-c on the left and a number on the right. Since you are trying to find (b-c)/2, this setup will get you most of the way to the answer.
3. Subtract the two equations:
Equation 1: a+b = 180
Equation 2: a+c = 300
Equation 1 - 2: a - a + b - c = 180 - 300 = -120
Equation 1 - 2 (cont.): b-c = -120
4. With b-c = -120, if you divide by two, you arrive at the equation that you are being asked for:
(b-c)/2 = -120/2 = -60
4. Method 2
1. Since the question asks for the value of (b-c)/2, solve the two equations for the variable a so that they can be combined.
a = 180 – b
a = 300 – c
2. Combine the two equations by setting them equal to each other (since a = a) and manipulate them such that (b-c)/2 is on one side of the equation and a number is on the other.
180 – b = 300 – c
180 – 300 = b – c
-120 = b-c
-60 = (b – c)/2