It is relatively easy to find the simple average of a set of numbers. Find the sum of the numbers and divide that by the number of terms in the set. For example, if Mary’s test scores are 90, 80, 100, and 90, you can find the average by adding 90 + 80 + 100 + 90. The sum is 360. Divide 360 by the number of test scores (4), so 360 / 4 = 90. Mary’s average is 90. What if one of the tests counted twice or three times as much as the others? This would be an example of a weighted average problem. Weighted average problems tend to appear on standardized tests such as the GRE or GMAT.
Instructions
1. Start with a sample problem. Say you have a group of x numbers with an average of A1, and another group of y numbers with an average of A2. Find the combined or weighted average of the numbers in groups x and y. [x (A1) + y (A2)] / (x + y).
2. Follow with a second sample problem. Suppose 20 percent of the class scored 80, 30 percent scored 85, and 50 percent scored 90. Find the average of the entire class. Since there are three sets of average, the combined or weighted average is [x (A1) + y (A2) + z (A3)] / (x + y + z). If the class size is 100, then [20 (80) + 30 (85) + 50 (90)] / 100. 1600 + 2550 + 4500 / 100. 8650 / 100 = 86.5.
3. Learn to work with large sets of numbers. Occasionally, a standardized test problem may give you the weighted average while asking you to find the average of a set within the group. For example, if an entire class average is 86.5, and we know that 20 percent averaged 80 and 30 percent averaged 85, what is the average of the remaining 50 percent? The combined weighted average is [x (A1) + y (A2) + z (A3)] / (x + y + z). 86.5 = [20 (80) + 30 (85) + 50A3] / (20 + 30 + 50). 86.5 = (1600 + 2550 + 50A3) / (20 + 30 + 50). 86.5 = (1600 + 2550 + 50A3) / 100. 86.5 = (4150 + 50A3) / 100. Multiply both sides by 100. 100 (86.5) = [(4150 + 50A3) / 100] 100. 8650 = 4150 + 50A3. 8650 – 4150 = 4150 – 4150 + 50A3. 4500 = 50A3. 4500 / 50 = 50A3 / 50. A3 = 90.
4. Eliminate answer choices. Going back to the example from step 1, if you are pressed for time on the GRE or GMAT, you may be able to eliminate answer choices by eyeballing and back-solving. If 20 percent averaged 80, 30 percent averaged 85, and 50 percent averaged 90, find the weighted average. You know that the weighted average has to be between 80 and 90. Also, 50 percent averaged 90 while only 20 percent averaged 80, so you know the weighted average is closer to 90 than 80. Further, we know that the weighted average must be less than 87.5 (the midpoint between 85 and 90). You have 50 percent at 90, and only 30 percent at 85, with 20 percent at 80 pulling it down from 87.5. Just by eyeballing, you know the answer is greater than 85 and less than 87.5. That should eliminate a few answer choices.
5. On standardized tests such as the GRE or GMAT, you may not always be able to solve numerically. Instead, you may have to solve in terms of variables. Suppose the average of 100 scores is N. If the average of 20 scores is Q, the average of 30 scores is W, what is the average of the other 50 scores? Let x = the average of the other 50 scores. N = (20Q + 30W + 50x) / 100. 100N = 100 (20Q + 30W + 50x) / 100. 100N = 20Q + 30W + 50x. 100N (– 20Q – 30W) = 20Q + 30W (-20Q – 30W) + 50x. 50x = 100N – 20Q – 30W. 50x / 50 = (100N -20Q – 30W) / 50. x = (100N – 20Q – 30W) / 50. x = 10 (10N – 2Q – 3W) / 50. x = (10N -2Q – 3W) / 5.
Tags: percent averaged, weighted average, 4150 50A3, 100N 100N, 1600 2550, answer choices, average scores