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AVERAGES
Problems involving averages are very common on the GRE. They can be classified into four major categories as follows.

I. The average of N numbers is their sum divided by N, that is, average = sum/N.

Example: The average of x, 2x, and 6 is

II. Weighted average: The average between two sets of numbers is closer to the set with more numbers.

Example: If on a test three people answered 90% of the questions correctly and two people answered 80% correctly, then the average for the group is not 85% but rather

Here, 90 has a weight of 3--it occurs 3 times. Whereas 80 has a weight of 2--it occurs 2 times. So the average is closer to 90 than to 80 as we have just calculated.

III. Using an average to find a number.

Sometimes you will be asked to find a number by using a given average. An example will illustrate.

Example: If the average of five numbers is -10, and the sum of three of the numbers is 16, then what is the average of the other two numbers?
(A) -33 (B) -1 (C) 5 (D) 20 (E) 25

Let the five numbers be a, b, c, d, e. Then their average is

Now three of the numbers have a sum of 16, say, a + b + c = 16. So substitute 16 for a + b + c in the average above:

Solving this equation for d + e gives d + e = -66. Finally, dividing by 2 (to form the average) gives

Hence, the answer is (A).

IV. Average Speed = Total Distance/Total Time

Although the formula for average speed is simple, few people solve these problems correctly because most fail to find both the total distance and the total time.

Example: In traveling from city A to city B, John drove for 1 hour at 50 mph and for 3 hours at 60 mph. What was his average speed for the whole trip?
(A) 50 (B) 53 1/2 (C) 55 (D) 56 (E) 57 1/2

The total distance is 1(50) + 3(60) = 230. And the total time is 4 hours. Hence,