# Averages

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

Note: The average of N numbers is their sum divided by N, that is, .

Example-1
x > 0
 Column A Column B The average of x, 2x, and 6 The average of x and 2x

By the definition of an average, Column A equals  and Column B equals .

Now, if x is small, then x + 2 is larger than 3x/2. But if x is large, then 3x/2 is larger. (Verify this by plugging in x = 1 and x = 100.)

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

Example-2

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

Note: 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-3

If the average of five numbers is â€“10, and the sum of three of the numbers is 16, then what is the average of the other two numbers?

A.  â€“33
B.  â€“1
C.  5
D.  20
E.  25

Let the five numbers be abcd, 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 = â€“66.
Finally, dividing by 2 (to form the average) gives
.

Note:

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-4
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 Â½
C.   55
D.   56
E.   57 Â½

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