# Averages

Problems involving averages are very common on the SAT.  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

What is the average of x, 2x, and 6?

1. x/2
2. 2x
3. x + 2
Solution

By the definition of an average, we get .

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

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?

1. â€“33
2. â€“1
3. 5
4. 20
5. 25
Solution

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
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?
1. 50
2. 53 Â½
3. 55
4. 56
5. 57 Â½
Solution

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