Algebra: Average Word Problems

There are three main types of average problems commonly encountered in school algebra: Average (Arithmetic Mean), Weighted Average and Average Speed.

Average (Arithmetic Mean)

The average (arithmetic mean) uses the formula:

The formula can also be written as

Example:

The average (arithmetic mean) of a list of 6 numbers is 20. If we remove one of the numbers, the average of the remaining numbers is 15. What is the number that was removed?

Solution:

Step 1: The removed number could be obtained by difference between the sum of original 6 numbers and the sum of remaining 5 numbers i.e. sum of original 6 numbers – sum of remaining 5 numbers

Step 2: Using the formula sum of original 6 numbers = 20 × 6 = 120 sum of remaining 5 numbers = 15 × 5 = 75

Step 3: Using the formula from step 1 Number removed = sum of original 6 numbers – sum of remaining 5 numbers

120 – 75 = 45

Answer: The number removed is 45.

Weighted Average

Another type of average problem involves the weighted average - which is the average of two or more terms that do not all have the same number of members. To find the weighted term, multiply each term by its weighting factor, which is the number of times each term occurs.

The formula for weighted average is:

Example:

A class of 25 students took a science test. 10 students had an average (arithmetic mean) score of 80. The other students had an average score of 60. What is the average score of the whole class?

Solution:

Step 1: To get the sum of weighted terms, multiply each average by the number of students that had that average and then sum them up.

80 × 10 + 60 × 15 = 800 + 900 = 1700

Step 2: Total number of terms = Total number of students = 25

Step 3: Using the formula Answer: The average score of the whole class is 68.

Be careful! You will get the wrong answer if you add the two average scores and divide the answer by two.

Average Speed

Computation of average speed is a trickier type of average problems. Average speed uses the formula:

Example:

John drove for 3 hours at a rate of 50 miles per hour and for 2 hours at 60 miles per hour. What was his average speed for the whole journey?

Solution:

Step 1: The formula for distance is

Distance = Rate × Time Total distance = 50 × 3 + 60 × 2 = 270

Step 2: Total time = 3 + 2 = 5

Step 3: Using the formula

Answer: The average speed is 54 miles per hour.

Be careful! You will get the wrong answer if you add the two speeds and divide the answer by two.