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Average Problems
Sums, Counts, and Means

Master the art of finding averages by understanding the relationship between sums and counts. A must-know skill for the SSAT.

SSAT Middle Level

What Is an Average?

An average (also called the mean) is a single number that represents the "center" of a group of values. It tells you what each value would be if all values were the same and their total stayed unchanged.

The Average Formula

\(\text{Average} = \dfrac{\text{Sum of all values}}{\text{Number of values}}\)

Visualizing the Mean

0 20 40 60 80 100 80 90 70 100 85 Avg = 85

The Magic Triangle

Sum
Average
Count
× ÷
Find the Sum
\(\text{Sum} = \text{Average} \times \text{Count}\)
Find the Average
\(\text{Avg} = \dfrac{\text{Sum}}{\text{Count}}\)
Find the Count
\(\text{Count} = \dfrac{\text{Sum}}{\text{Average}}\)

Mean vs. Median vs. Mode

Mean Median Mode
Definition Sum ÷ Count Middle value (sorted) Most frequent value
Example: 2, 3, 3, 7, 10 5 3 3
SSAT Focus Most tested Sometimes tested Rarely tested
Affected by outliers? Yes No No

How to Solve Average Problems

1
Identify what you know. Do you have the average, the sum, or the count? Label each piece.
2
Find the total sum. If given the average and count: \(\text{Sum} = \text{Average} \times \text{Count}\).
3
Use the sum strategically. Add or subtract known values to find the missing one.
4
Compute the answer. Divide back if needed to get the final average or missing value.

Example 1: Finding the Average

A student scored 82, 91, 76, and 95 on four math tests. What is the average score?

Step 1: Find the sum
\(82 + 91 + 76 + 95 = \) \(344\)
Step 2: Divide by the count
\(\dfrac{344}{4} = \) \(86\)

Example 2: Finding a Missing Value

The average of 5 numbers is 72. Four of the numbers are 65, 80, 74, and 69. What is the fifth number?

Step 1: Find the total sum
\(\text{Sum} = 72 \times 5 = \) \(360\)
Step 2: Add the known values
\(65 + 80 + 74 + 69 = \) \(288\)
Step 3: Subtract to find the missing value
\(360 - 288 = \) \(72\)

Example 3: Combining Two Groups

A class of 12 boys has an average height of 58 inches. A class of 8 girls has an average height of 55 inches. What is the average height of all 20 students?

Step 1: Find each group's total
Boys: \(58 \times 12 = \) \(696\)    Girls: \(55 \times 8 = \) \(440\)
Step 2: Combine the totals
\(696 + 440 = \) \(1136\)
Step 3: Divide by total count
\(\dfrac{1136}{20} = \) \(56.8\)

Example 4: Adding a New Value

After 4 quizzes, Maya's average is 88. What score does she need on the 5th quiz to raise her average to 90?

Step 1: Current total sum
\(88 \times 4 = \) \(352\)
Step 2: Needed total for new average
\(90 \times 5 = \) \(450\)
Step 3: Find the needed score
\(450 - 352 = \) \(98\)

Why the Sum Is Everything

1
The average is a compressed form of the sum. It "hides" the total inside a smaller number.
2
"Unpacking" the average means multiplying: \(\text{Average} \times \text{Count} = \text{Sum}\). This is the most powerful move.
3
Once you have the sum, you can add values, remove values, or combine groups freely.

Example 5: Removing a Value

The average of 6 numbers is 45. If one number (\(30\)) is removed, what is the average of the remaining 5 numbers?

Step 1: Original total sum
\(45 \times 6 = \) \(270\)
Step 2: Subtract the removed value
\(270 - 30 = \) \(240\)
Step 3: New average
\(\dfrac{240}{5} = \) \(48\)

Common Mistakes to Avoid

✕ Averaging Averages
You cannot simply average two group averages when the groups have different sizes. Always use the total sum method.
✕ Wrong Count
Miscounting the number of values is the #1 arithmetic error. Circle the count before dividing.
✕ Forgetting to Multiply
When given an average, students forget to unpack it: \(\text{Avg} \times \text{Count} = \text{Sum}\). They try to add/subtract from the average directly.

Summary of Problem Types

Problem Type Given Find Key Step
Basic Average All values Average Sum ÷ Count
Missing Value Avg + some values One value Avg × Count, then subtract
Combined Groups Two group averages Overall avg Sum both groups, divide by total
Target Average Current avg + goal Needed score Goal sum − current sum
Remove a Value Avg + removed value New avg Original sum − value, new count

Question 1 of 5

The average of 3, 7, 11, and 15 is:

Question 2 of 5

The average of 5 numbers is 20. What is their sum?

Question 3 of 5

The average of four numbers is 50. Three of the numbers are 40, 55, and 60. What is the fourth number?

Question 4 of 5

A group of 10 students has an average test score of 85. Another group of 10 students has an average of 75. What is the average score of all 20 students?

Question 5 of 5

After 3 games, Jordan's scoring average is 18 points. How many points must Jordan score in the 4th game to raise the average to 20?