Averages made clear.
An average shares a total equally across a group. Recovering the total is often the fastest route to a missing value, a replacement question or a combined average.
Key concepts in Averages
Arithmetic mean
Average = sum / count. Therefore sum = average × count.
Combined average
Use group totals: (n₁a₁+n₂a₂)/(n₁+n₂). A simple mean of the two averages works only for equal group sizes.
Replacement
With a fixed count n, a change of d in average changes the total by nd.
Adding a value
New total = old total + added value; new count = old count + 1.
Combine totals, not averages
| Group | Count | Average | Total |
|---|---|---|---|
| First | 10 | 60 | 600 |
| Second | 20 | 75 | 1500 |
| Combined | 30 | 70 | 2100 |
A reliable solving method
- Write the number of observations clearly.
- Convert every average into a total before combining groups.
- Account for both the total change and the count change.
- Check the mean lies between the smallest and largest values.
Step-by-step solved examples
Average of 12, 18, 20 and 30?
- Sum = 12+18+20+30 = 80.
- There are four numbers.
- Average = 80/4 = 20.
Five scores average 24. Four are 20,22,25,28. Find the fifth.
- Required total = 5×24 = 120.
- Known total = 20+22+25+28 = 95.
- Missing score = 120−95 = 25.
Ten students average 60 and twenty average 75. Combined average?
- Group totals are 600 and 1500.
- Combined total = 2100; count = 30.
- Average = 2100/30 = 70, not 67.5.
Averages: practice questions
Choose one answer before opening the explanation. Write down your reasoning as well as the option; a correct guess is different from a method you can repeat.
Mean of 4,6,8?
Show answer and explanation — question 1
Answer B: 6
18/3 = 6.
Eight numbers average 15. Their sum?
Show answer and explanation — question 2
Answer D: 120
8×15 = 120.
Four values average 10. Add 20. New mean?
Show answer and explanation — question 3
Answer C: 12
(40+20)/5 = 12.
Five values: replacing 8 with 18 changes mean by?
Show answer and explanation — question 4
Answer B: 2
The total rises 10; average rises 10/5 = 2.
Three values average 12. Two are 10 and 14. Third?
Show answer and explanation — question 5
Answer B: 12
36−24 = 12.
Equal-sized groups average 40 and 60. Combined?
Show answer and explanation — question 6
Answer C: 50
Equal weights allow (40+60)/2 = 50.
Frequently asked questions
Can an average be a decimal when all values are integers?
Yes. For example, the mean of 1 and 2 is 1.5.
How do I detect a wrong combined average?
It must lie between the group averages and be closer to the larger group’s average.
Write the number of observations clearly. Check the mean lies between the smallest and largest values.