About Average
An average is a total divided by a count, and almost every question in this topic is solved by writing down that total. The one place the simple idea breaks down is average speed, where the answer is not the average of the two speeds.
What you need to understand
- Average = sum of the values divided by the number of values.
- Total = average x number of values, and this rearrangement answers most questions in the topic.
- Adding a value above the running average raises it; adding one below the average lowers it.
- If every value is increased by the same amount, the average increases by exactly that amount.
- Average speed is total distance divided by total time, and that definition always works.
- When two equal distances are covered at different speeds, the average speed is 2xy/(x + y) and not (x + y)/2, because the slower leg takes longer.
Formulas to remember
How to work through these questions
- For any average question, first work out the total by multiplying the average by the count.
- When a value is missing, subtract the sum of the known values from that total.
- For average speed, write down the total distance and the total time separately, and divide at the end.
- Treat the two legs of a journey separately whenever the speeds differ, because the times differ as well.
- Check that the average lies between the smallest and the largest value. If it does not, something is wrong.
Mistakes that cost marks
- Averaging two speeds directly when the times are different, which understates the average speed.
- Dividing by the wrong count, for example by the number of values after one has been removed.
- Giving the total when the question asks for the average, or the other way round.
- Forgetting to multiply the average back by the count when working out a missing value.
- Assuming the average must equal one of the values in the list.
Worked example
The average of 5 numbers is 24. Four of them are 20, 26, 18 and 30. Find the fifth number.
- Total of all five numbers = average x count = 24 x 5 = 120.
- Sum of the four known numbers = 20 + 26 + 18 + 30 = 94.
- The fifth number is the difference: 120 - 94 = 26.
- Check: the average lies between the smallest value 18 and the largest 30, which it does.
Answer: 26
Practice questions with answers
A few Average questions with the full solution shown, so you can see
how the method is applied before you attempt the timed set.
Question 1
Find the average of the numbers 64, 52, 58, 64, 50, 54.
Answer: Option B — with explanation
Average = sum of the values / number of values.
Sum = 342 and there are 6 values.
Average = 342 / 6 = 57.
Common mistakes
- gave the range instead of the average: 14
- gave the sum instead of the average: 342
- divided by one value too few: 68.4
Question 2
Find the average of the numbers 23, 31, 31, 21, 24, 25, 33, 36.
Answer: Option A — with explanation
Average = sum of the values / number of values.
Sum = 224 and there are 8 values.
Average = 224 / 8 = 28.
Common mistakes
- gave the sum instead of the average: 224
- divided by one value too few: 32
- gave the range instead of the average: 15
Question 3
Find the average of the numbers 49, 42, 31, 36, 39, 47, 36.
Answer: Option C — with explanation
Average = sum of the values / number of values.
Sum = 280 and there are 7 values.
Average = 280 / 7 = 40.
Common mistakes
- gave the sum instead of the average: 280
- gave the range instead of the average: 18
- divided by one value too few: 46.67
Question 4
The average of 6 numbers is 47. All but one of them are 63, 51, 12, 77, 70. Find the missing number.
Answer: Option C — with explanation
Total of all 6 numbers = 47 x 6 = 282.
Sum of the known numbers = 273.
Missing number = 282 - 273 = 9.
Common mistakes
- gave the average itself instead of the missing number: 47
- gave the total instead of the missing value: 282
- gave the sum of the known numbers: 273
Question 5
What is the average of 44, 37, 32, 31, 28, 27, 46?
Answer: Option C — with explanation
Average = sum of the values / number of values.
Sum = 245 and there are 7 values.
Average = 245 / 7 = 35.
Common mistakes
- gave the sum instead of the average: 245
- gave the range instead of the average: 19
- divided by one value too few: 40.83
Frequently asked questions
Why can we not average two speeds directly?
Because the two speeds are held for different lengths of time. If one leg is slow it takes longer, so it should carry more weight in the average. Only when the times are equal does the arithmetic mean give the right answer.
What is the formula for average speed over equal distances?
Average speed = 2xy / (x + y) where x and y are the two speeds. For 30 km/hr and 60 km/hr the average is 2 x 30 x 60 / 90 = 40 km/hr, not 45.
How do I find a missing value when I know the average?
Multiply the average by the number of values to get the total, then subtract the values you already know. The remainder is the missing value.
Does the average always have to be one of the numbers in the list?
No. The average of 4 and 7 is 5.5, which does not appear in the list at all. The average only has to lie between the smallest and the largest value.
