These questions give you an arithmetic average and ask you to recover a total or an unseen value. Some also change the size of a set, remove a value, or combine groups of different sizes.
Work backwards through the total
Start by multiplying the stated average by the full number of values. This reconstructs the total that the complete set must have. Then add the values you know and subtract that known total from the required total. For example, if four readings average 9 and three readings are 6, 8 and 10, the required total is 36, the known total is 24, and the missing reading is 12.
| Question pattern | Reliable method |
|---|---|
| One value is missing | Required total minus the sum of the known values. |
| One value is added or removed | Build the total before and after the change, then find their difference. |
| Several equal values are missing | Find the total left for them, then share that amount equally among them. |
| Two groups are combined | Turn each group average into a group total, add the totals, then divide by the combined count. |
| Only the overall total is requested | Stop after multiplying the stated average by the count. |
A final check is quick: put the recovered value back into the set, add everything, and divide by the full count. The stated average should return exactly. For an added or removed value, check both the earlier and later totals. This catches a subtraction in the wrong direction or the use of the wrong count.
The traps
| Tempting route | How to reject it |
|---|---|
| Copy the stated average as the missing value | The missing value may equal the average, but only the reconstructed total can prove it. |
| Give only the gap from a known value to the average | A gap is a distance, not the second value. Use the full required total or move the same distance to the other side only in a two-value set. |
| Average only the values already shown | That answers a question about the incomplete list. The target belongs to the full count stated in the problem. |
| Subtract an average from a total | Only like quantities should be compared. Convert the average into a required total before subtracting the known total. |
| Average two group averages directly | That gives both groups equal weight. Rebuild each total so the larger group contributes more observations. |
Try it
Over one term, 8 pupils read a mean of 7 books each. How many books did the 8 pupils read altogether? Enter the number of books.
Five numbers have a mean of 8. A sixth number is added and the mean of all six numbers becomes 7. Which statements about this change are true? Select all that apply.
- The sixth number is 1, because the mean fell by 1.
- The first five numbers total 40.
- The sixth number must be negative, because the mean went down.
- All six numbers together total 42.
- The sixth number is 2.
Three numbers have a mean of 6. One of them is 4, and the other two are equal to each other. What is each of the two equal numbers?
- 14
- 6
- 7
- 9