Year 6: Using the mean to solve problems
Part of the Year 6 pie charts, line graphs and the mean chapter. Chapter overview · whole chapter on one page
Using the mean to solve problems
If you know the mean and the number of values, you can find the total. Multiply the mean by the number of values. This works because the mean is the total shared equally.
You can also find a missing value. Work out the total from the mean, then take away the values you know. What is left is the missing value.
When two groups are put together, do not simply average the two means. First find each group total, add the totals, and divide by the number of values in both groups.
A mean is one number, so it does not show every value. A single very high or very low value can change the mean a lot.
Key ideas
- Total = mean × the number of values.
- Missing value = total from the mean − the values you know.
- Two groups together: add the two totals, then divide by all the values.
- A mean does not show each value. It shows the average.
Key words. The mean is the same as the average. Add all the values to get the total, then share the total equally between the number of values.
Method. Multiply the mean by the number of values to get the total.
- Write the mean and the number of values. The mean is the average of the values.mean 9, number of values 6
- Multiply the mean by the number of values. Sharing the total equally gave the mean, so multiplying gives the total back.9 × 6 = 54
- Write the answer. This is the total of all the values.54
Check. 54 ÷ 6 = 9, so the mean is right. ✓
Watch out. Multiply to find the total from a mean. Divide only when you need the mean from a total.
Key words. The mean is the same as the average. Add all the values to get the total, then share the total equally between the number of values.
Method. Find the total from the mean, then take away the values you know.
- Work out the total from the mean. The total is the mean times the number of values.8 × 5 = 40
- Add the values you know. These values are already in the list.9 + 6 + 7 + 10 = 32
- Take the known total away from the total. The missing value is what is left over.40 − 32 = 8
- Write the answer. This is the missing value.8
Check. 32 + 8 = 40, and 40 ÷ 5 = 8. ✓
Watch out. Find the total first. Do not take away the mean from one value.
Key words. The mean is the same as the average. Add all the values to get the total, then share the total equally between the number of values.
Method. Turn each mean back into a total, add the totals, then divide by the number of values altogether.
- Find the total of the first group. Multiply its mean by its number of values.12 × 5 = 60
- Find the total of the second group. Multiply its mean by its number of values.15 × 5 = 75
- Add the two totals. Together the two groups have this total.60 + 75 = 135
- Count everybody and divide. There are 5 + 5 = 10 values altogether.135 ÷ 10 = 13.5
Check. 13.5 × 10 = 135, so the combined mean is right. ✓
Watch out. Do not just average the two means. That only works when both groups have the same number of values.
Watch out
Common mistake: averaging two means without weighting them. If the groups have different numbers of values, the mean of all the values is not the average of the two means. Always add the totals first.
Exercise 12G Practice: using the mean to solve problems
Non-calculatorKey words. The mean is the same as the average. Add all the values to get the total, then share the total equally between the number of values.
Method. Multiply the mean by the number of values to get the total.
- Write the mean and the number of values. The mean is the average of the values.mean 7, number of values 8
- Multiply the mean by the number of values. Sharing the total equally gave the mean, so multiplying gives the total back.7 × 8 = 56
- Write the answer. This is the total of all the values.56
Check. 56 ÷ 8 = 7, so the mean is right. ✓
Watch out. Multiply to find the total from a mean. Divide only when you need the mean from a total.
Key words. The mean is the same as the average. Add all the values to get the total, then share the total equally between the number of values.
Method. Find the total from the mean, then take away the values you know.
- Work out the total from the mean. The total is the mean times the number of values.7 × 4 = 28
- Add the values you know. These values are already in the list.5 + 8 + 9 = 22
- Take the known total away from the total. The missing value is what is left over.28 − 22 = 6
- Write the answer. This is the missing value.6
Check. 22 + 6 = 28, and 28 ÷ 4 = 7. ✓
Watch out. Find the total first. Do not take away the mean from one value.
Key words. The mean is the same as the average. Add all the values to get the total, then share the total equally between the number of values.
Method. Find the total from the mean, then take away the values you know.
- Work out the total from the mean. The total is the mean times the number of values.6 × 5 = 30
- Add the values you know. These values are already in the list.2 + 4 + 7 + 9 = 22
- Take the known total away from the total. The missing value is what is left over.30 − 22 = 8
- Write the answer. This is the missing value.8
Check. 22 + 8 = 30, and 30 ÷ 5 = 6. ✓
Watch out. Find the total first. Do not take away the mean from one value.
Key words. The mean is the same as the average. Add all the values to get the total, then share the total equally between the number of values.
Method. Add all the values, then share the total equally between the number of values.
- Write the values. These are all the values. Count every one, even a zero.3, 4, 5, 8, 10
- Add the values to find the total. The total is what all the values make together.3 + 4 + 5 + 8 + 10 = 30
- Count the values. Count how many numbers there are. Do not count the total.There are 5 values
- Divide the total by the number of values. Sharing the total equally between the values gives the mean.30 ÷ 5 = 6
Check. 6 × 5 = 30, so the mean is right. ✓
Watch out. Divide by how many values there are, not by the biggest value. Do not forget to count a zero.
Key words. The mean is the same as the average. Add all the values to get the total, then share the total equally between the number of values.
Method. Turn each mean back into a total, add the totals, then divide by the number of values altogether.
- Find the total of the first group. Multiply its mean by its number of values.8 × 5 = 40
- Find the total of the second group. Multiply its mean by its number of values.14 × 5 = 70
- Add the two totals. Together the two groups have this total.40 + 70 = 110
- Count everybody and divide. There are 5 + 5 = 10 values altogether.110 ÷ 10 = 11
Check. 11 × 10 = 110, so the combined mean is right. ✓
Watch out. Do not just average the two means. That only works when both groups have the same number of values.
Key words. The mean is the same as the average. Add all the values to get the total, then share the total equally between the number of values.
Method. Multiply the mean by the number of values to get the total.
- Write the mean and the number of values. The mean is the average of the values.mean 25, number of values 6
- Multiply the mean by the number of values. Sharing the total equally gave the mean, so multiplying gives the total back.25 × 6 = 150
- Write the answer. This is the total of all the values.150
Check. 150 ÷ 6 = 25, so the mean is right. ✓
Watch out. Multiply to find the total from a mean. Divide only when you need the mean from a total.
Key words. The mean is the same as the average. Add all the values to get the total, then share the total equally between the number of values.
Method. Change the total by the difference, then divide by the same number of values.
- Write the total before the change. The total is the mean times the number of values.60 (the total before the change)
- Change the total. Take away the old value 20 and add the new value 10.60 − 20 + 10 = 50
- Divide by the number of values. There are still 5 values, so share the new total equally between them.50 ÷ 5 = 10
Check. 10 × 5 = 50, so the new mean is right. ✓
Watch out. Change the total, not the mean. Only the value that changed moves the total.
Key words. The mean is the same as the average. Add all the values to get the total, then share the total equally between the number of values.
Method. Add all the values, then share the total equally between the number of values.
- Write the values. These are all the values. Count every one, even a zero.6, 9, 7, 10, 8
- Add the values to find the total. The total is what all the values make together.6 + 9 + 7 + 10 + 8 = 40
- Count the values. Count how many numbers there are. Do not count the total.There are 5 values
- Divide the total by the number of values. Sharing the total equally between the values gives the mean.40 ÷ 5 = 8
Check. 8 × 5 = 40, so the mean is right. ✓
Watch out. Divide by how many values there are, not by the biggest value. Do not forget to count a zero.
Key words. The mean is the same as the average. Add all the values to get the total, then share the total equally between the number of values.
Method. Change the total by the difference, then divide by the same number of values.
- Write the total before the change. The total is the mean times the number of values.40 (the total before the change)
- Change the total. Take away the old value 8 and add the new value 13.40 − 8 + 13 = 45
- Divide by the number of values. There are still 5 values, so share the new total equally between them.45 ÷ 5 = 9
Check. 9 × 5 = 45, so the new mean is right. ✓
Watch out. Change the total, not the mean. Only the value that changed moves the total.
Key words. The mean is the same as the average. Add all the values to get the total, then share the total equally between the number of values.
Method. Multiply the mean by the number of values to get the total.
- Write the mean and the number of values. The mean is the average of the values.mean 4, number of values 3
- Multiply the mean by the number of values. Sharing the total equally gave the mean, so multiplying gives the total back.4 × 3 = 12
- Write the answer. This is the total of all the values.12
Check. 12 ÷ 3 = 4, so the mean is right. ✓
Watch out. Multiply to find the total from a mean. Divide only when you need the mean from a total.
Key words. The mean is the same as the average. Add all the values to get the total, then share the total equally between the number of values.
Method. Find the total from the mean, then take away the values you know.
- Work out the total from the mean. The total is the mean times the number of values.140 × 4 = 560
- Add the values you know. These values are already in the list.136 + 142 + 150 = 428
- Take the known total away from the total. The missing value is what is left over.560 − 428 = 132
- Write the answer. This is the missing value.132
Check. 428 + 132 = 560, and 560 ÷ 4 = 140. ✓
Watch out. Find the total first. Do not take away the mean from one value.
Watch out
Watch out: a mean tells you the average, not every value. Two sets of values can have the same mean and be very different.
Exam tip
When you find a missing value, check it by putting it back into the list and working out the mean again.