Worked examples
The question
Question: Lena's test scores are 80, 85 and 90.
What score on the next test makes her mean 87?
1 Total needed
After the next test there will be 4 scores.
For a mean of 87, the 4 scores must add up to 87 times 4.
That is 348.
total needed = 87 × 4 = 348
The mean times how many numbers gives the total.
That is the key idea today.
mean × how many = total
Careful!
With the next test, there are 4 scores, not 3.
So we multiply by 4.
Careful: not 87 × 3 ✗
2 Total so far
Now we add the 3 scores we know, from left to right.
total so far = 80 + 85 + 90
So far, the total is 255.
total so far = 255
3 What is missing
The total must go from 255 up to 348.
The missing part is 348 minus 255.
That is 93.
next test = 348 − 255 = 93
4 Check
Let's check: 255 plus 93 is 348.
And 348 divided by 4 is 87, the mean we wanted.
So it is right!
255 + 93 = 348, 348 ÷ 4 = 87
Let's look back.
First: total needed.
Then: total so far.
Then: what is missing.
Last: check.
Answer next test = 348 − 255 = 93
▶ Watch it solved
The question
Question: Leo's test scores are 70, 82, 76 and 80.
What score on the next test makes his mean 78?
1 Total needed
After the next test there will be 5 scores.
For a mean of 78, the 5 scores must add up to 78 times 5.
That is 390.
total needed = 78 × 5 = 390
The mean times how many numbers gives the total.
That is the key idea today.
mean × how many = total
Careful!
With the next test, there are 5 scores, not 4.
So we multiply by 5.
Careful: not 78 × 4 ✗
2 Total so far
Now we add the 4 scores we know, from left to right.
total so far = 70 + 82 + 76 + 80
So far, the total is 308.
total so far = 308
3 What is missing
The total must go from 308 up to 390.
The missing part is 390 minus 308.
That is 82.
next test = 390 − 308 = 82
4 Check
Let's check: 308 plus 82 is 390.
And 390 divided by 5 is 78, the mean we wanted.
So it is right!
308 + 82 = 390, 390 ÷ 5 = 78
Let's look back.
First: total needed.
Then: total so far.
Then: what is missing.
Last: check.
Answer next test = 390 − 308 = 82
▶ Watch it solved
The question
Question: A class of 20 students has a mean score of 80.
A class of 30 students has a mean score of 70.
What is the mean score of all 50 students?
1 Total of each group
The first class has 20 students with a mean of 80.
Together they have 20 times 80.
That is 1,600.
class 1 total = 20 × 80 = 1,600
The mean times how many numbers gives the total.
That is the key idea today.
mean × how many = total
The second class has 30 students with a mean of 70.
Together they have 30 times 70.
That is 2,100.
class 2 total = 30 × 70 = 2,100
2 Put them together
All 50 students together: 1,600 plus 2,100 is 3,700.
total = 1,600 + 2,100 = 3,700
And 20 plus 30 is 50 students.
how many = 20 + 30 = 50
Careful!
The two classes are not the same size.
So the mean of everyone is not halfway between 80 and 70.
Careful: not (80 + 70) ÷ 2 = 75 ✗
3 Share equally
The mean shares the total equally: 3,700 divided by 50.
mean = 3,70050
So the mean of all 50 students is 74.
mean = 74
4 Check
Let's check: if all 50 had 74, together they would have 74 times 50.
That is 3,700, the total.
So it is right!
74 × 50 = 3,700
And 74 is closer to 70, the mean of the bigger class.
That makes sense.
Let's look back.
First: total of each group.
Then: put them together.
Then: share equally.
Last: check.
▶ Watch it solved
