Step by step
The question
Question: A class of 18 students has a mean score of 75.
A class of 12 students has a mean score of 85.
What is the mean score of all 30 students?
There are four choices.
First we solve it, the normal way.
Then we pick.
1 Total of each group
The first class has 18 students with a mean of 75.
Together they have 18 times 75.
That is 1,350.
The mean times how many numbers gives the total.
That is the key idea today.
The second class has 12 students with a mean of 85.
Together they have 12 times 85.
That is 1,020.
2 Put them together
All 30 students together: 1,350 plus 1,020 is 2,370.
And 18 plus 12 is 30 students.
Careful!
The two classes are not the same size.
So the mean of everyone is not halfway between 75 and 85.
3 Share equally
The mean shares the total equally: 2,370 divided by 30.
So the mean of all 30 students is 79.
4 Check
Let's check: if all 30 had 79, together they would have 79 times 30.
That is 2,370, the total.
So it is right!
And 79 is closer to 75, the mean of the bigger class.
That makes sense.
5 Pick the choice
Our answer is 79.
That is choice A.
6 Why not the others
B is 80.
Wrong: halfway between 75 and 85: the classes are not the same size.
That is the Careful mistake from before.
C is 81.
Wrong: gave the 85 to the bigger class.
D is 2370.
Wrong: that is the total: we still had to share it out.
Let's look back.
First: total of each group.
Then: put them together.
Then: share equally.
Then: check.
Then: pick the choice.
Last: why not the others.