Step by step
The question
Question: A class of 24 students has a mean score of 70.
A class of 16 students has a mean score of 80.
What is the mean score of all 40 students?
There are four choices.
First we solve it, the normal way.
Then we pick.
1 Total of each group
The first class has 24 students with a mean of 70.
Together they have 24 times 70.
That is 1,680.
The mean times how many numbers gives the total.
That is the key idea today.
The second class has 16 students with a mean of 80.
Together they have 16 times 80.
That is 1,280.
2 Put them together
All 40 students together: 1,680 plus 1,280 is 2,960.
And 24 plus 16 is 40 students.
Careful!
The two classes are not the same size.
So the mean of everyone is not halfway between 70 and 80.
3 Share equally
The mean shares the total equally: 2,960 divided by 40.
So the mean of all 40 students is 74.
4 Check
Let's check: if all 40 had 74, together they would have 74 times 40.
That is 2,960, the total.
So it is right!
And 74 is closer to 70, the mean of the bigger class.
That makes sense.
5 Pick the choice
Our answer is 74.
That is choice A.
6 Why not the others
B is 75.
Wrong: halfway between 70 and 80: the classes are not the same size.
That is the Careful mistake from before.
C is 76.
Wrong: gave the 80 to the bigger class.
D is 2960.
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.