Step by step
The question
Question: A group of 25 runners has a mean time of 84 minutes.
Another group of 15 runners has a mean time of 76 minutes.
What is the mean time of all 40 runners?
1 Total of each group
The first group has 25 runners with a mean of 84.
Together they have 25 times 84.
That is 2,100.
The mean times how many numbers gives the total.
That is the key idea today.
The second group has 15 runners with a mean of 76.
Together they have 15 times 76.
That is 1,140.
2 Put them together
All 40 runners together: 2,100 plus 1,140 is 3,240.
And 25 plus 15 is 40 runners.
Careful!
The two groups are not the same size.
So the mean of everyone is not halfway between 84 and 76.
3 Share equally
The mean shares the total equally: 3,240 divided by 40.
So the mean of all 40 runners is 81.
4 Check
Let's check: if all 40 had 81, together they would have 81 times 40.
That is 3,240, the total.
So it is right!
And 81 is closer to 84, the mean of the bigger group.
That makes sense.
Let's look back.
First: total of each group.
Then: put them together.
Then: share equally.
Last: check.