In a town, there are 33 families that own either 1, 2, or 3 bikes. The number of families that own 1 bike is equal to the number of families that own 3 bikes.
What is the number of the bikes in the town?
I know the answer is 66. If you test cases you will see that the number is always 66. FOr example:
15 own 1b, 15 own 3b, 3 own 2b = 66
10 own 1b, 10 own 3b, 13 own 2b = 66
My question is, how do you explain this result in a technical way, taking into consideration the mean?
On
If $n$ families own one bike, and $n$ families own $3$ bikes, then $33-2n$ families own $2$ bikes. The total number of bikes is then:
$$n\cdot 1+n\cdot 3+(33-2n)\cdot 2=4n+66-4n=66$$
On
Alternative approach:
Think about the what the total number of bikes would be if the average number of bikes per family was exactly $(2)$. Each family that owns $(1)$ bike rather than $(2)$ deducts $(1)$ from the overall total, and each family that owns $(3)$ bikes rather than $(2)$ adds $(1)$ to the overall total.
Since you are (in effect) given that the number of $(-1)$ changes equals the number of $(+1)$ changes, then you know that the total number of bikes in this town is the same number that it would be if the exact average number of bikes per family is $(2)$.
Probably the easiest way to make sense of this problem is to simply imagine that every family with 3 bikes gives one of them to a family with only a single bike. This would then mean that every family owns 2 bikes, and the total would be $33*2=66$ bikes.