AMC 10 · 2013 · #5
Grade 6 rate-ratioPick an answer.
To average two groups together I first need each group's total age. I break the work into three subproblems: the total age of the kids, the total age of the parents, then one division of the combined total by the combined head count. Tracking the units (years of age per person) keeps that final division honest.
Averages hide totals
Every average hides a total.
An average is a total split evenly, so to merge groups you first need the totals back.
An average is a total split evenly, so merging two groups needs their totals back first.
▸ Why?
Multiplying an average back by its count returns the total it came from.
▸ Why?
The merged group is exactly the two groups put together, so both totals and both counts add.
Total age of the fifth-graders
The first group's total is 363.
Multiplying the average by the head count rebuilds the group's whole pile of years.
5.NBT.B.5Identify SubproblemsTotal age of the parents
The second group's total is 1815.
Same move as the kids: average times count gives that group's total years.
5.NBT.B.5Identify SubproblemsCombine totals and people
Both totals and both counts combine.
One combined group has one combined total and one combined count.
4.NBT.B.4Identify SubproblemsDivide once to get the average
One division gives 24.75, choice (B).
Total age shared over every person is age per person, which is exactly the average.
6.NS.B.3Analyze The UnitsYou cannot average two averages when the groups are different sizes — turn each average back into a total, add everything up, then divide just once.
- Averages hide totals
- Total age of the fifth-graders
- Total age of the parents
- Combine totals and people
- Divide once to get the average