AMC 10 · 2013 · #6
Grade 6 rate-ratioPick an answer.
AMC 10 2013 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
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
An average is a total shared out evenly, so two averages merge only if the groups match in size — and 33 and 55 do not.
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 to merge two groups you first need the totals back.
▸ Why?
An average is a total shared over a count, so multiplying by the count undoes the sharing.
▸ Why?
The merged group's total is exactly the two totals added, and its count the two counts added.
Total age of the fifth-graders
The 33 fifth-graders each average 11 years, so their ages add up to 33 times 11, or 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 55 parents each average 33 years, so their ages add up to 55 times 33, or 1815.
Same move as the kids: average times count gives that group's total years.
5.NBT.B.5Identify SubproblemsCombine totals and people
Pour the totals together: 363 + 1815 = 2178 years, spread across 33 + 55 = 88 people.
One combined group has one combined total and one combined count.
4.NBT.B.4Identify SubproblemsDivide once to get the average
Share 2178 years over all 88 people: one division gives 24.75 years per person.
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