AMC 10 · 2011 · #4
Grade 6 rate-ratioPick an answer.
The three groups are not the same size, so you cannot just average 12, 15, and 10. Naming the number of fifth graders with one variable lets you write every group's size in terms of it. The variable then cancels, so you can even pretend there is just one fifth grader to keep the arithmetic light.
Name the group sizes
One letter names all three group sizes.
One unknown for the smallest group lets the ratio chain build every other group size from it.
6.RP.A.3Introduce A VariableWrite the overall average
The average is a total over a count.
Averaging a mix means pooling all the minutes and all the runners, not averaging the three group averages.
Averaging a mix means pooling all the minutes and all the runners, not averaging the three averages.
▸ Why?
An average is a total shared out over a count, so both the total and the count must be rebuilt first.
▸ Why?
The whole group is exactly the three groups put together, so their totals and counts simply add.
Add up the totals
Both sums carry the same unknown.
Weighting each average by its group's size is just size times rate for each group, then summed.
5.NBT.B.5Analyze The UnitsDivide and let f cancel
It cancels, leaving 88/7, choice (B).
Because both totals scale with the same f, the ratio is fixed, so the actual head count is irrelevant.
5.NF.B.3Solve An Easier Related ProblemTo average groups of different sizes, add up everyone's minutes and divide by everyone, not the three averages.
- Name the group sizes
- Write the overall average
- Add up the totals
- Divide and let f cancel