AMC 10 · 2005 · #5
Grade 6 arithmeticPick an answer.
Averages cannot be added together, but totals can, so Tool #16 (Change Focus) shifts attention from the two averages to the two hidden totals. Tool #7 (Identify Subproblems) then splits the work into three clean pieces: the total of the first group, the total of the second group, and the average of everything combined. Tool #3 (Eliminate Possibilities) guards the main trap, because simply averaging the two averages 30 and 20 gives 25, which is choice (C) but is wrong since the two groups are not the same size.
Turn the first average into a total
The first average hides a total of 600.
An average is just the pile shared out evenly, so multiplying it back by the count rebuilds the whole pile.
6.SP.B.5Change Focus Count The ComplementTurn the second average into a total
The second hides a total of 600 as well.
The same rebuild-the-pile move works for the second group regardless of its different size.
6.SP.B.5Identify SubproblemsAdd the totals and count all the numbers
Totals combine directly, giving 1200 over 50 numbers.
Totals and counts stack up cleanly when groups merge, even though averages do not.
Totals and counts stack up cleanly when two groups merge, even though the averages do not.
▸ Why?
An average is a total shared out over a count, so it can be turned back into a total and back again.
▸ Why?
The merged pile is nothing but the two piles put together, so its total is the two totals added.
Divide to get the combined average
Dividing gives 24, below the naive midpoint, choice (B).
The bigger group tugs the shared average toward its own value, so the mean leans toward the twenty-average side.
6.SP.A.3Eliminate PossibilitiesYou cannot average two averages when the groups are different sizes; turn each average back into a total, add the totals, and divide by how many numbers there are in all.
- Turn the first average into a total
- Turn the second average into a total
- Add the totals and count all the numbers
- Divide to get the combined average