Competition · AMC preparation · step 4 of 4
AMC 8 · 2004 · #9
Grade 6 arithmeticPick an answer.
AMC 8 2004 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
We are given averages but asked for another average. Tool #16 (Change Focus) says: stop looking at averages and look at totals instead. Each average comes with a count, so multiplying gives a sum — and sums split and combine cleanly across groups, while averages do not. Once we have the total of all 5 numbers and the total of the first 2, subtracting gives the total of the last 3, and dividing by 3 gives the average we want.
Turn the average into a total
Shift from average to total: multiply the overall average by the count of five to get the whole-list sum.
The Grade 6 mean formula reverses easily: if you know the mean and the count, the sum is just their product.
6.SP.B.5Change Focus Count The ComplementDo the same for the first two
Do the same for the first two: multiply their average by the count of two to get their total.
Same reversal of the mean formula, now with count 2 instead of 5.
6.SP.B.5Change Focus Count The ComplementSubtract to get the last three
The two groups don't overlap, so subtract the first-two total from the whole-list total to get the last three's total.
Totals across disjoint groups add up — the same part-whole reasoning used in Grade 4 multi-step word problems.
The last three numbers together total 174.
▸ Why?
The last three numbers are the whole list of five with the first two taken away, so their total is the total of all five (270) minus the total of the first two (96).
▸ Why?
The five numbers split into two groups that do not overlap and leave none out — the first two and the last three — so those two group totals add up to the total of the entire list.
▸ Why?
Since the first-two total and the last-three total add up to the whole total, subtracting the first-two total from the whole total gives back exactly the last-three total.
Divide back into an average
Convert the last three's total back to an average by dividing it by the count of three.
Reverse the reversal: once we have the sum and the count, dividing recovers the mean.
6.SP.B.5Change Focus Count The ComplementWhen a problem gives you several averages and asks for another, switch to totals first — totals add and subtract cleanly across groups, and you can always divide at the end to get the average you wanted.
- Turn the average into a total
- Do the same for the first two
- Subtract to get the last three
- Divide back into an average
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