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.
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.5Count The ComplementDo 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.5Count The ComplementThe 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.
4.OA.A.3Count The ComplementConvert 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.5Count 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.