Competition · AMC preparation · step 4 of 4
AMC 8 · 2007 · #7
Grade 6 arithmeticPick an answer.
AMC 8 2007 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Tool #7 (Identify Subproblems) fits because averages do not subtract directly — you cannot just take 30 - 18. Instead the question splits into three small subproblems, each one a single Grade 6 step: (a) recover the original sum of ages from the original average, (b) subtract the leaver's age to get the new sum, (c) divide the new sum by the new count. No variable or equation is needed — the mean formula run forward and backward is enough.
Recover the original total
Recover the original total: sum = average × count, so five people averaging 30 give a total of 150 years.
Grade 6 mean formula in reverse: knowing the average and the count pins down the total.
Because the five people average 30 years, their ages together must total 5 × 30 = 150 years.
▸ Why?
An average is the equal share each person would hold if the whole total were split evenly among them, so the total is just that one equal share built back up, once for every person.
▸ Why?
Splitting the total evenly among the five people is dividing by five, and stacking five equal shares back up is multiplying by five, which reverses that division and returns the original total.
▸ Why?
Building the equal share back up once for each of the five people is five equal groups of that share, and five equal groups gathered together is five times the share.
Subtract the person who left
Subtract the leaver: the 18-year-old walks out, so the total drops by exactly that, leaving 132 years.
Only the total changes here — the count change is handled in the next step.
4.OA.A.3Identify SubproblemsAverage the four left
Re-average: divide the new total by the new count of 4, giving 33.
Same mean formula, applied forward this time: sum ÷ count.
6.SP.B.5Identify SubproblemsAverages do not subtract. To handle a "someone leaves" problem, go back to the total, take the leaver's age out, then divide by the new count.
- Recover the original total
- Subtract the person who left
- Average the four left
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