Problem 1 · 2007 AMC 8
Easy
Arithmetic & Operations
average-target
Theresa's parents have agreed to buy her tickets to see her favorite band if she spends an average of 10 hours per week helping around the house for 6 weeks. For the first 5 weeks she helps around the house for 8, 11, 7, 12 and 10 hours. How many hours must she work for the final week to earn the tickets?
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Answer: D — 12 hours.
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Hint 1 of 2
An average is a promise about the total: averaging 10 over 6 weeks means 60 hours total, no matter how they're split up. So don't average — just chase the total.
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Hint 2 of 2
This is the target-total trick: average × count = total needed, then subtract what's banked.
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Approach: convert the average into a required total
- Turn the average into a goal: averaging 10 hours over 6 weeks means a total of 6 · 10 = 60 hours. That's the whole game — how the hours are spread doesn't matter.
- Bank so far: 8 + 11 + 7 + 12 + 10 = 48 hours.
- Final week = 60 − 48 = 12.
- You'll see this again: any 'what's needed to hit an average' problem — final exam scores, gas mileage — collapses the moment you write average × count = total.
Another way — balance the surpluses and deficits:
- Score each week against the goal of 10: −2, +1, −3, +2, 0. They net to −2.
- She's 2 hours behind, so the last week must beat 10 by 2: 10 + 2 = 12.
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