AMC 8 · 2007 · #1

Grade 6 arithmetic

Archetype Arithmetic Expression Decomposition └ Sub-archetype Weighted Group Aggregation

mean-median-mode-rangelinear-equations-one-varmulti-digit-arithmetic identify-subproblemsconvert-to-algebra ↑ Prerequisites: multi-digit-arithmeticmean-median-mode-range

📏 Short solution 💡 2 insights

Problem

Theresa needs to average 10 hours of housework per week over 6 weeks. In the first 5 weeks she logs 8, 11, 7, 12, and 10 hours. How many hours must she work in week 6 to hit the 10-hour average?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 8 2007 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Set Up an Equation

The condition "average over 6 weeks = 10" is a single equation about a single unknown — the week-6 hours. Tool #4 (Introduce a Variable) names that unknown x. Tool #3 (Set Up an Equation) turns the average rule (sum ÷ count = average) into (48+x)/6 = 10. The cleanest shortcut is the equivalent form: total hours needed = average × count = 60, so x = 60 - 48.

1STEP 1

Name the unknown: let x be the hours Theresa works in week 6.

x = hours in week 6

2STEP 2

Average × count = sum, so the 6 weeks must total 10 × 6 = 60 hours.

total needed = 10 × 6 = 60

3STEP 3

Add the five known weeks: 8 + 11 + 7 + 12 + 10 = 48 hours already banked.

8 + 11 + 7 + 12 + 10 = 48

4STEP 4

Total must be 60 and 48 is done, so 48 + x = 60 gives x = 12 → (D).

48 + x = 60 → x = 60 - 48 = 12 → (D)

Answer

Plug x = 12 back into the average:

\frac{8+11+7+12+10+12}{6}

\frac{60}{6}

= 10. The average lands exactly on 10, matching the goal. Also, 12 sits comfortably inside the existing weekly range (7 to 12), so it is realistic — not a wild outlier.

💡Key takeaway

An average target is really a total target in disguise: multiply average by count to get the total you owe, subtract what you've already done, and the rest is week 6.

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