Competition · AMC preparation · step 4 of 4

AMC 8 · 2007 · #1

Grade 6 arithmetic

Archetype Arithmetic Expression Decomposition └ Sub-archetype Weighted Group Aggregation

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📏 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

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

x = hours in week 6

2STEP 2

Find the total hours needed

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

total needed = 10 × 6 = 60

3STEP 3

Add the first five weeks

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

8 + 11 + 7 + 12 + 10 = 48

4STEP 4

Solve for week 6

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.

Name the unknown
Find the total hours needed
Add the first five weeks
Solve for week 6

A parent dashboard for the family lives at sensimlab.com.

Name the unknown

Find the total hours needed

Add the first five weeks

Solve for week 6

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