AMC 8 · 2007 · #1

Grade 6 arithmetic

Archetype Arithmetic Expression Decomposition

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'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?

$\mathrm{(A)}\ 9 \qquad\mathrm{(B)}\ 10 \qquad\mathrm{(C)}\ 11 \qquad\mathrm{(D)}\ 12 \qquad\mathrm{(E)}\ 13$

Pick an answer.

(A)

(B)

(C)

(D)

(E)

View mode:

Toolkit + CCSS Solution

Understand

Restated: 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?

Givens: Target average over $6$ weeks: $10$ hours per week; First $5$ weekly hours: $8, 11, 7, 12, 10$; Answer choices: (A) $9$, (B) $10$, (C) $11$, (D) $12$, (E) $13$

Unknowns: The number of hours $x$ Theresa must work in week $6$

Understand

Givens: Target average over $6$ weeks: $10$ hours per week; First $5$ weekly hours: $8, 11, 7, 12, 10$; Answer choices: (A) $9$, (B) $10$, (C) $11$, (D) $12$, (E) $13$

Plan

Primary tool: #3 Set Up an Equation

Secondary: #4 Introduce a Variable

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 $\div$ count $=$ average) into $\dfrac{48+x}{6} = 10$. The cleanest shortcut is the equivalent form: total hours needed $=$ average $\times$ count $= 60$, so $x = 60 - 48$.

Execute — Answer: D

#4 Introduce a Variable 6.EE.A.2 Step 1

Name the unknown.
Let $x$ be the number of hours Theresa works in week $6$.

$$x = \text{hours in week } 6$$

💡 Giving the unknown a letter is the Grade 6 "write expressions with variables" move.

#3 Set Up an Equation 6.SP.B.5 Step 2

Convert the average condition into the total hours required.
Average $\times$ count $=$ sum, so the $6$-week total must equal $10 \times 6 = 60$ hours.

$$\text{total needed} = 10 \times 6 = 60$$

💡 The Grade 6 mean rule rearranges to total $=$ average $\times$ count. That turns "average $10$" into the hard number $60$.

#3 Set Up an Equation 4.NBT.B.4 Step 3

Add the first $5$ weeks to see how much Theresa has banked so far.

$$8 + 11 + 7 + 12 + 10 = 48$$

💡 Just a straight sum of the five known weeks.

#3 Set Up an Equation 6.EE.B.7 Step 4

The total must be $60$, and $48$ is already done, so week $6$ supplies the rest.

$$48 + x = 60 \;\Rightarrow\; x = 60 - 48 = 12 \;\Rightarrow\; \textbf{(D)}$$

💡 One-step linear equation: subtract $48$ from both sides.

[1] #4 6.EE.A.2 Name the unknown. Let $x$ be the number of hours Theresa works in week $6$.

[2] #3 6.SP.B.5 Convert the average condition into the total hours required. Average $\times$ co

[3] #3 4.NBT.B.4 Add the first $5$ weeks to see how much Theresa has banked so far.

[4] #3 6.EE.B.7 The total must be $60$, and $48$ is already done, so week $6$ supplies the rest.

Review

Reasonableness: Plug $x = 12$ back into the average: $\dfrac{8+11+7+12+10+12}{6} = \dfrac{60}{6} = 10$. 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.

Alternative: Tool #11 (Find an Invariant) via deviations from $10$: weeks $1$-$5$ deviate by $-2, +1, -3, +2, 0$, summing to $-2$. To make the total deviation zero (so the average is exactly $10$), week $6$ must contribute $+2$, giving $10 + 2 = 12$ hours. Same answer (D).

CCSS standards used (min grade 6)

4.NBT.B.4 Fluently add and subtract multi-digit whole numbers (Adding $8 + 11 + 7 + 12 + 10 = 48$ and computing $60 - 48 = 12$.)
6.EE.A.2 Write, read, and evaluate expressions in which letters stand for numbers (Naming the unknown week-$6$ hours with the variable $x$.)
6.EE.B.7 Solve real-world problems by writing and solving equations of the form $x + p = q$ (Solving $48 + x = 60$ for $x$ in one step.)
6.SP.B.5 Summarize numerical data sets, including reporting the number of observations and measures of center (Using average $\times$ count $=$ total to convert "average $10$ over $6$ weeks" into the total $60$.)

⭐ 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 $6$.

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: D

Review

CCSS standards used (min grade 6)

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