AMC 8 · 2007 · #5

Grade 6 arithmetic

Archetype Arithmetic Expression Decomposition

linear-equations-one-varmulti-digit-arithmeticrate identify-subproblemsconvert-to-algebra ↑ Prerequisites: multi-digit-arithmetic

📏 Short solution 💡 2 insights

Problem

Chandler wants to buy a $500$ dollar mountain bike. For his birthday, his grandparents
send him $50$ dollars, his aunt sends him $35$ dollars and his cousin gives him $15$ dollars. He earns
$16$ dollars per week for his paper route. He will use all of his birthday money and all
of the money he earns from his paper route. In how many weeks will he be able
to buy the mountain bike?

$\mathrm{(A)}\ 24 \qquad\mathrm{(B)}\ 25 \qquad\mathrm{(C)}\ 26 \qquad\mathrm{(D)}\ 27 \qquad\mathrm{(E)}\ 28$

Pick an answer.

(A)

(B)

(C)

(D)

(E)

View mode:

Toolkit + CCSS Solution

Understand

Restated: Chandler wants a $\$500$ mountain bike. Birthday gifts give him $\$50 + \$35 + \$15$, and his paper route earns $\$16$ each week. He spends every dollar on the bike. How many weeks of paper-route work does he need?

Givens: Bike price: $\$500$; Birthday gifts: $\$50$ from grandparents, $\$35$ from aunt, $\$15$ from cousin; Paper-route earnings: $\$16$ per week; Answer choices: (A) $24$, (B) $25$, (C) $26$, (D) $27$, (E) $28$

Unknowns: The number of weeks of paper-route work Chandler needs

Understand

Plan

Primary tool: #7 Identify Subproblems

Secondary: #3 Write an Equation

Tool #7 (Identify Subproblems) breaks the question into three smaller pieces: (a) total birthday gift money, (b) the gap between gifts and the bike price, and (c) how many $\$16$ chunks fit into that gap. Each piece is a single Grade 4 step. Tool #3 (Write an Equation) shows up at the end as the unit-rate division: weeks $=$ gap $\div \$16$. We deliberately avoid full Tool #13 (Algebra) — no need to set up $500 = 100 + 16w$ when the same answer falls out of arithmetic.

Execute — Answer: B

#7 Identify Subproblems 4.NBT.B.4 Step 1

Add the three birthday gifts to see how much money Chandler starts with before any paper-route work.

$\$50 + \$35 + \$15 = \$100$

💡 Combining three gift amounts into one running total is a Grade 4 addition step.

#7 Identify Subproblems 4.OA.A.3 Step 2

Subtract the gift money from the bike price to find how much Chandler still owes himself — the amount the paper route must cover.

$\$500 - \$100 = \$400$

💡 Splitting the $\$500$ goal into "already have" and "still need" is the standard multi-step word-problem move.

#3 Write an Equation 6.RP.A.2 Step 3

Each week of paper-route work adds $\$16$. To find how many weeks fit into the $\$400$ gap, divide the gap by the weekly rate.

$\text{weeks} = \dfrac{\$400}{\$16 \text{ per week}} = 25 \text{ weeks} \;\Rightarrow\; \textbf{(B)}$

💡 Dollars per week is a unit rate. Dividing total dollars needed by dollars-per-week gives the count of weeks — exactly the Grade 6 unit-rate idea.

[1] #7 4.NBT.B.4 Add the three birthday gifts to see how much money Chandler starts with before a

[2] #7 4.OA.A.3 Subtract the gift money from the bike price to find how much Chandler still owes

[3] #3 6.RP.A.2 Each week of paper-route work adds $\$16$. To find how many weeks fit into the $

Review

Reasonableness: Run the plan forward to check: $25$ weeks $\times \$16/\text{week} = \$400$ from the paper route, plus the $\$100$ in gifts, gives $\$500$ exactly — the bike price. Quick bounds also fit: $24$ weeks would give $24 \times 16 = \$384 + \$100 = \$484$ (short by $\$16$), and $26$ weeks would give $26 \times 16 = \$416 + \$100 = \$516$ (too much). $25$ is the only weekly count that lands on $\$500$.

Alternative: Tool #6 (Guess and Check) on the answer choices: try the middle option (C) $26$ weeks. Total saved would be $\$100 + 26 \times \$16 = \$100 + \$416 = \$516$ — past $\$500$, so $26$ is too many. Step down to (B) $25$: $\$100 + 25 \times \$16 = \$100 + \$400 = \$500$, an exact hit. Answer (B).

CCSS standards used (min grade 6)

4.NBT.B.4 Fluently add and subtract multi-digit whole numbers using the standard algorithm (Adding the three birthday gifts $\$50 + \$35 + \$15 = \$100$ to get total gift money.)
4.OA.A.3 Solve multistep word problems with whole numbers using the four operations (Subtracting the gift total from the bike price, $\$500 - \$100 = \$400$, to find what the paper route must cover.)
6.RP.A.2 Understand the concept of a unit rate associated with a ratio (Dividing $\$400$ by the unit rate $\$16$ per week to get $25$ weeks.)

⭐ When you are saving for a big goal, find what you already have, subtract to see the gap, then divide by what you earn each week. The answer is just one division.

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: B

Review

CCSS standards used (min grade 6)

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