AMC 8 · 2014 · #5
Easy mode Grade 6Problem
Margie's car can drive miles using gallon of gas.
Gas costs for each gallon.
Margie has to spend on gas.
How many miles can she drive with that gas?
Pick an answer.
Toolkit + CCSS Solution
Understand
Restated: Margie's car gets $32$ miles for every gallon of gas. Gas costs $\$4$ per gallon. If she spends $\$20$ on gas, how many miles can she drive?
Givens: Fuel efficiency = $32$ miles per gallon; Price of gas = $\$4$ per gallon; Money available = $\$20$; Answer choices: (A) $64$, (B) $128$, (C) $160$, (D) $320$, (E) $640$ (miles)
Unknowns: The total number of miles Margie can drive on $\$20$ worth of gas
Understand
Restated: Margie's car gets $32$ miles for every gallon of gas. Gas costs $\$4$ per gallon. If she spends $\$20$ on gas, how many miles can she drive?
Givens: Fuel efficiency = $32$ miles per gallon; Price of gas = $\$4$ per gallon; Money available = $\$20$; Answer choices: (A) $64$, (B) $128$, (C) $160$, (D) $320$, (E) $640$ (miles)
Plan
Primary tool: #7 Identify Subproblems
Secondary: #8 Analyze the Units
The question "how many miles" links three quantities — dollars, gallons, and miles — so we use Tool #7 (Identify Subproblems) to split the work into two clean steps: first turn $\$20$ into gallons, then turn gallons into miles. Tool #8 (Analyze the Units) keeps us honest at each step: $\$ \div (\$/\text{gal}) = \text{gal}$, and $\text{gal} \times (\text{mi}/\text{gal}) = \text{mi}$. The units cancel correctly, so we know the operations are right.
Execute — Answer: C
4.NBT.B.6 Step 1 Subproblem 1: figure out how many gallons $\$20$ buys. Each gallon costs $\$4$, so divide total money by price per gallon.
💡 Dividing dollars by dollars-per-gallon cancels the dollars and leaves gallons. The arithmetic ($20 \div 4 = 5$) is a Grade 4 division fact.
4.NBT.B.5 Step 2 - Subproblem 2: turn those $5$ gallons into miles.
- The car covers $32$ miles for each gallon, so multiply gallons by the miles-per-gallon rate.
💡 The second subproblem reuses the answer to the first. Multiplying gallons by miles-per-gallon cancels gallons and leaves miles — exactly the Grade 4 multi-digit multiplication skill.
6.RP.A.3 Step 3 - Combine the two subproblems to read off the final answer.
- $160$ miles matches choice (C).
💡 The whole chain $\$ \to \text{gal} \to \text{mi}$ is a unit-rate problem: at $\$4$/gal and $32$ mi/gal, every dollar buys $8$ miles, and $\$20 \times 8 = 160$.
4.NBT.B.6 Subproblem 1: figure out how many gallons $\$20$ buys. Each gallon costs $\$4$, 4.NBT.B.5 Subproblem 2: turn those $5$ gallons into miles. The car covers $32$ miles for e 6.RP.A.3 Combine the two subproblems to read off the final answer. $160$ miles matches ch Review
Reasonableness: Quick sanity check: $1$ dollar buys $\tfrac{1}{4}$ gallon, and $\tfrac{1}{4}$ gallon goes $\tfrac{32}{4} = 8$ miles. So $\$1 = 8$ mi, and $\$20 = 20 \times 8 = 160$ mi. Same answer, no setup needed. The magnitude is right too — a small car driving a couple hundred miles on $\$20$ is realistic at $\$4$/gal.
Alternative: Tool #6 (Guess and Check) on the answer choices: the miles per dollar must be $32/4 = 8$, so the total miles must equal $20 \times 8 = 160$. Of the five choices only (C) $160$ fits; (A) $64$ would mean $3.2$ mi/$\$$, (B) $128$ would mean $6.4$ mi/$\$$, (D) $320$ would mean $16$ mi/$\$$, and (E) $640$ would mean $32$ mi/$\$$ — none match the $8$ mi/$\$$ rate the problem forces.
CCSS standards used (min grade 6)
4.NBT.B.6Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors (Dividing $\$20$ by $\$4$ per gallon to get $5$ gallons.)4.NBT.B.5Multiply a whole number of up to four digits by a one-digit whole number (Multiplying $5$ gallons by $32$ miles per gallon to get $160$ miles.)4.MD.A.2Solve word problems involving distances, time, liquid volumes, and money (Setting up the dollars-to-gallons-to-miles word-problem chain in a single real-world scenario.)6.RP.A.3Use ratio and rate reasoning to solve real-world and mathematical problems (Combining the two rates ($\$4$/gal and $32$ mi/gal) into a single unit rate ($8$ mi/$\$$) and applying it to $\$20$.)
⭐ This AMC 8 problem only needs Grade 4 division and multiplication, with a Grade 6 "rate" idea on top — split it into two small steps and the answer falls out.
⭐ This AMC 8 problem only needs Grade 4 division and multiplication, with a Grade 6 "rate" idea on top — split it into two small steps and the answer falls out.