AMC 8 · 2014 · #5

Grade 6 rate-ratio

Archetype Arithmetic Expression Decomposition └ Sub-archetype Linear Rate Accumulation

rateunit-conversionmulti-digit-arithmetic dimensional-analysisidentify-subproblems ↑ Prerequisites: multi-digit-arithmeticrate

📏 Short solution 💡 2 insights

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Problem

Margie's car gets 32 miles for every gallon of gas. Gas costs

4 per gallon</span>. If she spends <span class="mk-c">

20 on gas, how many miles can she drive?

Pick an answer.

(A)

(B)

128

(C)

160

(D)

320

(E)

640

AMC 8 2014 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Identify Subproblems

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 $20 into gallons, then turn gallons into miles. Tool #8 (Analyze the Units) keeps us honest at each step:$ ÷ ($/gal) = gal, and gal × (mi/gal) = mi. The units cancel correctly, so we know the operations are right.

1STEP 1

Subproblem 1 — how many gallons does $20 buy? Divide money by the$ 4 price: 5 gallons.

20/(

4/gal) = 5 gal

2STEP 2

Subproblem 2 — turn gallons into miles: 5 gal × 32 mi/gal = 160 miles.

5 gal × 32 mi/gal = 160 mi

3STEP 3

Chain the two subproblems — 160 miles matches choice (C).

160 mi → (C)

Answer

160

Quick sanity check: 1 dollar buys

\frac{1}{4}

gallon, and

\frac{1}{4}

gallon goes

\frac{32}{4}

= 8 miles. So

1 = 8 mi, and

20 = 20 × 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.

💡Key takeaway

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.

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