AMC 8 · 2016 · #14

Grade 5 rate-ratioarithmetic

Archetype Arithmetic Expression Decomposition └ Sub-archetype Successive Multiplicative Steps

rateunit-conversionmulti-digit-arithmetic identify-subproblemsdimensional-analysis ↑ Prerequisites: ratemulti-digit-arithmetic

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Problem

Karl's car gets 35 miles per gallon, and a full tank is 14 gallons. He left with a full tank, drove 350 miles, bought 8 gallons, then drove until he arrived with a half-full tank. How many miles did he drive in total that day?

Pick an answer.

(A)

525

(B)

560

(C)

595

(D)

665

(E)

735

AMC 8 2016 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Identify Subproblems

The trip naturally splits into two legs separated by the gas-station stop. Tool #7 (Identify Subproblems) tells us to track fuel state at three checkpoints — start, refuel, arrival — and solve each leg separately, then add the distances. Tool #8 (Analyze the Units) keeps the conversions clean: divide miles by 35 mi/gal to get gallons used, and multiply gallons by 35 mi/gal to get miles driven.

1STEP 1

Divide 350 miles by the 35 mi/gal rate to get the fuel burned on Leg 1: 10 gallons.

(350 mi)/(35 mi/gal) = 10 gal

2STEP 2

Karl started full and used 10, so before refueling the tank holds 4 gallons.

14 - 10 = 4 gal

3STEP 3

Add the 8 gallons bought to the remaining 4, giving 12 gallons — still under the 14-gallon cap.

4 + 8 = 12 gal

4STEP 4

A half-full tank is 7 gallons, so Leg 2 burned 12 minus 7, or 5 gallons.

12 - 7 = 5 gal

5STEP 5

Multiply the 5 gallons by 35 mi/gal to convert Leg 2 fuel into 175 miles.

5 gal × 35 mi/gal = 175 mi

6STEP 6

Add the two legs — 350 plus 175 — to reach the day's total of 525 miles.

350 + 175 = 525 mi → (A)

Answer

525

Sanity-check the fuel budget. Total gallons burned = 10 + 5 = 15 gal, total miles = 15 × 35 = 525 mi, which matches our answer. Also, the tank never overflowed: after refueling it held 12 gallons (less than the 14 capacity). And 525 is the smallest answer choice, which is consistent with Leg 2 being shorter than Leg 1.

💡Key takeaway

This AMC 8 problem only needs Grade 5 multiplication, division, and a half — keep a clean fuel diary at each checkpoint and the answer falls out!

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