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

📏 Medium solution 💡 3 insights

📘 View easy version →

Problem

Karl's car uses a gallon of gas every $35$ miles, and his gas tank holds $14$ gallons when it is full. One day, Karl started with a full tank of gas, drove $350$ miles, bought $8$ gallons of gas, and continued driving to his destination. When he arrived, his gas tank was half full. How many miles did Karl drive 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.

Try it yourself first — the explanation is most useful after you’ve attempted it.

View mode:

Toolkit + CCSS Solution

Understand

Restated: 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?

Givens: Fuel efficiency: $35$ miles per gallon; Tank capacity: $14$ gallons (full); Started full; First leg: drove $350$ miles; Bought $8$ gallons mid-trip; Ended with a half-full tank ($7$ gallons); Answer choices: (A) $525$, (B) $560$, (C) $595$, (D) $665$, (E) $735$

Unknowns: Total miles Karl drove that day

Understand

Plan

Primary tool: #7 Identify Subproblems

Secondary: #8 Analyze the Units

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 $35 \text{ mi/gal}$ to get gallons used, and multiply gallons by $35 \text{ mi/gal}$ to get miles driven.

Execute — Answer: A

#8 Analyze the Units 5.NBT.B.6 Step 1

Leg 1 fuel used.
Karl drove $350$ miles at $35$ miles per gallon, so the gallons used equal miles divided by mi/gal.

$$\dfrac{350 \text{ mi}}{35 \text{ mi/gal}} = 10 \text{ gal}$$

💡 Dividing miles by miles-per-gallon cancels the "miles" unit and leaves "gallons" — a Grade 5 division of whole numbers.

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

Fuel state after Leg 1.
Karl began with $14$ gallons and used $10$, so $4$ gallons remain before the refuel.

$$14 - 10 = 4 \text{ gal}$$

💡 Tracking a quantity that changes step by step is a Grade 4 multi-step word-problem skill.

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

Fuel state after refueling.
Karl adds $8$ gallons to the remaining $4$, so the tank now holds $12$ gallons (still within the $14$-gallon limit).

$$4 + 8 = 12 \text{ gal}$$

💡 The subproblem checkpoint after refueling sets up Leg 2 cleanly.

#7 Identify Subproblems 5.NF.B.4 Step 4

Leg 2 fuel used.
Karl arrived with a half-full tank, which is $\tfrac{1}{2} \times 14 = 7$ gallons.
So Leg 2 burned the difference between $12$ and $7$ gallons.

$$12 - 7 = 5 \text{ gal}$$

💡 Taking half of $14$ is a Grade 5 fraction-times-whole-number computation.

#8 Analyze the Units 5.NBT.B.5 Step 5

Convert Leg 2 fuel into miles.
Multiplying gallons by mi/gal gives miles.

$$5 \text{ gal} \times 35 \text{ mi/gal} = 175 \text{ mi}$$

💡 The "gallons" cancels, leaving "miles" — exactly what the question asks for.

#7 Identify Subproblems 4.OA.A.3 Step 6

Add the two legs to get the total distance for the day.

$$350 + 175 = 525 \text{ mi} \;\Rightarrow\; \textbf{(A)}$$

💡 Combining the subproblem results is the final assembly step of Tool #7.

[1] #8 5.NBT.B.6 Leg 1 fuel used. Karl drove $350$ miles at $35$ miles per gallon, so the gallons

[2] #7 4.OA.A.3 Fuel state after Leg 1. Karl began with $14$ gallons and used $10$, so $4$ gallo

[3] #7 4.OA.A.3 Fuel state after refueling. Karl adds $8$ gallons to the remaining $4$, so the t

[4] #7 5.NF.B.4 Leg 2 fuel used. Karl arrived with a half-full tank, which is $\tfrac{1}{2} \tim

[5] #8 5.NBT.B.5 Convert Leg 2 fuel into miles. Multiplying gallons by mi/gal gives miles.

[6] #7 4.OA.A.3 Add the two legs to get the total distance for the day.

Review

Reasonableness: Sanity-check the fuel budget. Total gallons burned = $10 + 5 = 15$ gal, total miles = $15 \times 35 = 525$ 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.

Alternative: Tool #11 (Work Backwards): start from the final state of $7$ gallons. Before Leg 2, the tank had $7 + (\text{Leg 2 gallons})$. The refuel added $8$, so before refueling it had that minus $8$. Before Leg 1, it had $14$. Working back: Leg 1 used $14 - (\text{before-refuel}) = 10$ gal $\Rightarrow 350$ mi (matches given). Then Leg 2 used $5$ gal $\Rightarrow 175$ mi. Same total: $525$.

CCSS standards used (min grade 5)

4.OA.A.3 Solve multistep word problems with whole numbers (Tracking the tank's gallon count across the start, refuel, and arrival checkpoints, and adding the two leg distances at the end.)
5.NBT.B.5 Fluently multiply multi-digit whole numbers (Converting Leg 2 gas ($5$ gal) back into miles via $5 \times 35 = 175$.)
5.NBT.B.6 Divide multi-digit whole numbers (Computing Leg 1 fuel from distance: $350 \div 35 = 10$ gal.)
5.NF.B.4 Multiply a fraction by a whole number (Finding the arrival tank level as $\tfrac{1}{2} \times 14 = 7$ gal.)

⭐ 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!

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: A

Review

CCSS standards used (min grade 5)

More like this