Competition · AMC preparation · step 4 of 4
AMC 8 · 2016 · #14
Grade 5 rate-ratioarithmeticPick an answer.
AMC 8 2016 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
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.
Find the fuel used on Leg 1
Divide 350 miles by the 35 mi/gal rate to get the fuel burned on Leg 1: 10 gallons.
Dividing miles by miles-per-gallon cancels the "miles" unit and leaves "gallons" — a Grade 5 division of whole numbers.
5.NBT.B.6Analyze The UnitsFind the fuel left
Karl started full and used 10, so before refueling the tank holds 4 gallons.
Tracking a quantity that changes step by step is a Grade 4 multi-step word-problem skill.
4.OA.A.3Identify SubproblemsAdd the refuel
Add the 8 gallons bought to the remaining 4, giving 12 gallons — still under the 14-gallon cap.
The subproblem checkpoint after refueling sets up Leg 2 cleanly.
4.OA.A.3Identify SubproblemsFind the fuel used on Leg 2
A half-full tank is 7 gallons, so Leg 2 burned 12 minus 7, or 5 gallons.
Taking half of 14 is a Grade 5 fraction-times-whole-number computation.
5.NF.B.4Identify SubproblemsConvert fuel into miles
Multiply the 5 gallons by 35 mi/gal to convert Leg 2 fuel into 175 miles.
The "gallons" cancels, leaving "miles" — exactly what the question asks for.
The 5 gallons of gas burned on the second leg carry the car exactly 175 miles.
▸ Why?
At a fixed 35 miles per gallon, each gallon covers the same 35 miles, so 5 gallons is 5 equal 35-mile stretches, and 5 × 35 = 175.
▸ Why?
The mileage rate never changes: one gallon always covers exactly 35 miles, so an amount of gas can be renamed as the number of miles it drives.
▸ Why?
Five gallons is 5 of those equal 35-mile groups, and gathering 5 equal groups of 35 is just the multiplication 5 × 35.
Add the two legs
Add the two legs — 350 plus 175 — to reach the day's total of 525 miles.
Combining the subproblem results is the final assembly step of Tool #7.
4.OA.A.3Identify SubproblemsThis 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!
- Find the fuel used on Leg 1
- Find the fuel left
- Add the refuel
- Find the fuel used on Leg 2
- Convert fuel into miles
- Add the two legs
A parent dashboard for the family lives at sensimlab.com.