Competition · AMC preparation · step 4 of 4
AMC 8 · 2014 · #5
Grade 6 rate-ratioPick an answer.
AMC 8 2014 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
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.
Find how many gallons
Subproblem 1 — how many gallons does $20 buy? Divide money by the $4 price: 5 gallons.
Dividing dollars by dollars-per-gallon cancels the dollars and leaves gallons. The arithmetic (20 ÷ 4 = 5) is a Grade 4 division fact.
At $4 per gallon, $20 buys exactly 5 gallons of gas.
▸ Why?
Gas costs a steady $4 for each gallon, so the number of gallons in $20 is just how many $4 amounts add up to $20.
▸ Why?
$4 per gallon means every gallon trades for the same fixed $4, so one gallon and $4 name the same purchase and $20 can be re-counted in gallons.
▸ Why?
Five equal $4 groups pile up to $20 (since 5 × 4=20), so $20 holds exactly five $4 groups — which is what 20 ÷ 4=5 finds.
▸ Why?
Reading 5 × 4 as five equal groups of four counts out $20 from five $4 groups.
▸ Why?
Because 5 × 4=20, dividing 20 back by 4 must return 5 — dividing undoes the multiplying.
Turn gallons into miles
Subproblem 2 — turn gallons into miles: 5 gal × 32 mi/gal = 160 miles.
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.
4.NBT.B.5Identify SubproblemsRead off the answer
Chain the two subproblems — 160 miles matches choice (C).
The whole chain 4/gal and 32 mi/gal, every dollar buys 8 miles, and $20 × 8 = 160.
6.RP.A.3Identify SubproblemsThis 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.
- Find how many gallons
- Turn gallons into miles
- Read off the answer
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