AMC 10 · 2020 · #4

Grade 6 rate-ratio

Archetype Arithmetic Expression Decomposition └ Sub-archetype Multi-Segment Path or Count Sum

rateunit-conversiondimensional-analysismulti-digit-arithmetic dimensional-analysisidentify-subproblems ↑ Prerequisites: rate

📏 Medium solution 💡 2 insights

Problem

A driver goes 60 mph for 2 hours in a car that gets 30 miles per gallon. She is paid

0.50 per mile</span>, and her only cost is gasoline at <span class="mk-c">

2.00 per gallon. What is her net pay rate in dollars per hour?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 10 2020 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Analyze the Units

Tool #8 (Analyze the Units) is perfect — the question is built entirely from rates (mile/hour, mile/gallon, /mile, /gallon, /hour). Track units through and the formula writes itself. Tool #7 (Subproblems) splits the work into earnings rate, gas-cost rate, and net rate — all per hour, so we don't even need the total 120 miles. Tool #3 (Eliminate) checks the magnitude: gross is $30/hr (since 60 mi/hr ×$ 0.50/mi = $30/hr), so the net must be a bit less than 30 — answers ≤ 20 are out.

1STEP 1

Gross pay: speed times pay-per-mile cancels the miles, leaving $30 per hour.

60 mi/hr ×

0.50/mi =

30 per hour

2STEP 2

Gas cost: each hour's 60 miles burns 2 gallons, costing $4 per hour.

60 mi/hr × (1 gal)/(30 mi) ×

2.00/gal =

4 per hour

3STEP 3

Net pay: gross minus gas, $30 -$ 4 = $26 per hour, which is choice (E).

30/hr -

4/hr = $26/hr → (E)

4STEP 4

Sanity check: gross is $30 per hour and gas only$ 4, so net lands at $26 — no smaller choice fits.

gross =

30/hr, gas =

4/hr → (E)

Answer

Whole-trip check (units in dollars). Distance = 60 × 2 = 120 mi. Earnings = 120 × 0.50 =

60. Gas used =

\frac{120}{30}

= 4 gal, costing 4 × 2 =

8. Net = 60 - 8 =

52 over 2 hours =

26/hr. Matches (E).

💡Key takeaway

This AMC 10 problem only needs Grade 6 "ratios and rates" you already know — earnings $30/hr, gas$ 4/hr, net $26/hr.

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