AMC 10 · 2022 · #2

Grade 6 rate-ratio

Archetype Arithmetic Expression Decomposition └ Sub-archetype Linear Rate Accumulation

rateratio-proportionestimation dimensional-analysisidentify-subproblems ↑ Prerequisites: rateratio-proportion

📏 Short solution 💡 2 insights

Problem

Mike rode 15 laps in 57 minutes at a steady pace. About how many laps did he finish in the first 27 minutes?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 10 2022 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Analyze the Units

Constant pace = constant rate, and "rate" is what tool #8 (Analyze the Units) is built for. Track laps/minute and the units carry us straight to the answer. Once we hit $\frac{135}{19}$ , tool #6 (Guess and Check) pins the integer down with one multiplication (19 × 7 = 133), and tool #3 (Eliminate Possibilities) confirms by comparing to the five spaced-out choices. Algebra (#13) is unnecessary — the units do the work.

1STEP 1

Since 27 is just under half of 57, the laps land just under half of 15 — about 7.5.

\frac{27}{57}

≈

\frac{1}{2}

→ laps ≈

\frac{15}{2}

≈ 7.5

2STEP 2

Pace is 15 laps per 57 min; multiply by 27 min and the minutes cancel, leaving $\frac{15 × 27}{57}$ laps.

\frac{15 laps}{57 min}

× 27 min =

\frac{15 × 27}{57}

laps

3STEP 3

Both 15 and 57 share a factor of 3, so the rate simplifies and the product shrinks to $\frac{135}{19}$ .

\frac{15 × 27}{57}

\frac{5 × 27}{19}

\frac{135}{19}

4STEP 4

Guess and check: 19 × 7 = 133 leaves remainder 2, so $\frac{135}{19}$ = 7 $\frac{2}{19}$ ≈ 7.1 — only 7 fits the choices.

19 × 7 = 133, 135 - 133 = 2 →

\frac{135}{19}

≈ 7.1 → (B)

Answer

Pace check: 15 laps in 57 minutes is roughly one lap per 4 minutes. In 27 minutes you would expect

\frac{27}{4}

≈ 6.75 laps — practically 7. The estimate, the exact computation 7

\frac{2}{19}

, and the choice (B) 7 all agree.

💡Key takeaway

This AMC 10 problem only needs Grade 6 rate sense — 27 minutes is just under half of 57, so the laps are just under half of 15, which lands on 7.

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