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AMC 10 · 2022A · #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 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

Estimate with half

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

27/57 ≈ 1/2 → laps ≈ 15/2 ≈ 7.5

2STEP 2

Set up the exact rate

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

(15 laps)/(57 min) × 27 min = (15 × 27)/57 laps

3STEP 3

Simplify the fraction

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

(15 × 27)/57 = (5 × 27)/19 = 135/19

4STEP 4

Pick the closest whole number

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 → 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.

Estimate with half
Set up the exact rate
Simplify the fraction
Pick the closest whole number

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Estimate with half

Set up the exact rate

Simplify the fraction

Pick the closest whole number

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