AMC 8 · 2020 · #11

Grade 6 rate-ratio

Archetype Arithmetic Expression Decomposition └ Sub-archetype Fraction / Expression Sum Evaluation

rategraph-readingunit-conversion identify-subproblemsdimensional-analysis ↑ Prerequisites: ratefraction-arithmetic

📏 Medium solution 💡 3 insights 📊 Diagram

Problem

Maya and Naomi both travel the same 6-mile route from school to the beach. A distance-vs-time graph shows Naomi's bus reaching the beach at the 10-minute mark and Maya's bike reaching it at the 30-minute mark. We want the difference of their average speeds in miles per hour (mph).

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 8 2020 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Analyze the Units

This is a rate problem with a unit mismatch: distance is in miles, the graph's time axis is in minutes, but the answer wants mph. Tool #8 (Analyze the Units) forces us to convert minutes to hours before dividing, so the result automatically carries the right units. Tool #1 (Draw a Diagram) is already half done for us — we just read the two endpoints off the graph. Tool #7 (Identify Subproblems) splits the work into three clean pieces: Naomi's speed, Maya's speed, and the difference — solve each, then combine.

1STEP 1

Read each endpoint off the graph: Naomi covers 6 miles in 10 minutes, Maya covers 6 miles in 30 minutes.

Naomi: (t, d) = (10 min, 6 mi) Maya: (t, d) = (30 min, 6 mi)

2STEP 2

Convert times to hours so speed comes out in mph: 10 min = $\frac{1}{6}$ hr and 30 min = $\frac{1}{2}$ hr.

10 min =

\frac{10}{60}

hr =

\frac{1}{6}

hr 30 min =

\frac{30}{60}

hr =

\frac{1}{2}

3STEP 3

Naomi's speed = 6 miles divided by $\frac{1}{6}$ hr = 36 mph (dividing by $\frac{1}{6}$ is times 6).

Naomi's speed =

\frac{6 mi}{\frac{1}{6} hr}

= 6 × 6 = 36 mph

4STEP 4

Maya's speed = 6 miles divided by $\frac{1}{2}$ hr = 12 mph (dividing by $\frac{1}{2}$ is times 2).

Maya's speed =

\frac{6 mi}{\frac{1}{2} hr}

= 6 × 2 = 12 mph

5STEP 5

Subtract the speeds: 36 mph minus 12 mph = 24 mph, which is choice (E).

Difference = 36 mph - 12 mph = 24 mph → (E)

Answer

A bus traveling 6 miles in 10 minutes is moving at 36 mph, a normal in-town bus speed. A bike covering 6 miles in half an hour is moving at 12 mph, a normal kid's biking pace. The bus is 3 times faster, and 36 - 12 = 24 mph fits choice (E). The units (mph) match the question, so the answer is consistent.

💡Key takeaway

This AMC 8 problem only needs Grade 6 rate reasoning — distance divided by time — that you already know!

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