AMC 8 · 2020 · #11

Grade 6 rate-ratio

Archetype Arithmetic Expression Decomposition

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

📏 Medium solution 💡 3 insights 📊 Diagram

Problem

After school, Maya and Naomi headed to the beach, $6$ miles away. Maya decided to bike while Naomi took a bus. The graph below shows their journeys, indicating the time and distance traveled. What was the difference, in miles per hour, between Naomi's and Maya's average speeds?

$\textbf{(A) }6 \qquad \textbf{(B) }12 \qquad \textbf{(C) }18 \qquad \textbf{(D) }20 \qquad \textbf{(E) }24$

Pick an answer.

(A)

(B)

(C)

(D)

(E)

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Toolkit + CCSS Solution

Understand

Restated: 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).

Givens: Both Maya and Naomi cover the same total distance of $6$ miles; From the graph: Naomi's total time = $10$ minutes (dashed line ends at $(10, 6)$); From the graph: Maya's total time = $30$ minutes (solid line ends at $(30, 6)$); Answer choices: (A) $6$, (B) $12$, (C) $18$, (D) $20$, (E) $24$ (mph)

Unknowns: Naomi's average speed minus Maya's average speed, in mph

Understand

Plan

Primary tool: #8 Analyze the Units

Secondary: #1 Draw a Diagram, #7 Identify Subproblems

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.

Execute — Answer: E

#1 Draw a Diagram 5.G.A.2 Step 1

Read the two endpoints off the distance-vs-time graph.
Naomi's dashed line goes from $(0, 0)$ to $(10, 6)$, so she travels $6$ miles in $10$ minutes.
Maya's solid line goes from $(0, 0)$ to $(30, 6)$, so she travels $6$ miles in $30$ minutes.

$$\text{Naomi}: (t, d) = (10 \text{ min}, 6 \text{ mi}) \quad\quad \text{Maya}: (t, d) = (30 \text{ min}, 6 \text{ mi})$$

💡 Reading an ordered pair $(\text{time}, \text{distance})$ off a coordinate graph is a Grade 5 coordinate-plane skill.

#8 Analyze the Units 5.MD.A.1 Step 2

Convert each travel time from minutes to hours so the speed will come out in miles per hour.
$10$ min is $\tfrac{10}{60} = \tfrac{1}{6}$ hr, and $30$ min is $\tfrac{30}{60} = \tfrac{1}{2}$ hr.

$$10 \text{ min} = \dfrac{10}{60} \text{ hr} = \dfrac{1}{6} \text{ hr} \qquad 30 \text{ min} = \dfrac{30}{60} \text{ hr} = \dfrac{1}{2} \text{ hr}$$

💡 Switching from minutes to hours within the same time system is exactly the Grade 5 "convert standard measurement units" standard.

#7 Identify Subproblems 6.RP.A.3 Step 3

Compute Naomi's average speed using $\text{speed} = \dfrac{\text{distance}}{\text{time}}$.
Dividing by $\tfrac{1}{6}$ is the same as multiplying by $6$.

$$\text{Naomi's speed} = \dfrac{6 \text{ mi}}{\tfrac{1}{6} \text{ hr}} = 6 \times 6 = 36 \text{ mph}$$

💡 Computing a unit rate (miles per hour) from a distance and a time is Grade 6 rate reasoning — and it's the first of our two subproblems.

#7 Identify Subproblems 6.RP.A.3 Step 4

Compute Maya's average speed the same way.
Dividing by $\tfrac{1}{2}$ is the same as multiplying by $2$.

$$\text{Maya's speed} = \dfrac{6 \text{ mi}}{\tfrac{1}{2} \text{ hr}} = 6 \times 2 = 12 \text{ mph}$$

💡 Same rate reasoning, second subproblem — Maya rides slowly because biking is slower than a bus.

#8 Analyze the Units 4.NBT.B.4 Step 5

Subtract Maya's speed from Naomi's speed to get the difference asked for.

$$\text{Difference} = 36 \text{ mph} - 12 \text{ mph} = 24 \text{ mph} \;\Rightarrow\; \textbf{(E)}$$

💡 A simple two-digit subtraction whose units (mph) match what the problem asks for.

[1] #1 5.G.A.2 Read the two endpoints off the distance-vs-time graph. Naomi's dashed line goes

[2] #8 5.MD.A.1 Convert each travel time from minutes to hours so the speed will come out in mil

[3] #7 6.RP.A.3 Compute Naomi's average speed using $\text{speed} = \dfrac{\text{distance}}{\tex

[4] #7 6.RP.A.3 Compute Maya's average speed the same way. Dividing by $\tfrac{1}{2}$ is the sam

[5] #8 4.NBT.B.4 Subtract Maya's speed from Naomi's speed to get the difference asked for.

Review

Reasonableness: 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.

Alternative: Tool #5 (Look for a Pattern) shortcut: Maya takes $3$ times as long as Naomi over the same distance, so her speed is $\tfrac{1}{3}$ of Naomi's. If Naomi's speed is $v$, then Maya's is $\tfrac{v}{3}$ and the difference is $\tfrac{2v}{3}$. From the graph we read $v = 36$ mph, giving $\tfrac{2 \times 36}{3} = 24$ mph — same answer with less arithmetic.

CCSS standards used (min grade 6)

5.G.A.2 Represent real-world and mathematical problems by graphing points (Reading the ordered pairs $(10, 6)$ for Naomi and $(30, 6)$ for Maya off the distance-vs-time coordinate graph.)
5.MD.A.1 Convert among different-sized standard measurement units within a given system (Converting $10$ minutes to $\tfrac{1}{6}$ hour and $30$ minutes to $\tfrac{1}{2}$ hour so the final speed comes out in miles per hour.)
6.RP.A.3 Use ratio and rate reasoning to solve real-world and mathematical problems (Computing Naomi's and Maya's average speeds as the unit rate $\text{speed} = \text{distance} / \text{time}$ in miles per hour.)
4.NBT.B.4 Fluently add and subtract multi-digit whole numbers (Subtracting Maya's speed from Naomi's: $36 - 12 = 24$ mph.)

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

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: E

Review

CCSS standards used (min grade 6)

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