AMC 8 · 2020 · #11
Grade 6 rate-ratioProblem
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?
Pick an answer.
AMC 8 2020 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Try it yourself first — the explanation is most useful after you’ve attempted it.
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
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)
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
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.
💡 Reading an ordered pair $(\text{time}, \text{distance})$ off a coordinate graph is a Grade 5 coordinate-plane skill.
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.
💡 Switching from minutes to hours within the same time system is exactly the Grade 5 "convert standard measurement units" standard.
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 $6$.
💡 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.
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 $2$.
💡 Same rate reasoning, second subproblem — Maya rides slowly because biking is slower than a bus.
4.NBT.B.4 Step 5 Subtract Maya's speed from Naomi's speed to get the difference asked for.
💡 A simple two-digit subtraction whose units (mph) match what the problem asks for.
5.G.A.2 Read the two endpoints off the distance-vs-time graph. Naomi's dashed line goes 5.MD.A.1 Convert each travel time from minutes to hours so the speed will come out in mil 6.RP.A.3 Compute Naomi's average speed using $\text{speed} = \dfrac{\text{distance}}{\tex 6.RP.A.3 Compute Maya's average speed the same way. Dividing by $\tfrac{1}{2}$ is the sam 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.2Represent 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.1Convert 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.3Use 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.4Fluently 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!
⭐ This AMC 8 problem only needs Grade 6 rate reasoning — distance divided by time — that you already know!
More like this
Same archetype — closest grade level first.