← Sensim Lab Sensim Math

AMC 8 · 2005 · #17

Grade 6 rate-ratio
Archetype Coordinate Plane Figure Computation
rateslope-interceptgraph-readingcoordinate-geometry identify-subproblems ↑ Prerequisites: rategraph-reading
📏 Short solution 💡 2 insights 📊 Diagram

Problem

The results of a cross-country team's training run are graphed below. Which student has the greatest average speed?

Pick an answer.

(A)
Angela
(B)
Briana
(C)
Carla
(D)
Debra
(E)
Evelyn
View mode:

Toolkit + CCSS Solution

Understand

Restated: Five students each ran a cross-country training run. Each student is plotted as one point on a graph with time on the horizontal axis and distance on the vertical axis. Find the student whose average speed is the greatest.

Givens: The horizontal axis is time; the vertical axis is distance; Each student is one dot $(t, d)$, where $t$ is the time they ran and $d$ is the distance they covered; Approximate dot positions: Evelyn $(1.25, 4.5)$, Briana $(2.5, 2.2)$, Carla $(4.25, 5.2)$, Debra $(5.6, 2.8)$, Angela $(6.8, 1.4)$; Answer choices: (A) Angela, (B) Briana, (C) Carla, (D) Debra, (E) Evelyn

Unknowns: The student with the greatest average speed

Understand

Restated: Five students each ran a cross-country training run. Each student is plotted as one point on a graph with time on the horizontal axis and distance on the vertical axis. Find the student whose average speed is the greatest.

Givens: The horizontal axis is time; the vertical axis is distance; Each student is one dot $(t, d)$, where $t$ is the time they ran and $d$ is the distance they covered; Approximate dot positions: Evelyn $(1.25, 4.5)$, Briana $(2.5, 2.2)$, Carla $(4.25, 5.2)$, Debra $(5.6, 2.8)$, Angela $(6.8, 1.4)$; Answer choices: (A) Angela, (B) Briana, (C) Carla, (D) Debra, (E) Evelyn

Plan

Primary tool: #1 Draw a Diagram

Secondary: #3 Eliminate Possibilities

The problem already gives a diagram, so Tool #1 (Draw a Diagram) means using that diagram the right way: add a line from the origin to each dot and compare how steep those lines are. Steeper line $=$ more distance per unit of time $=$ greater average speed. Once each candidate has a line, Tool #3 (Eliminate Possibilities) finishes the job — for a multiple-choice question, we can rank the five candidates by steepness and eliminate anyone whose line is clearly flatter than another's.

Execute — Answer: E

#1 Draw a Diagram 6.RP.A.3 Step 1
  • Turn average speed into something visible on the graph.
  • Each dot $(t, d)$ stands for one student's total time and total distance, so their average speed is $\dfrac{d}{t}$.
  • That same ratio is the steepness (rise over run) of the line from the origin $(0,0)$ to the dot.
  • Bigger ratio means a steeper line.
$$\text{average speed} = \dfrac{d}{t} = \text{steepness of the line from }(0,0)\text{ to }(t,d)$$

💡 Grade 6 reads a ratio $d : t$ as a unit rate "distance per unit time." On a graph, that unit rate shows up as how steeply the line climbs.

#1 Draw a Diagram 6.RP.A.2 Step 2
  • Compute each student's ratio $\dfrac{d}{t}$ from the dot positions.
  • The exact numbers do not have to be perfect — the goal is to rank them.
$$\begin{array}{l|c|c|c} \text{student} & t & d & d/t \\\hline \text{Evelyn} & 1.25 & 4.5 & 3.6 \\ \text{Briana} & 2.5 & 2.2 & 0.88 \\ \text{Carla} & 4.25 & 5.2 & \approx 1.22 \\ \text{Debra} & 5.6 & 2.8 & 0.5 \\ \text{Angela} & 6.8 & 1.4 & \approx 0.21 \end{array}$$

💡 Grade 6 unit rate: divide distance by time to get "how much distance per one unit of time." The student with the biggest unit rate is the fastest.

#3 Eliminate Possibilities 6.RP.A.3 Step 3
  • Eliminate by ranking.
  • Evelyn's ratio $3.6$ is bigger than every other ratio in the table.
  • So Evelyn's line from the origin is the steepest, and her average speed is the greatest.
  • Every other choice loses to Evelyn:
$$3.6 > 1.22 > 0.88 > 0.5 > 0.21 \;\Rightarrow\; \text{Evelyn} > \text{Carla} > \text{Briana} > \text{Debra} > \text{Angela}$$

💡 On a multiple-choice problem, once one candidate beats all the others, every other answer is eliminated.

#3 Eliminate Possibilities 6.RP.A.3 Step 4
  • Read off the answer.
  • The greatest average speed belongs to Evelyn.
$$\textbf{(E)}\;\text{Evelyn}$$

💡 The largest unit rate names the winner.

[1] #1 6.RP.A.3 Turn average speed into something visible on the graph. Each dot $(t, d)$ stands
[2] #1 6.RP.A.2 Compute each student's ratio $\dfrac{d}{t}$ from the dot positions. The exact nu
[3] #3 6.RP.A.3 Eliminate by ranking. Evelyn's ratio $3.6$ is bigger than every other ratio in t
[4] #3 6.RP.A.3 Read off the answer. The greatest average speed belongs to Evelyn.

Review

Reasonableness: A quick visual sanity check: Evelyn's dot is close to the vertical axis (small time) but high up (big distance). Every other dot is either lower, farther right, or both, so the line from the origin to Evelyn rises faster than the line to anyone else. The numbers agree: Evelyn's $\dfrac{d}{t} \approx 3.6$ is nearly three times Carla's $\approx 1.22$, and Carla's dot is the next-steepest by eye. The ranking matches what the graph shows, so answer (E) is consistent.

Alternative: Tool #16 (Change Focus): instead of comparing $\dfrac{d}{t}$ values, ask "who covered the most distance in the least time?" Evelyn has the second-largest distance ($4.5$) and the smallest time ($1.25$), so she beats every other student on at least one of the two measures. No student has both more distance and less time than Evelyn, so no one can be faster — confirming (E).

CCSS standards used (min grade 6)

  • 6.RP.A.2 Understand the concept of a unit rate associated with a ratio (Reading each dot $(t, d)$ as a ratio $d : t$ and computing the unit rate $d/t$ as the student's average speed.)
  • 6.RP.A.3 Use ratio and rate reasoning to solve real-world and mathematical problems (Comparing the five unit rates to rank the students by average speed and pick the largest.)

⭐ Distance over time is the rate hidden in every dot — and on a distance-time graph, that rate is exactly how steeply a line from the origin climbs to the dot. Steepest line, fastest runner.

⭐ Distance over time is the rate hidden in every dot — and on a distance-time graph, that rate is exactly how steeply a line from the origin climbs to the dot. Steepest line, fastest runner.

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