AMC 8 · 2005 · #17

Grade 6 rate-ratio

Archetype Coordinate Plane Figure Computation └ Sub-archetype Coordinate Graph Slope as Rate

rateslope-interceptgraph-readingcoordinate-geometry identify-subproblems ↑ Prerequisites: rategraph-reading

📏 Short solution 💡 2 insights 📊 Diagram

Problem

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.

Pick an answer.

(A)

Angela

(B)

Briana

(C)

Carla

(D)

Debra

(E)

Evelyn

AMC 8 2005 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Draw a Diagram

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.

1STEP 1

Each dot (t, d) is a student's time and distance, so average speed is $\frac{d}{t}$ — the steepness of the line from the origin to the dot.

average speed =

\frac{d}{t}

= steepness of the line from (0,0) to (t,d)

2STEP 2

Divide d by t for each dot to get its ratio; exact values do not matter, only the ranking. The top ratio is about 3.6.

student & t & d &

\frac{d}{t}

; Evelyn & 1.25 & 4.5 & 3.6 ; Briana & 2.5 & 2.2 & 0.88 ; Carla & 4.25 & 5.2 & ≈ 1.22 ; Debra & 5.6 & 2.8 & 0.5 ; Angela & 6.8 & 1.4 & ≈ 0.21

3STEP 3

Rank the ratios: 3.6 beats every other, so that dot's line from the origin is the steepest and its speed the greatest.

3.6 ＞ 1.22 ＞ 0.88 ＞ 0.5 ＞ 0.21 → Evelyn ＞ Carla ＞ Briana ＞ Debra ＞ Angela

4STEP 4

Read off the answer: the greatest average speed belongs to Evelyn.

(E) Evelyn

Answer

Evelyn

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

\frac{d}{t}

≈ 3.6 is nearly three times Carla's ≈ 1.22, and Carla's dot is the next-steepest by eye. The ranking matches what the graph shows, so answer (E) is consistent.

💡Key takeaway

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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