AMC 8 · 2019 · #5

Grade 8 rate-ratioalgebra

Archetype Meeting Time with Piecewise Rates └ Sub-archetype Staggered Start or Schedule Table

rategraph-readingslope-intercept identify-subproblemspattern-recognition ↑ Prerequisites: graph-readingrate

📏 Medium solution 💡 3 insights 📊 Diagram

Problem

A tortoise walks the whole race at a single slow steady pace, while a hare sprints out ahead, stops for a nap, then sprints again — and yet finishes after the tortoise. Five distance-vs-time graphs (A)-(E) are offered; pick the one whose two curves match this story.

Pick an answer.

(A)

(distance-time graph A) hare and tortoise — variant interpretation

(B)

(distance-time graph B) tortoise straight line; hare steep up, flat (nap), steep up — arrives after tortoise

(C)

(distance-time graph C) variant interpretation

(D)

(distance-time graph D) variant interpretation

(E)

(distance-time graph E) variant interpretation

AMC 8 2019 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Eliminate Possibilities

There are only five candidate graphs, so Tool #3 (Eliminate Possibilities) is the natural move: translate each clue from the story into a visual feature that a correct graph MUST have, then knock out any graph that fails that feature. Tool #1 (Draw a Diagram) is the partner — before looking at the choices, sketch in your head what each racer's curve should look like (tortoise = one straight ramp; hare = steep ramp, flat shelf, second steep ramp ending later than the tortoise's). Then matching becomes a checklist.

1STEP 1

Tortoise = slow steady pace, so its curve is one gentle straight line through the origin — never bending.

d_tortoise(t) = (constant speed) × t

2STEP 2

Hare = run, nap, run: its curve is steep up, then a flat shelf, then steep up again.

d_hare(t) = steep up ; flat ; steep up

3STEP 3

Eliminate (D): the tortoise's path is a curve, and a curve means changing speed — not a steady pace.

4STEP 4

Eliminate (C) and (E): neither has the flat nap shelf, and (C) even dips downward (running backward).

5STEP 5

Between (A) and (B), use the last clue: whoever's curve hits the finish line at the smaller time wins.

6STEP 6

In (A) the hare finishes first (wrong); in (B) the tortoise's line hits the finish first — (B) satisfies every clue.

t_tortoise, finish ＜ t_hare, finish → (B)

Answer

(distance-time graph B) tortoise straight line; hare steep up, flat (nap), steep up — arrives after tortoise

Check that the winning graph (B) honors every clue: (1) tortoise's curve is a single straight line from the origin — yes, matches "steady pace"; (2) hare's curve goes steep-up, flat, steep-up — yes, matches "run, nap, run"; (3) the hare's curve never decreases (no backward running) — yes; (4) the tortoise's curve hits the top horizontal finish line at an earlier time than the hare's curve does — yes, matches "the tortoise was already there." All four story facts are satisfied.

💡Key takeaway

This AMC 8 problem only needs Grade 8 graph-reading — turning each phrase of the story into a slope (steep, flat, gentle) — that you already know!

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