AMC 8 · 2019 · #5

Grade 8 rate-ratioalgebra

Archetype Meeting Time with Piecewise Rates

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

📏 Medium solution 💡 3 insights 📊 Diagram

Problem

A tortoise challenges a hare to a race. The hare eagerly agrees and quickly runs ahead, leaving the slow-moving tortoise behind. Confident that he will win, the hare stops to take a nap. Meanwhile, the tortoise walks at a slow steady pace for the entire race. The hare awakes and runs to the finish line, only to find the tortoise already there. Which of the following graphs matches the description of the race, showing the distance $d$ traveled by the two animals over time $t$ from start to finish?

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

View mode:

Toolkit + CCSS Solution

Understand

Restated: 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.

Givens: The horizontal axis is time $t$; the vertical axis is distance traveled $d$; Tortoise: one slow, steady (constant) pace from start to finish; Hare: fast run, then a complete stop (nap), then fast run again; Result: the tortoise reaches the finish distance before the hare does

Unknowns: Which graph (A through E) shows both racers correctly

Understand

Plan

Primary tool: #3 Eliminate Possibilities

Secondary: #1 Draw a Diagram

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.

Execute — Answer: B

#1 Draw a Diagram 8.F.A.3 Step 1

Sketch what the tortoise's curve must look like.
"Slow steady pace" means constant speed, and on a distance-time graph constant speed is a straight line through the origin with a gentle (not steep) slope, never bending.

$$d_{\text{tortoise}}(t) = (\text{constant speed}) \times t$$

💡 Constant speed shows up as a straight line because the same distance is added in every equal slice of time.

#1 Draw a Diagram 8.F.B.5 Step 2

Sketch what the hare's curve must look like.
Three phases stitched together: a steep straight piece up (fast run), then a flat horizontal piece (nap — time passes, distance does not change), then another steep straight piece up to the finish line.

$$d_{\text{hare}}(t) = \begin{cases} \text{steep up} \\ \text{flat} \\ \text{steep up} \end{cases}$$

💡 Each verb in the hare's story ("runs", "naps", "runs") becomes one piece of the graph, and the flat shelf is the giveaway sign of a nap.

#3 Eliminate Possibilities 8.F.B.5 Step 3

Eliminate graph (D): in (D) the tortoise's path is a curve, not a straight line.
A curve means the speed is changing, which contradicts "slow steady pace".

💡 A bending curve says "speeding up or slowing down" — the tortoise does neither.

#3 Eliminate Possibilities 8.F.B.5 Step 4

Eliminate graphs (C) and (E): neither contains the flat horizontal shelf that represents the hare's nap.
(In (C), the hare's distance actually drops at one point — which would mean running backward — and in (E) there is no rest segment at all.)

💡 No flat shelf $=$ no nap; a downward dip would mean the hare un-ran part of the race, which is impossible.

#3 Eliminate Possibilities 6.RP.A.2 Step 5

Decide between the survivors (A) and (B) using the last clue: "the tortoise is already at the finish line when the hare arrives." Find where each animal's curve crosses the top (finish-distance) line and see whose crossing is earlier in time.

💡 Whoever's line hits the finish-distance line at the smaller $t$-value is the winner.

#3 Eliminate Possibilities 8.F.B.5 Step 6

In graph (A), the hare's piecewise curve reaches the finish line first — that would mean the hare wins, which contradicts the story.
In graph (B), the tortoise's straight line reaches the finish line first and the hare's second sprint arrives later.
Graph (B) is the only one that satisfies every clue.

$$t_{\text{tortoise, finish}} < t_{\text{hare, finish}} \;\Rightarrow\; \textbf{(B)}$$

💡 The tortoise winning means its finish-time mark sits to the LEFT of the hare's finish-time mark on the time axis.

[1] #1 8.F.A.3 Sketch what the tortoise's curve must look like. "Slow steady pace" means consta

[2] #1 8.F.B.5 Sketch what the hare's curve must look like. Three phases stitched together: a s

[3] #3 8.F.B.5 Eliminate graph (D): in (D) the tortoise's path is a curve, not a straight line.

[4] #3 8.F.B.5 Eliminate graphs (C) and (E): neither contains the flat horizontal shelf that re

[5] #3 6.RP.A.2 Decide between the survivors (A) and (B) using the last clue: "the tortoise is a

[6] #3 8.F.B.5 In graph (A), the hare's piecewise curve reaches the finish line first — that wo

Review

Reasonableness: 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.

Alternative: Tool #1 (Draw a Diagram) on its own: before peeking at the choices, freehand a $d$-vs-$t$ picture from the story — one gentle slanted line all the way across for the tortoise, and a tall-flat-tall staircase for the hare that ends to the right of the tortoise's finish point. Now compare your sketch to (A)-(E); only (B) is a topological match, no elimination grid needed.

CCSS standards used (min grade 8)

8.F.A.3 Interpret the equation y = mx + b as defining a linear function (Recognizing that the tortoise's "constant speed" must appear as a straight-line graph (a linear function $d = mt$) through the origin.)
8.F.B.5 Describe qualitatively the functional relationship between two quantities (Translating each phrase of the race story (run, nap, run, finishes later) into qualitative graph features (steep, flat, steep, later finish time) and rejecting the graphs whose shape contradicts the story.)
6.RP.A.2 Understand the concept of a unit rate and use rate language (Reading the slope of each distance-time segment as the racer's speed, so that "flat = stopped" and "steeper = faster" are meaningful comparisons.)

⭐ 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!

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: B

Review

CCSS standards used (min grade 8)

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