Competition · AMC preparation · step 4 of 4
AMC 8 · 2019 · #5
Grade 8 rate-ratioalgebra
Pick an answer.
AMC 8 2019 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
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.
Sketch the tortoise's graph
Tortoise = slow steady pace, so its curve is one gentle straight line through the origin — never bending.
Constant speed shows up as a straight line because the same distance is added in every equal slice of time.
8.F.A.3Draw A DiagramSketch the hare's graph
Hare = run, nap, run: its curve is steep up, then a flat shelf, then steep up again.
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.
The hare's run, then nap, then run must appear on the distance-time graph as a steep line, then a flat line, then a steep line.
▸ Why?
In each running phase the hare holds a fast, unchanging speed, so it adds the same distance in every equal slice of time, and adding the same amount each equal slice makes the graph climb in equal steady steps — a steep straight line.
▸ Why?
At a fixed speed the distance covered is exactly that speed multiplied by the time elapsed, so every equal stretch of time adds the same distance and the total rises at one steady rate.
▸ Why?
During the nap time keeps passing but the hare gains no distance, so its distance total is left unchanged and the graph neither rises nor falls — a flat, level line.
▸ Why?
Adding zero distance moment after moment leaves the running total exactly as it was, so the height on the graph stays where it is.
▸ Why?
The hare's whole race is just these three stretches laid end to end with no gap and no overlap, so its full curve is the steep piece, the flat piece, and the steep piece joined in that order.
▸ Why?
A complete trip is exactly its parts with nothing left out and nothing counted twice, so the whole path is those three stretches put back together in sequence.
Rule out the curved tortoise line
Eliminate (D): the tortoise's path is a curve, and a curve means changing speed — not a steady pace.
A bending curve says "speeding up or slowing down" — the tortoise does neither.
8.F.B.5Eliminate PossibilitiesRule out graphs with no nap
Eliminate (C) and (E): neither has the flat nap shelf, and (C) even dips downward (running backward).
No flat shelf = no nap; a downward dip would mean the hare un-ran part of the race, which is impossible.
8.F.B.5Eliminate PossibilitiesCompare the two finish times
Between (A) and (B), use the last clue: whoever's curve hits the finish line at the smaller time wins.
Whoever's line hits the finish-distance line at the smaller t-value is the winner.
6.RP.A.2Eliminate PossibilitiesDrop the graph where hare wins
In (A) the hare finishes first (wrong); in (B) the tortoise's line hits the finish first — (B) satisfies every clue.
The tortoise winning means its finish-time mark sits to the LEFT of the hare's finish-time mark on the time axis.
8.F.B.5Eliminate PossibilitiesThis 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!
- Sketch the tortoise's graph
- Sketch the hare's graph
- Rule out the curved tortoise line
- Rule out graphs with no nap
- Compare the two finish times
- Drop the graph where hare wins
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