AMC 10 · 2017 · #9

Grade 7 rate-ratio
Archetype Meeting Time with Piecewise Rates └ Sub-archetype Piecewise-Leg Time Balance
rateunit-conversionfraction-arithmetic identify-subproblems ↑ Prerequisites: rate
📏 Medium solution 💡 2 insights
Problem
Two riders cover the same triangular loop of three roads — a 10 km hill, a 15 km hill, and a 20 km flat — but in opposite directions, each with their own flat, uphill, and downhill speeds. Find how many more minutes one rider's whole loop takes than the other's.

Pick an answer.

(A)
45
(B)
60
(C)
65
(D)
90
(E)
95

AMC 10 2017 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Analyze the Units

The units carry the whole problem: time equals distance divided by speed, so km ÷ km/h gives hours. Tool #8 (Analyze the Units) fixes that relationship and reminds us to finish in minutes. Tool #7 (Identify Subproblems) splits each loop into its three legs, each a one-line division. Tool #1 (Draw a Diagram) tracks which leg is uphill, downhill, or flat for each rider once the directions are reversed.

1STEP 1

Time is distance over speed

On any leg a rider holds one speed, so its time is distance divided by speed — km ÷ km/h leaves hours. Convert to minutes at the end.

time = distance/speed (km÷km/h=h)
2STEP 2

Minnie's three legs

Minnie: 10 km uphill at 5 kph, 15 km downhill at 30 kph, 20 km flat at 20 kph, giving 2 + ½ + 1 = 3.5 hours = 210 minutes.

10/5+15/30+20/20=2+1/2+1=3.5 h=210 min
3STEP 3

Flip the slopes for Penny

Penny rides backwards A→C→B→A: the flat stays flat, but Minnie's downhill B→C now climbs for Penny and her uphill A→B now falls.

A→ C:flat, C→ B:uphill, B→ A:downhill
4STEP 4

Penny's three legs

Penny: 20 km flat at 30 kph, 15 km uphill at 10 kph, 10 km downhill at 40 kph, giving 2/3 + 3/2 + 1/4 = 29/12 hours = 145 minutes.

20/30+15/10+10/40=8/12+18/12+3/12=29/12 h=145 min
5STEP 5

Subtract the two times

Minnie needs 210 minutes, Penny 145, so Minnie takes 210 − 145 = 65 minutes more. The answer is (C).

210-145=65 min → (C)
Answer
65
Penny should be faster: she is quicker on every kind of road, and she spends the long climb on the shorter 15 km hill rather than the same hill Minnie crawls up. So Minnie taking more time fits. The numbers are clean — 210 and 145 minutes — and their gap, 65 minutes, is one of the choices and sits between the smaller and larger options.
💡Key takeaway

Split a trip into legs, find each leg's time as distance divided by speed, and remember that riding the loop backwards turns every hill into the opposite slope.

  • Time is distance over speed
  • Minnie's three legs
  • Flip the slopes for Penny
  • Penny's three legs
  • Subtract the two times

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