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2017 AMC 10A Problem 9

2017 AMC 10A Problem 9AMC 10 Step by Step

Name: Date: September 13, 2026

Linear Equations & Word Problems · difficulty 2 of 5 · about 3 min · position P1-10

Minnie rides on a flat road at $20$ kilometers per hour (kph), downhill at $30$ kph, and uphill at $5$ kph. Penny rides on a flat road at $30$ kph, downhill at $40$ kph, and uphill at $10$ kph. Minnie goes from town $A$ to town $B$ , a distance of $10$ km all uphill, then from town $B$ to town $C$ , a distance of $15$ km all downhill, and then back to town $A$ , a distance of $20$ km on the flat. Penny goes the other way around using the same route. How many more minutes does it take Minnie to complete the $45$ -km ride than it takes Penny?

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

Answer

Key insight

Compute each leg as distance over speed; Penny rides the loop backwards, so her uphill leg is the 15 km stretch and her downhill leg is the 10 km one.

Solution

Compute each rider's time leg by leg with $\text{time} = \text{distance}/\text{speed}$ .

Minnie: $A \to B$ is $10$ km uphill at $5$ kph, $2$ hours. $B \to C$ is $15$ km downhill at $30$ kph, $\tfrac12$ hour. $C \to A$ is $20$ km flat at $20$ kph, $1$ hour. Total $3\tfrac12$ hours $= 210$ minutes.

Penny rides $A \to C \to B \to A$ , so the slopes are reversed. $A \to C$ is $20$ km flat at $30$ kph, $\tfrac23$ hour $= 40$ minutes. $C \to B$ is $15$ km, now uphill, at $10$ kph, $1\tfrac12$ hours $= 90$ minutes. $B \to A$ is $10$ km, now downhill, at $40$ kph, $\tfrac14$ hour $= 15$ minutes. Total $145$ minutes.

Difference: $210 - 145 = 65$ minutes.

The answer is $\boxed{\textbf{(C)}\ 65}$ .

Why this works

Multi-leg trips are handled one leg at a time, because speed changes at each leg; there is no shortcut through average speed. The one conceptual step is that a hill climbed in one direction is descended in the other, so reversing the loop reassigns the terrain to each segment.

The trap

Giving Penny the same uphill and downhill legs as Minnie; reversing direction swaps which segments are up and down.

Common mistakes

Giving Penny the same uphill and downhill legs as Minnie; reversing direction swaps which segments are up and down.
Mixing hours and minutes, for example adding $\tfrac12$ hour as $50$ minutes.

Techniques

Set up the equation/formula and compute; no special trick needed