2017 AMC 10A Problem 9

Problem 9 of 25EasierAlgebra

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?

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Solution

It will take Minnie

10 \div 5 = 2

hours to travel the uphill distance. It will take her

15 \div 30 = \frac{1}{2}

hours to travel the downhill distance. Finally, it will take her

20 \div 20 = 1

hour to travel the flat. This will take her a total of

\begin{aligned}60\left(2 + \frac{1}{2} + 1\right) &= 60 \cdot \frac{7}{2} \\&= 210 \end{aligned}

minutes. It will take Penny

20 \div 30 = \frac{2}{3}

hours to travel the flat. It will take her another

15 \div 10 = \frac{3}{2}

hours to travel the uphill. Finally, it will take her

10 \div 40 = \frac{1}{4}

hours to travel the downhill. This is a total of

\begin{aligned} 60\left(\dfrac{2}{3} + \dfrac{3}{2} + \dfrac{1}{4}\right) &= 60 \cdot \dfrac{29}{12} \\&= 145 \end{aligned}

minutes. The trip takes Minnie

210 - 145 = 65

more minutes to travel than Penny. Thus, C is the correct answer.