2017 AMC 10A Problem 9

Question

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?

45

60

65

90

95

Compute Minnie’s three segment times in hours

Compute Penny’s three segment times on the same route in the opposite direction

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.

Problem 9 in other years