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?

$45$

$60$

$65$

$90$

$95$

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 $\begin{align*}60\left(2 + \frac{1}{2} + 1\right) &= 60 \cdot \frac{7}{2} \\&= 210 \end{align*}$ 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 $\begin{align*} 60\left(\dfrac{2}{3} + \dfrac{3}{2} + \dfrac{1}{4}\right) &= 60 \cdot \dfrac{29}{12} \\&= 145 \end{align*}$ minutes. The trip takes Minnie $210 - 145 = 65$ more minutes to travel than Penny.

Thus, C is the correct answer.

2017 AMC 10A Problem 9

Solution:

Problem 9 in Other Years