2021 AMC 10A Problems/Problem 6

Problem

Chantal and Jean start hiking from a trailhead toward a fire tower. Jean is wearing a heavy backpack and walks slower. Chantal starts walking at $4$ miles per hour. Halfway to the tower, the trail becomes really steep, and Chantal slows down to $2$ miles per hour. After reaching the tower, she immediately turns around and descends the steep part of the trail at $3$ miles per hour. She meets Jean at the halfway point. What was Jean's average speed, in miles per hour, until they meet?

$\textbf{(A)} ~\frac{12}{13} \qquad\textbf{(B)} ~1 \qquad\textbf{(C)} ~\frac{13}{12} \qquad\textbf{(D)} ~\frac{24}{13} \qquad\textbf{(E)} ~2$

Solution 1 (Generalized Distance)

Let $2d$ miles be the distance from the trailhead to the fire tower. When Chantal meets Jean, she has traveled for $\frac d4 + \frac d2 + \frac d3 = d\left(\frac 14 + \frac 12 + \frac 13\right) =d\left(\frac{3}{12} + \frac{6}{12} + \frac{4}{12}\right)=\frac{13}{12}d$ hours. Jean has also traveled for $\frac{13}{12}d$ hours, and he travels for $d$ miles. So, his average speed is $\frac{d}{\left(\frac{13}{12}d\right)}=\boxed{\textbf{(A)} ~\frac{12}{13}}$ miles per hour.

~MRENTHUSIASM

Solution 2 (Convenient Distance)

We will use the same template as shown in Solution 1, except that we will replace $\boldsymbol{d}$ with a concrete number.

Let $24$ miles be the distance from the trailhead to the fire tower. When Chantal meets Jean, she has traveled for $\frac{12}{4} + \frac{12}{2}+\frac{12}{3}=3+6+4=13$ hours. Jean has also traveled for $13$ hours, and he travels for $12$ miles. So, his average speed is $\boxed{\textbf{(A)} ~\frac{12}{13}}$ miles per hour.

~MRENTHUSIASM

Video Solution 2(Simple and Quick)

https://youtu.be/vwtGZVJ0TbI

~ Education, the Study of Everything

Video Solution (Using Speed, Time, Distance)

https://youtu.be/hRFMsxhXQd0

~ pi_is_3.14

Video Solution 3

https://youtu.be/LonrTlNRk94

~savannahsolver

Video Solution by TheBeautyofMath

https://youtu.be/cckGBU2x1zg

~IceMatrix

2021 AMC 10A (Problems • Answer Key • Resources)
Preceded by Problem 5	Followed by Problem 7
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All AMC 10 Problems and Solutions

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