Difference between revisions of "2021 AMC 10A Problems/Problem 6"

Revision as of 21:57, 11 February 2021

Problem 6

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 (Generalized Distance)

Let $2d$ miles be the distance from the start 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 also has 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)

Let $12$ miles be the distance from the start to the fire tower. When Chantal meets Jean, she travels for $\frac{12}{4} + \frac{12}{2}+\frac{12}{3}=3+6+4=13$ hours. Jean also has 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 (Using Speed, Time, Distance)

https://youtu.be/hRFMsxhXQd0

~ pi_is_3.14

@@ Line 1: / Line 1: @@
+==Problem 6==
+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 <math>4</math> miles per hour. Halfway to the tower, the trail becomes really steep, and Chantal slows down to <math>2</math> miles per hour. After reaching the tower, she immediately turns around and descends the steep part of the trail at <math>3</math> miles per hour. She meets Jean at the halfway point. What was Jean's average speed, in miles per hour, until they meet?
+<math>\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</math>
+== Solution (Generalized Distance) ==
+Let <math>2d</math> miles be the distance from the start to the fire tower. When Chantal meets Jean, she has traveled for <cmath>\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</cmath> hours. Jean also has traveled for <math>\frac{13}{12}d</math> hours, and he travels for <math>d</math> miles. So, his average speed is <cmath>\frac{d}{\left(\frac{13}{12}d\right)}=\boxed{\textbf{(A)} ~\frac{12}{13}}</cmath> miles per hour.
+~MRENTHUSIASM
+== Solution 2 (Convenient Distance) ==
+Let <math>12</math> miles be the distance from the start to the fire tower. When Chantal meets Jean, she travels for <cmath>\frac{12}{4} + \frac{12}{2}+\frac{12}{3}=3+6+4=13</cmath> hours. Jean also has traveled for <math>13</math> hours, and he travels for <math>12</math> miles. So, his average speed is <math>\boxed{\textbf{(A)} ~\frac{12}{13}}</math> miles per hour.
+~MRENTHUSIASM
 == Video Solution (Using Speed, Time, Distance) ==
 https://youtu.be/hRFMsxhXQd0
 ~ pi_is_3.14

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