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2021 AMC 10A Problem 6

2021 AMC 10A Problem 6AMC 10 Step by Step

Name: Date: September 13, 2026

Linear Equations & Word Problems · difficulty 2 of 5 · about 3 min · position P1-10

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?

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

Answer

Key insight

Name the half-trail length d; Chantal's three legs take d/4 + d/2 + d/3 = 13d/12 hours, during which Jean covers exactly d.

Solution

Let the halfway point be $d$ miles from the trailhead, so the tower is $2d$ miles away.

Chantal's trip until the meeting has three legs, each of length $d$ :
- trailhead to halfway at $4$ mph: $\frac{d}{4}$ hours;
- halfway to tower at $2$ mph: $\frac{d}{2}$ hours;
- tower back to halfway at $3$ mph: $\frac{d}{3}$ hours.

Total time: $\frac{d}{4} + \frac{d}{2} + \frac{d}{3} = \frac{3d + 6d + 4d}{12} = \frac{13d}{12}$ hours.

In that same time Jean walks from the trailhead to the halfway point, a distance of $d$ . His average speed is
$\frac{d}{13d/12} = \frac{12}{13}\text{ mph}.$

The answer is $\boxed{\textbf{(A)}\ \frac{12}{13}}$ .

Why this works

Both hikers walk for the same amount of time, so computing Chantal's total time is the whole problem. Average speed is total distance over total time, never the average of speeds. The unknown $d$ cancels, so you may also set $d = 12$ (the lcm of the speeds) to avoid fractions.

Alternative approach

Let the halfway distance be $12$ miles. Chantal's times are $3 + 6 + 4 = 13$ hours, and Jean covers $12$ miles in $13$ hours: $\frac{12}{13}$ mph.

The trap

Averaging Chantal's three speeds, or forgetting that Jean only reaches the halfway point, not the tower.

Common mistakes

Averaging Chantal's three speeds, or forgetting that Jean only reaches the halfway point, not the tower.
Inverting the final fraction and answering $\frac{13}{12}$ , choice (C).

Techniques

Set up the equation/formula and compute; no special trick needed · Substitute to simplify (u = x+1/x, shifting, scaling)