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?

Answer: A

Concepts:distance rate and time

Difficulty rating: 1140

AMC 8AMC 10AMC 12AIME

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Written solution:

Let $2d$ be the distance from the trailhead to the fire tower, where $d > 0.$

Then Chantal hiked for $$\dfrac{d}{4} + \dfrac{d}{2} + \dfrac{d}{3} = \dfrac{13d}{12}$$ hours.

If Jean travelled $d$ miles in $\dfrac{13d}{12}$ hours, then his speed was $$d \div \dfrac{13d}{12} = \dfrac{12}{13}$$ miles per hour.

Thus, A is the correct answer.