Work through 2026 AMC 8 Problem 19 on rate problems with a written solution, hints, and a visual explanation.
AlgebraLevel 4 · Challenge
Miguel is walking with his dog, Luna. When they reach the entrance to a park, Miguel throws a ball straight ahead and continues to walk at a steady pace. Luna sprints toward the ball, which stops by a tree. As soon as the dog reaches the ball, she brings it back to Miguel. Luna runs 5 times faster than Miguel walks. What fraction of the distance between the entrance and the tree does Miguel cover by the time Luna brings him the ball?
Diagram 1
A. 1/6
B. 1/5
C. 1/4
D. 1/3
E. 2/5
Hints
Assign a variable to the distance Miguel walks during the whole event, and remember Luna's speed is 5 times his.
Figure out the total distance Luna covers: she runs all the way to the tree, then runs back to meet Miguel partway.
Set Luna's total distance equal to 5 times Miguel's distance and solve for Miguel's distance as a fraction of the distance to the tree.
Read the step-by-step solution
Answer: D · 1/3
Set up variables
Let Miguel's speed be v and the distance to the tree be d. Luna's speed is 5v. Let x be the distance Miguel walks during the entire event.
Use total distance
Luna runs d to the tree and then (d − x) back to Miguel, so Luna runs d + (d − x) = 2d − x. Since Luna runs 5x in the same time Miguel walks x: 5x = 2d − x, giving 6x = 2d, so x = d/3.
