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2021 Fall AMC 10A Problem 11

Problem 11 of 25IntermediateAlgebra

Emily sees a ship traveling at a constant speed along a straight section of a river. She walks parallel to the riverbank at a uniform rate faster than the ship. She counts 210210 equal steps walking from the back of the ship to the front. Walking in the opposite direction, she counts 4242 steps of the same size from the front of the ship to the back. In terms of Emily’s equal steps, what is the length of the ship?

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Solution

Let xx be the length of the ship. Then in the time that Emily moves 210210 steps, the ship moves 210−x210 - x steps. In the time that Emily moves 4242 steps, the ship moves x−42x - 42 steps. Since the ship and Emily move at a constant rate 210210−x=42x−42. \dfrac{210}{210 - x} = \dfrac{42}{x - 42}. Cross-multiplying yields 210x−210⋅42=210⋅42−42x 210x - 210 \cdot 42 = 210 \cdot 42 - 42x 42⋅6x=2⋅42⋅210 42 \cdot 6 x = 2 \cdot 42 \cdot 210 x=70. x = 70. Thus, A is the correct answer.

More practice

Concepts: relative speed · distance rate and time · ratio and proportion

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.