Question

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 $210$ equal steps walking from the back of the ship to the front. Walking in the opposite direction, she counts $42$ 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?

$70$

$84$

$98$

$105$

$126$

Solution:

Let $x$ be the length of the ship. Then in the time that Emily moves $210$ steps, the ship moves $210 - x$ steps.

In the time that Emily moves $42$ steps, the ship moves $x - 42$ steps. Since the ship and Emily move at a constant rate $\dfrac{210}{210 - x} = \dfrac{42}{x - 42}.$ Cross-multiplying yields $210x - 210 \cdot 42 = 210 \cdot 42 - 42x$ $42 \cdot 6 x = 2 \cdot 42 \cdot 210$ $x = 70.$

Thus, A is the correct answer.

2021 AMC 10A Fall Problem 11

Solution:

Problem 11 in Other Years