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Problem 293

Algebra Difficulty 2.1 Find the answer CEMC Cayley

Carl and André are running a race. Carl runs at a constant speed of x m/sx \mathrm{~m} / \mathrm{s}. André runs at a constant speed of y m/sy \mathrm{~m} / \mathrm{s}. Carl starts running, and then André starts running 20 s later. After André has been running for 10 s, he catches up to Carl. What is the ratio y:xy: x?

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

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Official solution

André runs for 10 seconds at a speed of y m/sy \mathrm{~m} / \mathrm{s}. Therefore, André runs 10y m10y \mathrm{~m}. Carl runs for 20 seconds before André starts to run and then 10 seconds while André is running. Thus, Carl runs for 30 seconds. Since Carl runs at a speed of x m/sx \mathrm{~m} / \mathrm{s}, then Carl runs 30x m30x \mathrm{~m}. Since André and Carl run the same distance, then 30x m=10y m30x \mathrm{~m} = 10y \mathrm{~m}, which means that yx=3\frac{y}{x} = 3. Thus, y:x=3:1y: x = 3: 1.

Source: Omni-MATH, licensed Apache-2.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.