2021 WSMO Team Round Problems/Problem 5

Problem

Two runners are running at different speeds. The first runner runs at a consistent 12 miles per hour. The second runner runs at $t+4$ miles per hour, where $t$ is the number of hours that have passed. After $n$ hours, the runners have run the same distance, where $n$ is positive. Find $n$ .

Proposed by pinkpig

Solution

After $n$ hours, the first runner has run $12n$ miles. The second runner runs at $t+4$ miles per hour, so his average speed over $n$ hours is $\frac{(n+4)+4}{2} = \frac{n+8}{2}.$

Thus, the second runner travels $\frac{n+8}{2} \cdot n = \frac{n^2+8n}{2} \text{ miles}.$

Setting the distances equal: $\frac{n^2+8n}{2} = 12n \Rightarrow n^2+8n = 24n \Rightarrow n^2 = 16n \Rightarrow n = \boxed{16}.$ ~pinkpig

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2021 WSMO Team Round Problems/Problem 5

Problem

Solution