The ring chase

Algebra Level 3

Frodo and Gollum are 900 meters apart in an infinitely long, lighted tunnel in Middle-Earth. Frodo has the ring, which makes him invisible. He can wear the ring for 15 seconds until it falls off, and then it takes him 5 seconds to put it back on. Frodo runs at a rate of 8 meters per second away from Gollum, but he cannot run while he puts the ring on. Gollum runs at a rate of 30 meters per second towards Frodo, but Gollum can only run if Frodo doesn't have the ring on. In seconds, how long does it take for Gollum to catch up with Frodo?

Frodo has the ring on to begin with for 15 seconds as normal.

The answer is 600.

3 solutions

Frodo Baggins
Mar 29, 2014

In a period of 20 seconds, Frodo runs 8 m/s for 15 seconds for a total of 120 meters. Gollum runs 30 m/s for 5 seconds for a total of 150 meters. Therefore, Gollum catches up with Frodo 30 meters every 20 seconds. 900/30=30, so it will take 30 periods of 20 seconds to catch up, totaling 600 seconds.

Nanayaranaraknas Vahdam
Apr 1, 2014

First, take the first 15 seconds. Frodo moves for 15 seconds, at 8 m/s. Thus, he moves 120 m in the first 15 seconds.

Then, for the next 5 seconds, Frodo cannot move, and Gollum moves for 5 seconds, at 30 m/s. Thus, he moves 150 m in the next 5 seconds.

So, in a span of 20 seconds, Gollum moves closer to Frodo by 150-120=30 m

Thus, to bridge the gap of 900 m, it would take (900/30)*20 seconds. Thus, it takes 600 seconds.

Venture Hi
Mar 29, 2014

there are two conditions which the problem can start: one with him with the ring on and one without the ring, which the answer would have been 505 seconds.

The ring chase

The answer is 600.

3 solutions

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