Mark went rock climbing on Saturday on Mt. Rockytop. He started at 10:00 a.m. and climbed at the rate of 4 miles per hour. His friend Paul began climbing at noon, climbing at a rate of 5 miles per hour. At what time would Paul catch up to Mark
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Let t=Mark's time spent going up the mountain.
Then Mark's distance up the mountain is 4t. (rate * time=distance).
Since Paul's started two hours later, t-2 is the amount of time that Paul spends going up the mountain. (He is 2 hours behind Mark)
So Paul's distance up the mountain is 5(t-2).
Set the two distances equal: 4t=5t-10. So t=10.
Since t is the amount of time it took Mark to go up the mountain, we can say that t=0 is at 10:00AM, when he started.
So t=10 is 8:00PM.
And to check, after 10 hours, Mark is 4*10, or 40 miles up the mountain. And after 8 hours, Paul is 5*8=40 miles up the mountain.