Question

Mark Keaton’s workout consists of jogging for 3 miles and then riding his bike
for 5 miles at a speed 4 miles per hour faster than he jogs. If his total workout time is one hour, find his jogging speed and his biking speed.

Answer 1

First figure out what you're being asked to find. His jogging speed, which we'll call j, and his biking speed, which we'll call b.

What do we know about these two? We know that the biking speed is 4 mph faster than his jogging speed:

\[b = j + 4\]
We also know that the time he spent jogging, which we'll call t, and the time he spent biking add up to an hour. So the biking time = 1 - t. We know that he jogged for 3 miles and biked for 5, so:

\[jt = 3\]\[b(1 - t) = 5\]

We now have everything we need to solve the problem. First we substitute our equation for b in the last equation:

\[(j + 4)(1 - t) = 5\]
Next we solve for either user our second equation to solve for either j or t (it doesn't matter):

\[t = \frac{3}{j}\]
Substitute this into the equation before it and solve for j:

\[(j + 4)(1 - \frac{3}{j}) = 5\]\[j + 4 - \frac{3j}{j} - \frac{12}{j} = 5\]\[j + 4 - 3 - \frac{12}{j} = 5\]\[j - 4 - \frac{12}{j} = 0\]\[j^{2} -4j -12 = 0\]\[(j -6)(j + 2) = 0\]
So j = 6 or j = -2. Remember that j is the jogging speed and a negative jogging speed doesn't make any sense, so his jogging speed is 6 mph, and because his biking speed is 4 mph faster than his jogging speed, his biking speed is 10 mph. You can (and should) plug these back into the equations to check your work.