Markus jogs 4 mi around a track at an average rate of r mi/h THen he cools down by walking 1 mi at a rate 3 mi/h slower. His whole workout lasts 1 h. At what rates did Markus jog and walk?

To find the rates at which Markus jogged and walked, we can set up a system of equations based on the given information.

Let's denote the rate at which Markus jogged as "r" mi/h.

Given that Markus jogs 4 miles at an average rate of r mi/h, we can write:
Time taken to jog = Distance / Rate
Time taken to jog = 4 / r

Next, we are told that Markus cools down by walking 1 mile at a rate 3 mi/h slower. Therefore, his walking rate will be (r - 3) mi/h.
The time taken to walk 1 mile at a rate of (r - 3) mi/h is:
Time taken to walk = Distance / Rate
Time taken to walk = 1 / (r - 3)

We know that Markus's whole workout lasts 1 hour. So the total time taken for jogging and walking should add up to 1 hour:
Time taken to jog + Time taken to walk = 1

Substituting the values we found above, we get:
4 / r + 1 / (r - 3) = 1

To solve this equation, we can simplify it by getting rid of the denominators:

Multiply the entire equation by r(r - 3) to eliminate the fractions:
4(r - 3) + r = r(r - 3)

Now let's expand and simplify:
4r - 12 + r = r^2 - 3r

Bringing all the terms to one side, we get:
r^2 - 8r + 12 = 0

This is a quadratic equation which can be factored as:
(r - 6)(r - 2) = 0

Setting each factor equal to zero, we find two possible solutions for r:
r - 6 = 0 or r - 2 = 0

Solving each equation, we get:
r = 6 or r = 2

Therefore, the rates at which Markus jogged and walked are:
Jogging rate (r) = 6 mi/h
Walking rate (r - 3) = 6 - 3 = 3 mi/h

Alternatively, if we choose the other solution:
Jogging rate (r) = 2 mi/h
Walking rate (r - 3) = 2 - 3 = -1 mi/h

Since a negative speed does not make sense in this context, the valid rates for jogging and walking are:
Jogging rate = 6 mi/h
Walking rate = 3 mi/h