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 solve this problem, let's break it down into two parts: Markus's jogging and his walking during the cooldown.

Let's start by finding the rate at which Markus jogged, denoted by "r mi/h."

The total distance Markus jogged is 4 miles, and he jogged for a certain amount of time. Since we know that distance = rate × time, we can form an equation for the jogging part of the workout:

4 = r × t, where "t" is the time taken to jog.

Next, let's consider the cooldown walk. Markus walked 1 mile at a rate 3 mi/h slower than his jogging speed. Therefore, the walking rate is (r - 3) mi/h.

To calculate the time Markus spent walking, we can use the formula:

1 = (r - 3) × (1 - t), where "1 - t" is the remaining time after jogging.

Now, we know that the total workout lasted 1 hour. So, the sum of the jogging and walking times should be equal to 1:

t + 1 - t = 1

Simplifying, we get:

1 = 1

This equation is true; hence, our equations for jogging and walking are correct. Now, we can solve the system of equations:

4 = r × t (equation 1)
1 = (r - 3) × (1 - t) (equation 2)

We can rearrange equation 2 to solve for "t":

1 - t = 1 / (r - 3)
t = 1 - 1 / (r - 3)

Now, substitute this value of "t" in equation 1:

4 = r × (1 - 1 / (r - 3))
4 = r - r / (r - 3)
4(r - 3) = r(r - 3) - r
4r - 12 = r^2 - 3r - r
0 = r^2 - 7r + 12
0 = (r - 4)(r - 3)

Now, we have two possible solutions:

1) r - 4 = 0 => r = 4
2) r - 3 = 0 => r = 3

Since the walking rate is "3 mi/h slower than his jogging speed," we discard the second solution (r = 3). Therefore, the rate at which Markus jogged is r = 4 mi/h.

To find the walking rate, subtract 3 mi/h from the jogging rate:

Walking rate = 4 mi/h - 3 mi/h = 1 mi/h

Therefore, Markus jogged at a rate of 4 mi/h and walked at a rate of 1 mi/h during the cooldown.