A man bicycles 5mph faster than he can walk. He bicycles 24 miles and walks back along the same route for a charity race in 11 hours. How fast does he walk?
To find the speed at which the man walks, we'll use the formula:
Speed = Distance / Time
Let's assign variables to the unknowns:
Let the man's walking speed be 'W' mph
His biking speed will then be 'W + 5' mph
The distance of one leg of the trip (either biking or walking) is 24 miles.
Now, let's calculate the time it takes for the man to bike 24 miles.
Time = Distance / Speed
Time for biking = 24 / (W + 5)
Next, let's determine the time it takes for the man to walk back the same distance.
Time for walking = Distance / Speed
Time for walking = 24 / W
According to the problem, the total time for the round trip is 11 hours. So we can set up an equation:
Time for biking + Time for walking = Total time
24 / (W + 5) + 24 / W = 11
We can solve this equation to find the value of W, which represents the man's walking speed.
To do so, we'll combine the fractions on the left side:
(24W + 24(W + 5)) / (W(W + 5)) = 11
Simplifying further:
[24W + 24W + 120] / (W^2 + 5W) = 11
Combining like terms:
48W + 120 = 11(W^2 + 5W)
Expanding:
48W + 120 = 11W^2 + 55W
Rearranging the equation:
11W^2 + 55W - 48W - 120 = 0
11W^2 + 7W - 120 = 0
To solve this quadratic equation, you can try factoring or use the quadratic formula. Here, I will use factoring.
Factoring the equation:
(11W + 24)(W - 5) = 0
Setting each factor equal to zero:
11W + 24 = 0, or W - 5 = 0
Solving for W:
11W = -24 -> W = -24/11 (which is not applicable in this context)
W = 5
So, the man walks at a speed of 5 mph.
He walks 24 miles in 11 hours.
24/11 = 2.18 mph