Answer: Let's call Jean's speed running "r" and her speed riding her bicycle "b". We know that she runs 8 miles and rides 4 miles, and we also know that her speed on the bike is 8 mph faster than her speed running. We can set up two equations based on this information:
Equation 1: Time running + Time biking = 1.25 hr
Equation 2: Distance running + Distance biking = 8 + 4 = 12 miles
To solve for Jean's speed running and biking, we need to use a bit of algebra to eliminate one of the variables. We can start by using the formula:
time = distance / speed
For the running and biking portions of the race, this gives us:
Time running = 8 / r
Time biking = 4 / b
Substituting these expressions into Equation 1, we get:
8 / r + 4 / b = 1.25
Multiplying both sides by rb (to eliminate the denominators), we get:
8b + 4r = 1.25rb
We also know that b = r + 8, so we can substitute this expression into the equation above:
8(r + 8) + 4r = 1.25r(r + 8)
Simplifying and rearranging, we get:
1.25r^2 - 6r - 64 = 0
We can use the quadratic formula to solve for r:
r = [6 ± sqrt(6^2 + 4(1.25)(64))] / 2(1.25) ≈ 4 or 10.4
The negative root doesn't make sense in this context (it would mean Jean is running backwards!), so we can discard it and conclude that Jean's speed running is about 4 mph.
To find her speed biking, we can use the equation b = r + 8:
b = 4 + 8 = 12 mph
Therefore, Jean's speed running is 4 mph and her speed biking is 12 mph.
Explanation: