Gabriel can ride his bike three times faster than he can run. He runs 9 miles and
bikes 45 miles. If the total time for both activities is four hours, how fast can he run ?
Quite fast.
Let's assume Gabriel's running speed is R miles per hour. Since Gabriel can ride his bike three times faster than he can run, his biking speed would be 3R miles per hour.
We can use the given information to set up two equations based on the distances Gabriel ran and biked:
Equation 1: Distance = Speed × Time (for running)
9 = R × Time (1)
Equation 2: Distance = Speed × Time (for biking)
45 = 3R × Time (2)
We also know that the total time for both activities is four hours:
Time (running) + Time (biking) = 4
Now, let's solve the equations to find the value of R, Gabriel's running speed:
From Equation (1):
Time (running) = 9 / R
From Equation (2):
Time (biking) = 45 / (3R) = 15 / R
Using the information about the total time:
Time (running) + Time (biking) = 4
Substituting the values of Time (running) and Time (biking) in terms of R:
9 / R + 15 / R = 4
Combining the fractions and simplifying:
(9 + 15) / R = 4
24 / R = 4
Cross-multiplying:
24 = 4R
Finally, solving for R (running speed):
R = 24 / 4 = 6 miles per hour
Therefore, Gabriel can run at a speed of 6 miles per hour.