please help!

Question

On a particular day the wind added 5 miles per hour to Alfonso's rate when he was cycling with the wind and subtracted 5 miles per hour from his rate on his return trip. Alphonso found that the same amount of time he could cycle 75 miles with the wind, he could go only 45 miles against the wind. What was his normal bicycling speed with no wind?

Answer 1

Remember that distance = rate × time. Let r = normal bicycling speed with no wind. We'll write one equation for Alfonso's trip with the wind and one for Alfonso's trip against the wind.

 

 

 

WITH THE WIND:

 

He bikes a distance of 75 miles with the wind. He bikes at a rate of r + 5, his normal rate plus the speed of the wind. We're not told how long he bikes, so call the time t. So:

 

75 = t(r + 5)

 

 

 

AGAINST THE WIND:

 

He bikes a distance of 45 miles against the wind. He bikes at a rate of r - 5, his normal rate minus the speed of the wind. We're not told how long he bikes, but we know he bikes as long against the wind as with it, so call the time t. So:

 

45 = t(r - 5)

 

 

 

Now you have a system of equations:

 

75 = t(r + 5)

45 = t(r - 5)

 

Solve each equation for t, then set them equal to each other to solve for r:

 

75 / (r+5) = t

45 / (r-5) = t

 

75 / (r+5) = 45 / (r-5)

75(r-5) = 45(r+5)

75r - 375 = 45r + 225

30r = 600

r = 20

 

So, Alfonso's normal biking speed is r = 20 MPH.