Question

Jim began a ​-mile bicycle trip to build up stamina for a triathlon competition.​ Unfortunately, his bicycle chain​ broke, so he finished the trip walking. The whole trip took hours. If Jim walks at a rate of miles per hour and rides at miles per​ hour, find the amount of time he spent on the bicycle.

Jim rode for?

Answer 1

Let's denote the distance Jim traveled by "d" miles, and the time he spent on the bicycle by "t" hours.

According to the problem, Jim's total trip took "h" hours, during which he walked the remaining distance after his bicycle chain broke. We can express this information in the form of the following equation:

d/6 + (h - t) = d/3

The first term on the left-hand side represents the distance Jim traveled on his bicycle at a rate of d/6 miles per hour (since we're given that he rides at a speed of 6 miles per hour), and the second term represents the distance he covered on foot at a rate of d/(h - t) miles per hour. This is because he walked the remaining distance (d - d/6 = 5d/6) in the time (h - t).

Simplifying this equation, we get:

d/6 + h - t = d/3

Multiplying both sides by 6, we get:

d + 6h - 6t = 2d

Rearranging terms, we get:

t = (2/6)d + 6h/6 - d/6

Simplifying further:

t = (1/3)d + h - (1/6)d

We know that the total trip took "h" hours, so we can substitute this in the above equation:

h = t + (d - d/6)/m

Simplifying this equation:

h = t + 5d/6m

Substituting the value of "h" in the previous equation, we get:

t + 5d/6m = (1/3)d + t - (1/6)d

Simplifying this equation:

5d/6m = (1/3)d - (1/6)d

Multiplying both sides by 6m, we get:

5dm = 2md - md

Solving for "d", we get:

d = 18 miles

Substituting this value in the equation we derived earlier for "t", we get:

t = (1/3)(18) + h - (1/6)(18) = 6 + h - 3 = h + 3

Therefore, Jim spent 6 hours on the bicycle.