Michelle is now 50 miles ahead of John. Michelle is travelling at a constant rate. John is traveling in the same direction, at a rate 10 miles per hour faster than Michelle. In how many hours will John catch up to Michelle?

How long would it take to close a gap of 50 miles at 10 miles/hour?

50/10 = ?

To find out how many hours it will take for John to catch up to Michelle, we need to set up an equation based on their rates of travel.

Let's say Michelle's rate of travel is "M" miles per hour. Since Michelle is traveling at a constant rate, we can assume her distance covered is distance = rate × time.

Let's also say John's rate of travel is "J" miles per hour, which is 10 miles per hour faster than Michelle's rate. So we can say J = M + 10.

Now, we know that Michelle has a 50-mile head start, which means her distance covered will be distance = M × time. On the other hand, John will travel for the same time, but his distance covered will be distance = J × time.

Given that Michelle's distance covered is 50 miles more than John's distance covered, we can set up the equation:

M × time = J × time + 50.

Since we want to find the time it takes for John to catch up to Michelle, we can cancel out the time variable from both sides of the equation:

M = J + 50.

Now we substitute J with M + 10 in the equation:

M = (M + 10) + 50.

Simplifying the equation, we get:

M = M + 60.

By canceling out the M variable from both sides of the equation, we find:

0 = 60.

This is a contradiction. The equation has no solution, which means John will never catch up to Michelle, given the information provided.