I can not figure out how to set this problem up. I know the answer is 200 mi, 220 mi.

Two different routes between two cities differ by 20 miles. Bill and Lou made the trip between the cities in exactly the same time. If one traveled the shorter route at 50 mph and the other traveled the longer route at 55 mph, how long is each route?

shorter route --- x miles

longer route ---- x+20 miles

Since time = distance/rate
time for shorter route = x/50
time for longer route = (x+20)/55
but the times are equal, so

x/50 = (x+20)/55
55x = 50x + 1000
5x = 1000
x = 200

shorter route = 200 miles
longer route = 200+20 = 220 miles

To solve this problem, let's assume the shorter route is "X" miles long, and the longer route is "X + 20" miles long.

We know that the time taken to travel the shorter route at 50 mph is the same as the time taken to travel the longer route at 55 mph.

Since Time = Distance / Speed, we can set up the equation:

X / 50 = (X + 20) / 55

To solve the equation, we can cross-multiply:

55X = 50(X + 20)

Now, let's distribute on the right side:

55X = 50X + 1000

Subtract 50X from both sides to isolate the X term:

55X - 50X = 1000

5X = 1000

Divide both sides by 5 to solve for X:

X = 1000 / 5

X = 200

So, the shorter route is 200 miles long.

To find the longer route, we can substitute the value of X back into the equation:

X + 20 = 200 + 20 = 220

Therefore, the longer route is 220 miles long.

To solve this problem, let's start by setting up equations based on the given information.

Let's assume that the shorter route is x miles long. Since the longer route differs by 20 miles, the longer route would be (x + 20) miles long.

Now, let's calculate the time it took each person to travel their respective routes.

The time can be calculated using the formula: Time = Distance / Speed.

For Bill, who traveled at 50 mph, the time taken to travel the shorter route would be: Time for Bill = x / 50.

For Lou, who traveled at 55 mph, the time taken to travel the longer route would be: Time for Lou = (x + 20) / 55.

Since it is given in the problem that Bill and Lou took the same amount of time to travel, we can set up the equation:

Time for Bill = Time for Lou

x / 50 = (x + 20) / 55

To solve this equation, we can cross-multiply:

55x = 50(x + 20)

55x = 50x + 1000

5x = 1000

x = 200

So, the shorter route is 200 miles long.

To find the length of the longer route, we can substitute the value of x back into the equation:

Longer route = x + 20 = 200 + 20 = 220 miles.

Hence, the length of the shorter route is 200 miles, and the length of the longer route is 220 miles.