It took 4 hours for a biker to travel from one city to another going at a certain speed. On the return trip, the biker traveled at the same speed for the first 100-km and then for the rest of the trip he traveled at a speed which was 10 km/hour slower than the original speed, and thus the return trip took him 30 min longer. Find the distance between the two cities.

200, 160

Ah, the biker and his need for speed! Let's calculate this scenic journey step by step.

We know that the biker took 4 hours to travel from one city to another at a certain speed. Let's call this original speed "x" km/h.

Now, on the return trip, the biker initially traveled at the same speed of x km/h for the first 100 km. After that, though, things got interesting.

For the remainder of the trip, the biker decided to slow down a bit and traveled at a speed 10 km/h slower than the original speed. So, the speed for the remainder of the trip would be (x - 10) km/h.

This change in speed resulted in the return trip taking 30 minutes longer than the initial trip. Since the time difference is given in minutes, let's convert it to hours.

30 minutes is equal to 0.5 hours. I know, time flies when you're biking!

Now, let's dive into the equations.

For the initial trip:
Time = Distance / Speed
4 = Distance / x

For the return trip:
Time = Distance / Speed
4.5 = (Distance - 100) / (x - 10)

We have two equations, so we can solve for two variables.

Cross-multiplying, we get:
4x = Distance

For the return trip equation, we can distribute and simplify:
4.5(x - 10) = Distance - 100
4.5x - 45 = 4x - 100
0.5x = 55
x = 110

Now that we've found x, we can substitute it back into one of our equations to find the distance. Let's use the equation 4x = Distance.

Distance = 4 * 110
Distance = 440 km

So, the distance between the two cities is 440 kilometers. That's one "wheely" long bike ride!

To find the distance between the two cities, we need to determine the speed of the biker and the total time taken for the return trip.

Let's start by finding the speed of the biker on the first trip. We know that it took 4 hours for the biker to travel from one city to another. Let's assume the speed of the biker for the first trip is "x" km/h.

Using the formula: speed = distance / time, we can calculate the distance covered in the first trip. Since distance = speed * time, the distance covered in the first trip is:

Distance = x * 4

Now, let's consider the return trip. We know that the biker traveled at the same speed (x km/h) for the first 100 km. So, the time taken to cover the first 100 km is given by:

Time taken for the first 100 km = distance / speed = 100 / x

For the remaining distance (let's call it D), the biker traveled at a speed 10 km/h slower than the original speed (x - 10) km/h. Using the same formula as before, we can calculate the time taken for the remaining distance:

Time taken for the remaining distance = D / (x - 10)

We are also given that the return trip took 30 minutes longer, which is equal to 0.5 hours. So, the total time taken for the return trip can be obtained by adding the time for the first 100 km and the time for the remaining distance:

Total time taken for the return trip = Time taken for the first 100 km + Time taken for the remaining distance

Since the total time taken for the return trip is 30 minutes (0.5 hours) longer than the time taken for the first trip (which was 4 hours), we can set up the equation:

4 + 0.5 = 100 / x + D / (x - 10)

Simplifying the equation, we get:

4.5 = 100 / x + D / (x - 10)

To find the value of D (remaining distance), we need to use the given information that the total distance between the two cities remains the same for both trips. Therefore, the distance for the return trip is the total distance minus the 100 km already covered in the first trip. So, we can express D in terms of the distance:

D = Total distance - 100

Substituting this value into the equation, we get:

4.5 = 100 / x + (Total distance - 100) / (x - 10)

Simplifying the equation further, we can multiply both sides by x(x - 10) to eliminate the fractions:

4.5 * x(x - 10) = 100(x - 10) + x(Total distance - 100)

Expanding and rearranging the equation, we get:

4.5x^2 - 45x = 100x - 1000 + Total distance - 100x

Simplifying, we get:

4.5x^2 - 45x = -1000 + Total distance

Since we want to find the value of Total distance, we need to isolate it. Rearranging the equation, we get:

Total distance = 4.5x^2 - 45x + 1000

Now, we have an equation that allows us to calculate the distance between the two cities given the speed of the biker. To find the actual value of Total distance, we need to know the value of x (speed). Assuming a specific value for x would allow us to calculate the corresponding distance.

Once we have the speed value, we can substitute it into the equation to find the distance between the two cities.

let the "certain" speed be x km/h

then the distance between the two cities is 4x km

time for return trip
= 100/x + (4x-100)/(x-10)

100/x + (4x-100)/(x-10) - 4 = 1/2
100/x + (4x-100)/(x-10) = 9/2
multiply each term by 2x(x-10), the LCD
200(x-10) + 2x(4x-100) = 9x(x-10)
200x - 2000 + 8x^2 - 200x = 9x^2 - 90x
x^2 - 90x + 2000 = 0
(x-50)(x-40) = 0
x = 50 or x = 40

The distance between the two cities is either 200 km or 160 km, depending on the speed.
How can that be??

Case 1: x = 50
then the distance between the two cities is 4(50) or 200 km
and the speed during the first part was 50 km/h
check:
time for first trip = 200/50 = 4 hrs
time for return trip
= 100/50 + 100/40 = 4.5
so it took 1/2 hour longer
Answer is correct

Case 2: x = 40
then the distance between the two cities is 4(40) or 160 km
and the speed during the first part was 40 km/h
check:
time for first trip = 100/40 + 60/30
= /40 = 4 hours
time for return trip = 100/40 + 60/30
= 4.5
so it took 1/2 hour longer
This answer is also correct

69, 420 pp long

Read the question.