The road connecting two mountain villages goes only uphill or downhill. A bus always travels 15 mph uphill and 30 mph downhill. Find the distance between the villages if it takes exactly four hours for the bus to complete a round trip.

time(up) = distance/rate = d/15

time(down) = d/r = d/30
Add time up to time down to get
(d/15)+(d/30) = total time = 4.
Solve for d.
[Remember: time(up) + time(down) = 4.0 hr]

To find the distance between the villages, we need to consider the time it takes for the bus to travel uphill and downhill separately. Let's assume the distance between the villages is 'd' miles.

When the bus travels uphill, it covers the distance at a speed of 15 mph, so the time taken to travel uphill is d/15 hours.

When the bus travels downhill, it covers the same distance at a speed of 30 mph, so the time taken to travel downhill is d/30 hours.

According to the given information, the total time taken for the round trip is exactly 4 hours. So the equation we can form is:

d/15 + d/30 = 4

To solve this equation, let's find a common denominator:

2d/30 + d/30 = 4

Combining the fractions:

3d/30 = 4

Now, cross-multiply to solve for 'd':

3d = 4 * 30

3d = 120

Dividing both sides of the equation by 3:

d = 40

Therefore, the distance between the villages is 40 miles.