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.
I always had my students do these questions by organizing the information in a chart.
let the time take to go uphill be t hours
then the time taken to go downhill is 4-t hours
.......... Dist.....rate..time
uphill: ....15t......15....t
downhill....30(4-t) ..30...4-t
(I hope this somehow lined up)
but the distance is the same, so
15t = 30(4-t)
15t = 120 - 30t
45t = 120
t = 8/3
distance = 15t = 15(8/3)
= 40 km
To find the distance between the villages, we need to consider the time it takes for the bus to travel uphill and downhill on the round trip.
Let's assume the distance between the villages is 'd' miles.
When the bus is traveling uphill, it takes 15 mph to cover the distance 'd', so the time taken is given by: time uphill = distance uphill / speed uphill
On the other hand, when the bus is traveling downhill, it takes 30 mph to cover the distance 'd', so the time taken is given by: time downhill = distance downhill / speed downhill
Since the round trip takes exactly four hours, we can write the equation: time uphill + time downhill = 4
Substituting the formulas for time uphill and time downhill into the equation, we get:
distance uphill / speed uphill + distance downhill / speed downhill = 4
Now, let's solve the equation:
d / 15 + d / 30 = 4
To simplify the equation, we can multiply all terms by the least common multiple of the denominators (30):
2d + d = 120
Combining like terms:
3d = 120
Dividing both sides by 3:
d = 40
Therefore, the distance between the two villages is 40 miles.