A black bus left Town A for Town B at the same time when a white bus left Town B to Town A. The average speed of the black bus and the white bus were 56 km/h and 72 km/h respectively. The two buses passed each other at a point 24 km from the midway of the two towns. How far apart are these two towns?
If the towns are a distance 2x km apart, then
(x-24)/56 = (x+24)/72
solve for x and then get 2x
To find the distance between the two towns, we need to first determine the time it takes for the buses to meet.
Let's assume the distance between the two towns is 'x' kilometers. Since the midpoint of the two towns is halfway between them, it would be located at a distance of x/2 from each town.
Now, let's calculate the time it takes for the black bus to meet the white bus. The black bus travels at an average speed of 56 km/h, so in the time it takes to meet, it would have covered 56t kilometers, where 't' is the time in hours.
Similarly, the white bus travels at an average speed of 72 km/h, so in the same time 't', it would have covered 72t kilometers.
Together, the black and white buses cover a total distance of 56t + 72t = 128t km. However, this distance is equal to the total distance between the two towns minus the 24 km distance from the midway point.
So, we have the equation: 128t = x - 24.
Now, let's consider the time it takes for the black bus to reach the midway point. Since the midway point is x/2 km away, the time it takes for the black bus to reach it would be (x/2) / 56, as the average speed is given in km/h.
Similarly, the time it takes for the white bus to reach the midway point would be (x/2) / 72.
Since the two buses meet at the point 24 km from the midway point, we have the equation: (x/2) / 56 + (x/2) / 72 = t.
Now, we can solve these two equations to find the value of 'x' (the distance between the two towns):
128t = x - 24
(x/2) / 56 + (x/2) / 72 = t
Solve these equations simultaneously, and you will find the value of 'x', which represents the distance between the two towns.
Let's call the distance between Town A and Town B as "d".
Since the two buses passed each other 24 km from the midway of the two towns, we can determine the distance traveled by each bus based on their average speeds and the time it took for them to meet.
Let's assume that the time taken for the buses to meet is "t" hours.
The distance traveled by the black bus can be calculated using the formula: distance = speed × time.
So, the distance traveled by the black bus is given by 56t km.
Similarly, the distance traveled by the white bus is given by 72t km.
Since they met 24 km from the midway point, the sum of their distances should be equal to the total distance between the two towns.
Thus, we can set up the equation: 56t + 72t = d.
Combining like terms, we get: 128t = d.
Now, let's find the expression for "t" in terms of "d".
Dividing both sides of the equation by 128, we get: t = d/128.
Since the distance traveled by the black bus is equal to 56 times "t", we can substitute the value of "t" from above into the expression: distance = 56 × t.
Substituting, we get: distance = 56 × (d/128).
Simplifying further, we have: distance = 7d/16.
Similarly, the distance traveled by the white bus is equal to 72 times "t". Substituting the value of "t" from above into the expression: distance = 72 × t.
Substituting, we get: distance = 72 × (d/128).
Simplifying further, we have: distance = 9d/16.
Since the buses meet 24 km from the midway point, the sum of their distances should be 2 times this distance or 48 km, given by the equation: 7d/16 + 9d/16 = 48.
Combining like terms, we get: 16d/16 = 48.
Simplifying further, we have: d = 48 × 16.
Therefore, the distance between the two towns, Town A and Town B, is 768 km.