Answer: Let's assume the rate of the slower train is x km/h.
Then, the rate of the faster train would be x + 20 km/h, as it is traveling 20 km/h faster.
When the two trains meet, their combined distance traveled is equal to the total distance between the towns, which is 1136 kilometers.
Since distance = rate × time, the time taken for both trains to meet is t hours.
For the slower train, the distance traveled is x km/h × t hours = xt kilometers.
For the faster train, the distance traveled is (x + 20) km/h × t hours = (x + 20)t kilometers.
As they meet, their combined distance is 1136 kilometers, so we can write the equation:
xt + (x + 20)t = 1136
Now, let's solve for t:
xt + xt + 20t = 1136
2xt + 20t = 1136
t(2x + 20) = 1136
t = 1136 / (2x + 20)
Now that we have the value of t, we can find the rate of the slower train (x) by substituting the value of t back into either of the original equations. Let's use the equation for the slower train:
xt = 1136
Substitute the value of t:
x * (1136 / (2x + 20)) = 1136
Now, solve for x:
1136x = 1136(2x + 20)
1136x = 2272x + 22720
2272x - 1136x = 22720
1136x = 22720
x = 22720 / 1136
x = 20
So, the rate of the slower train is 20 km/h.
Now, to find the rate of the faster train, we can use the fact that it is 20 km/h faster than the slower train:
Rate of faster train = Rate of slower train + 20
Rate of faster train = 20 + 20
Rate of faster train = 40 km/h
Therefore, the rate of the slower train is 20 km/h, and the rate of the faster train is 40 km/h.