The average speed of an express train is 20km/hour more than the average speed of a passenger train .To cover a distance of 360kms the express train takes 3 hours less than that of the passenger train. Find the average speed of both passenger and express train.
speed of passenger train ---- x km/h
speed of express ------ x+20 km/h
Time = distance/rate
360/x - 360/(x+20) = 3 , saying the difference in their times is 3 hours
multiply each term by x(x+20), the LCD
360(x+20) - 360x = 3x(x+20)
3x^2 + 60x - 7200 = 0
x^2 + 20x - 2400 = 0
(x - 40)(x+60) = 0
carry on
To find the average speeds of the passenger and express trains, we can start by setting up two equations based on the given information.
Let's assume the average speed of the passenger train is 'x' km/h.
According to the problem, the average speed of the express train is 20 km/h more than that of the passenger train, so the average speed of the express train is 'x + 20' km/h.
Now, let's focus on the information about the time it takes to cover a distance of 360 km:
1. Express Train: It takes 3 hours less than the passenger train, so the time taken by the express train is 't - 3' hours.
2. Passenger Train: It takes 't' hours for the passenger train to cover the same distance.
Now, we know that the average speed is equal to the distance divided by the time taken. So, we can set up the following equations:
1. Express Train: Average Speed = Distance / Time
(x + 20) km/h = 360 km / (t - 3) hours
2. Passenger Train: Average Speed = Distance / Time
x km/h = 360 km / t hours
Now, we can solve these equations to find the values of 'x' and 't'.
First, let's simplify the equations by cross-multiplication:
1. (x + 20) * (t - 3) = 360
2. x * t = 360
Expanding the first equation:
xt - 3x + 20t - 60 = 360
xt + 20t - 3x - 60 = 360
xt + 20t = 360 + 3x + 60
xt + 20t = 420 + 3x
Now, we can substitute the value of xt from the second equation into the expanded form of the first equation:
360 + 20t = 420 + 3x
20t = 420 + 3x - 360
20t = 60 + 3x
Divide the equation by 20:
t = (60 + 3x) / 20
Now, substitute the value of t in the second equation:
x * ((60 + 3x) / 20) = 360
60x + 3x^2 = 7200
Rearrange the equation:
3x^2 + 60x - 7200 = 0
Divide the equation by 3:
x^2 + 20x - 2400 = 0
Now, we can solve this quadratic equation to find the value of 'x' (the average speed of the passenger train).
Factoring or using the quadratic formula, we find:
(x - 40)(x + 60) = 0
So, x = 40 or x = -60
Since negative speed doesn't make sense in this context, we can disregard x = -60.
Therefore, the average speed of the passenger train (x) is 40 km/h.
To find the average speed of the express train, we can substitute this value back into one of the equations:
Average speed of the express train = Average speed of the passenger train + 20 km/h
= 40 km/h + 20 km/h
= 60 km/h
Hence, the average speed of the passenger train is 40 km/h, and the average speed of the express train is 60 km/h.