two trains start from the same point simultaneously and in the same direction. the first train travels at 40 km/h, and the speed of the second train is 25 percent more than the speed of the first train. thirty minutes later, a third train starts form the same point and in the same direction. it overtakes the second train 90 minutes later than it overtook the first train. what is the speed of the third train
Answers
Let the speed of the third train be x.
The distances covered by the first two trains in 30 minutes is.
Train 1:
Speed = 40 km/h
Distance = 40 × 0.5 = 20 km
The time taken to overtake = 20/(x - 40)
Train 2 :
Speed = 1.25 × 40 = 50 km/h
Distance = 50 × 0.5 = 25 km
The time taken to overtake = 25/(x - 50)
The difference between the two times is : 90/60 = 1.5 hrs
Substituting this :
25/(x - 50) - 20/(x - 40) = 1.5
25(x - 40) - 20(x - 50) = 1.5(x - 40)(x - 50)
25x - 1000 - 20x + 1000 = 1.5(x² - 40x - 50x + 2000)
5x = 1.5(x² - 90x + 2000)
5x = 1.5x² - 135x + 3000
1.5x² - 140x + 3000 = 0
We solve by quadratic formula :
{130 +/- √(140² - 4 × 1.5 × 3000)} / (2 × 1.5)
= {140 +/-√(19600 - 18000)} / 3
= (140 +/- √1600) / 3
= (140 +/- 40) / 3
As from the solution :
x = 180/3 or 100/3
x = 60 or 33.3
We take 60 given that the third train is moving faster than the two trains.
Therefore the speed of the third train is :
60 km/h
The options for this question are missing:
A. 20 km/h
B. 40 km/h
C. 60 km/h
D. 50 km/h
E. 80 km/h
The answer for this question is Option C: 60 Km/h
Both the points will start from the point A and then the point B where the third train overtook the first train.
In order to make the first train over take the third train the distance which should be covered by the first train is:
1(1/2) h = distance covered by third train in 1 h.
We know that the third train left the station 30 minutes later.
This situation will lead to a distance which will be constant.
So from,
S*T = D
we get S α 1/T;
(t+1/2)/t = S/40
(2t+1/t) = S/40 --- (1)
the time is in the ratio 3:2 and if we consider the speed in ratios it will be 2:3 ratio.
So the final answer for the third train is 60 Km/h
Speed of 1st train =40km/hr
Speed of second train = 40*125/100 = 50km/hr
Distance covered by 1st train =40*1/2 = 20km/hr
Distance covered by 2nd train =50*1/2=25km/hr
Speed of the third train =x km/hr
40/20-x-50/25-x=90/60
x= 60km/hr