An express train takes 1 hour less than a passenger train

Q. An express train takes 1 hour less than a passenger train to travel 132 km between Mysore and Bangalore (without taking into consideration the time they stop at intermediate stations). If the average speed of the express train is 11 km/h more than that of the passenger train, find the average speed of the two trains.

Ans: Let consider speed of the Express train be x km/h and speed of the passenger train be y km/h.

We know that the time ‘T’ taken to cover a distance ‘D’ by an object moving at a speed ‘S’ is given by:

T = $\frac{D}{S}$

Hence, Time taken by Express train to cover distance 132 km at a speed of x km/h will be:

$T_E = \frac{132}{x}$

Similarly, Time taken by Passenger train to cover distance 132 km at a speed of y km/h will be:

$T_P = \frac{132}{y}$

By given 1st condition: express train takes 1 hour less than a passenger train

$\therefore T_E = T_P - 1$

$\therefore \frac{132}{x} = \frac{132}{y} - 1$

$\therefore \frac{132}{x}= \frac{132 - y}{y}$

$\therefore 132 y = (132 - y) (x)$

$\therefore 132 y = 132 x - x y$ …………… (i)

By given 2nd condition, we have: average speed of the express train is 11 km/h more than that of the passenger train

$\therefore S_E = S_P + 11$

$\therefore x = y + 11$ …….. (ii)

By substituting value of x from equation (ii) in equation (i), we get:

$132 y = 132 x - x y$

$\therefore 132 y = 132 (y + 11) - (y + 11) y$

$\therefore 132 y = 132 y + 1452 - y^2 - 11 y$

$\therefore y^2 + 11 y - 1452 = 0$

$\therefore y^2 + 44 y - 33 y - 1452 = 0$

$\therefore y (y + 44) - 33( y + 44) = 0$

$\therefore (y + 44) ( y - 33) = 0$

Therefore, y = – 44 and y = 33

Here, we reject y = – 44 because speed can not be negative,

and we accept y = 33

By substituting the value of y in equation (ii), we get:

x = y + 11

x = 33 + 11 = 44

Hence, speed of the Express train is 44 km/h and Passenger train is 33 km/h.

Check:

If speed of express train is 44 km/h, time taken to cover 132 kms distance, it will take $\frac{132}{44} = 3$ hrs.

If speed of passenger train is 33 km/h, time taken to cover 132 kms distance, it will take $\frac{132}{33} = 4$ hrs.

Since time taken by express train is 4 – 3 = 1 hour less than time taken by passenger train, given 1st condition is met

Next, the difference in their speeds: 44 – 33 = 11 kmh,

Since our 2nd condition is also met. hence our answer is correct.

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