PROBLEM ON SPEED & TIME –

Example.1) Richard traveled 320 km by train and 180 km by car, taking 7 hours. But if he traveled by 160 km by train and 240 km by car, he take 1 hour less. Find the speed of the train and that car.

Ans.) Let the speed of the train and that of the car be ‘x’ km per hour and ‘y’ km per hour respectively. Then –

If, train travel x km in 1 hour

1

Then train takes time to travel 1 km in -----

x

320

So, now the train will take the time to travel 320 km = -------

x

If, car travel y km in 1 hour

1

Then car takes time to travel 1 km in ------

y

180

So, now the car will take the time to travel 180 km = -------

Y

As per the given condition,

320          180

-------- + --------  = 7 …………….....(i)

x             y

If, train travel x km in 1 hour

1

Then train takes time to travel 1 km in ------

x

160

So, now the train will take the time to travel 160 km = -------

x

If, car travel y km in 1 hour

1

Then car takes time to travel 1 km in ------

y

240

So, now the car will take the time to travel 240 km = --------

Y

As per the given condition,

160           240

--------- + -------- = 6 …………….....(ii)

x              y

putting,

1                  1

------ = u, and ------ = v, the given questions become, so we will find –

x                  y

320u + 180v = 7 ………………..(iii)

160u + 240v = 6 ………………..(iv)

Multiplying (iv) by 2 and, we get –

320u + 480v = 12  ………………(v)

Now, we will subtract (iii) from (v), and we will get –

320u + 480v = 12

320u + 180v =  7

--------------------

300v = 5

Or,                v = 1/60   [now we will replace v = 1/y]

Or,             1/y = 1/60

Or,                y = 60

Now, we will put the value of y in (i), and we will get –

320           180

--------- + -------- = 7

x              y

320           180

=>  --------- + --------- = 7

x             60

320

=>  --------- +  3   =  7

x

320

=>  -------- = 7 – 3 = 4

x

320

=> -------- =   x

4

=>   x  =  80

Hence, speed of the train is 80 km/hour, and speed of the car is = 60 km/hour            (Ans.)

Example.2) A train covered a certain distance at a uniform speed. If the train had been 6 km/h faster, it would have taken 4 hours less than the schedule time. And if the train had been slower by 6 km/h, it would have taken 6 hours more than the schedule time. Find the length of the journey.

Ans.) Let the original speed of the train be x km/h and time taken by y hours. Then, distance = (xy) km

Case.1) When speed = (x + 6) km/h and time taken = (y – 4) hours

So, Distance = (x + 6) (y – 4) km

So,   xy =  (x + 6) (y – 4) km

=>    xy = xy + 6y – 4x – 24

=>    4x – 6y =  - 24

=>    2x – 3y = - 12  ………………………(i)

Case.2) When speed = (x – 6) km/h and time taken = (y + 6) hours

So, distance = (x – 6) (y + 6) km

So,  xy = (x – 6)(y + 6)

=>   xy = xy + 6x – 6y – 36

=>   6 (x – y)  =  36

=>    x – y  =  6 ………………………..(ii)

Now, we will multiply (ii) by 2, and we get –

2x – 2y = 12 ……………………….(iii)

Now, we will subtract (i) from (iii), and we get –

2x – 2y = 12

2x – 3y = - 12

------------------

y = 24

Now, we will substitute the value of y in (i), and we get –

x – y  =  6

=>       x – 24 =  6

=>        x  =   30

So, the original speed of the train 30 km/h and time taken by 24 hours.

Then, distance = (xy) km = 30 X 24 = 720 km

So, the obtained distance is 720 km      (Ans.)