The relationships between distance, speed and time :

**Time = Distance / Speed**

**Speed = Distance / Time**

**Distance = Time ⋅ Speed**

Let us look at some examples to understand how to solve word problems involving speed.

**Example 1 :**

A passenger train takes 3 hours less than a slow train for journey of 600 km. If the speed of the slow train is 10 km/hr less than that of the passenger train, find the speed of two trains.

**Solution :**

Here speed of slow train is compared with speed of passenger train.

Let x be the speed of the of the passenger train.

Speed of the slow train is 10 km/hr less than that of the passenger train.

So, x - 10 be the speed of the slow train.

Distance has to be covered = 600 km

Time = Distance/speed

Let t_{1} and t_{2 }be the time taken by passenger train and slow train respectively.

The differences of time taken by both trains are 3 hours

t_{1} = 600 / x

t_{2} = 600 / (x - 10)

t_{1} - t_{2 }= 3 hours

[600 / (x – 10)] – [600 / x] = 3

600[ (1/(x - 10) - (1/x)] = 3

200 [ x - (x - 10) / x(x - 10) ] = 1

200 [10/x^{2} – 10x] = 1

2000 = x^{2} – 10x

x^{2} – 10x - 2000 = 0

(x + 40) (x – 50) = 0

x + 40 = 0 x = -40 |
x - 50 = 0 x = 50 |

Therefore the speed of passenger train = 50 km/hr

Speed of slow train = 40 m/hr

**Example 2 :**

Mr John traveled by scooter from his house to a town that was some distance away. If he traveled at a constant speed of 50 km/hr, he would have arrived at 11 am. If he traveled at 75 km/hr, he would have arrived at 10 am. At what speed should be riding he wanted to arrive at 10.30 am ?

**Solution :**

If he travels the distance at the speed of 50 km/hr, he can reach the spot at 11 am.

If he travels the distance at the speed of 75 km/hr, then he will reach the spot at 10 am. The difference between two timings is 1 hour.

Bu we do not know at what time he started his travelling.

Time = Distance / Speed

Time = Distance / 50 -----(1)

Time = Distance / 75 -----(2)

(1) - (2)

1 = (D / 50) - (D / 75)

1 = (3D - 2D)/150

D = 150

150 km is the required distance. By applying the value of D in (1), we get

Time = 150/50

Time = 3 hours

He is taking 3 hours to cover the distance 150 km. So, he started his travelling at 8 am. But he should cover the distance with in 2 1/2 hours.

2 1/2 = 150 / Speed

5/2 = 150 / Speed

Speed = 150 (2/5)

Speed = 60 km/hr

The required speed is 60 km/hr.

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