Speed, Time and Distance concepts with solved examples including shortcut method

Speed, Time, and Distance:

1.Speed: The distance covered in a unit time interval is known as speed.

Speed = Distance travelled/ Time taken

Distance = Speed * Time

Time = Distance/Spedd

Unit conversion:

i. Km/hr to m/sec.

A km/hr =  A * (5/18) m/s

ii. m/sec to Km/hr.

A m/s = A * (18/5) Km/hr

how to remember:  To reduce Unit from Km/hr to m/s we divide it by 18 i.e 5/18.

To increase the unit from m/s to km/hr we multiply it by 18 i.e 18/5.

Ex.1 A train is moving at a speed of 180 km/hr. Its speed in meters per second?

Sol.1 Km/hr = 5/18 m/sec 

180 * (5/18) = 50 m/sec Ans.

Points to remember :

1. If the time taken is constant, then the distance traveled is directly proportional to the speed.

    S ∝ D

2.If the speed is constant, then the distance traveled is directly proportional to the time taken.

   D ∝ T

3.If the distance traveled is constant, the speed is inversely proportional to the time taken.

  S ∝ 1/T

Ex.2 In covering a certain distance the speed of A and B are in the ratio of 3:4/ A takes 30 minutes more than B to reach the destination. The time taken by A to reach the destination is?

Sol. Let the distance is D Km.

The speed of A is 3x km/hr

The speed of B is 4x km/hr

According to the question:

T1-T2 = 30min = 1/2

D/3x - D/4x = 0/2

D/12x = 1/2

D/x = 6

Time Taken by A = D/3x =6/3 = 2 hours.

Method 2.

Speed Ratio = 3:4

Time Ratio = 4:3

1 = 30 min

4 = 120 min = 2 hours

Ex.3 If a person walks at 14 km/hr instead of 10 km/hr. he would have walked 20 km more. The actual distance traveled by the person is?

Sol. Method 1 

Let the actual distance traveled be x km.

Then,

Speed = Distance/ Time

x/20 = (x+20)/14 

14x = 10x + 200

4x = 200

x = 50km

Method 2.

14-10 = 20/T

T =5 hrs

Distance = 10* 5 = 50 km Ans. 

Average Speed:

The average speed of an object is a measure of the distance covered by that object in a set period of time.

Average speed = Total Distance /Total time taken

a. If A covers a distance d₁ km at s₁ km/hr and then d₂ km at s₂ km/hr. Then the average speed during the whole journey is 

Average speed = s₁s₂(d₁+d₂)/(s₁d₁+s₂d₂)

b.A person goes a certain distance A to B at speed of s₁ and returns B to A at speed of s₂ km/hr. If he takes T hrs in all, then

Average speed = 2s₁s₂/(s₁+s₂)

Distance between A and B = t * [s₁s₂/(s₁+s₂)]

c.If a person traveled three equal Distance by three different Speed s₁,s₂,s₃ then

Average speed = 3s₁s₂s₃/(s₁s₂+s₂s₃+s₃s₁)

Ex4. A man goes from place A to B at a speed of 12 km/hr. and return at the speed of 18 km/hr. The average speed for the whole journey is.

Sol. Average speed = 2*12*18/(12+18)

2*12*18/30 =  72/5 km/hr

Ex5. One-third of a certain journey is covered at the rate of 25 km/hr/One-fourth at the rate of 30 km/hr. and the rest at 50 km/hr. The average speed for the whole journey.

Sol. Let the total journey = x km/hr

so according to the question

Total time = x/3*25 + x/4*30 + 5x/12*50

Average speed = x/(3x/100) = 100/3km/hr

Previous year questions of Time and Distance :

Ex.1 A student goes to school at a rate of 2.5 km/hr and reaches 6 minutes late. If he travels at a speed of 3 km/hr. he is 10 minutes early. What is the distance to the school?

Sol. Let the distance of the school be x km.

Diffrence of time = 6+10 = 16/60 hours = 4/15

Time = Distance/Speed

x/(5/2) - x/3 = 4/15

2x/75-x/3 = 4/15

x= 4 km

Method 2.

 Required distance = (product of two speed/ difference of two speed) * Difference between the arrival time

[{(5/2)*3}/(3 - 5/2)] * (6+10)/60  = 4 km/hr 

Ex. 2 The distance between 2 places R and S is 42 km. Anita starts from R with a uniform speed of 4 km/hr towards S and At some speed, Romita starts from S towards R. Also with some uniform speed they meet each other after 6 hours. The speed of Romita is?

Sol. Let the speed of Romita is x km/hr

Distance = speed * Time

The distance traveled by Romita and Anita together is 42 km.

4*6 + x*6 = 42

6x = 42-24 

x = 3 km/hr Ans

Ex. 3.A is twice as fast as B and B is thrice as fast as C is. The journey covered by C in 1.5 hours will be covered by A in?

Sol. Let the speed of B, C, and A is x km/hr, 2x km/hr, x/3 km/hr.

Speed of A/Speed of B = 2x/(x/3) = 6

Required time = 1/6 * 3/2 hrs, = 1/4 hour = 15 minutes

Method 2

The ratio of speed A: B: C = 2: 1: 1/3

Multiply ratio by 3

Speed ratio A:B:C = 6:3:1

Time ratio A:B:C = 1:3:6

C covered journey in 1.5 hours means,

6 = 1.5 = 3/2 

1 = 3/2 * 1/6 = 1/4 = 15 minutes

Ex.4 A boy goes to his school from his house at a speed of 3 km/hr and returns at a speed of 2 km/hr if he takes 5 hours in going and coming. The distance between his house and school?

Sol. Let the distance is x km

then 

x/3 + x/2 = 5

(2x+3x)/6 = 5

x= 6 km

Method 2.

Distance =  total time taken * (product of two speed)/ addition of two speed

5 + (3*2)/(3+2) = 6 km

Ex.5 A man traveled a distance of 50 km in 7 hrs partly on foot at the rate of 8 km/hr. and partly on bicycle at 16 km/hr. The distance traveled on the foot?

Sol. Journey on foot = x km

Journey on cycle = 80 - x km

x/8 + (80-x)/16 = 7

On solving the above equation we get.

x= 32 km

Ex.6 Walking 6/7 of his usual speed a man is 12 minutes late the usual time taken by him to cover that distance is?

Sol. Time and speed are inversely proportional.

Therefore,

Usual time * 7/6 - usual times = 12 minutes

Usual time * 1/6 = 12 minutes

Usual time = 72 minutes = 1 hour 12 minutes

Method 2.

Usual time = late time / {1/(6/7) -1} 

= 12/{(7/6) - 1} = 12/(1/6) = 72 minutes = 1 hour 12 minutes 

Ex.7 Running 4th of his usual speed, a person improves his timing by 10 min. Fin the usual time to cover the distance?

Sol. Usual time = improved time /  {1- 1/(4/3)}

10/(1-3/4) = 40 minutes Ans.

Ex. 8 A car covers four successive 7 km distances at speeds 10 km/hr,  20 km/hr, 30 km/hr, and 60 km/hr respectively. Its average speed over this distance is.

Sol. Total distance = 7*4 = 28 km

Total time = 7/10 +7/20 + 7/30 + 7/60

(42+21+17+7)/60 = 84/60 = 7/5

Average speed = total distance / total time = 28/(7/5) = 20 km/hr Ans.

For question-based on relative speed and trains click here.