Tips and Tricks to solve the Question based on Time Speed and Distance

Time Speed and distance

Speed

The rate at which a body or an object travels to cover a certain distance is called speed of that body.

Speed is the distance covered by an object in unit time.

Time

The duration in hours, minutes or seconds spent to cover a certain distance is called the time.

Distance

The length of the path travelled by any object or a person between two places is known as distance

Time Speed Distance Triangle

Time (SI unit - seconds) and distance (SI unit - meters) are fundamental quantities of measurement. For a distance ‘d’ covered in a time duration ‘t', the average speed ‘s’ or simply the speed is defined as the rate of covering the given distance or distance covered per unit time. Therefore, the formula for Time Speed Distance is:  

Speed = Distance / Time

This means,

Distance = Speed x Time 

Time = Distance / Speed

Note: SI unit of speed is meters per second or m/sec. The other most commonly used unit of speed is kilometer per hour (kmph or km/h).

To convert a speed given in m/s into a speed in km/h, multiply with 18/5.

To convert a speed given in km/h into a speed in m/s, multiply with 5/18.

A simple way to remember the multiplication factor is to recall that particular speed, when expressed in km/h, is numerically larger than the same speed expressed in m/s. For other units like m/min or km/min, it is sufficient to remember that 1 km = 1000 m and 1 min = 60 sec.

 Relation between Time Speed Distance

From the above, the following can be concluded about the relations between Time Speed Distance

1. If two bodies move at the same speed, the distances covered by them are (directly) proportionality to the times of travel. i.e. when s is constant,
2. If two bodies move for the same time period, the distances covered by them are (directly) proportional to the speeds of travel. i.e. when t is constant,
3. If two bodies move for the same distance, their times of travel are inversely proportional to the speeds of travel. i.e. when d is constant,

Time Speed Distance Formula

Here are some important time speed distance formula you need to know to solve problems quickly.

a) Average Speed

When a body travels different distances at different speeds, the average speed is the amount of time taken to travel the total distance in total time, Therefore, the formula for Time Speed Distance is:

When the body travels at speeds of u and v units for equal distances, i.e. the distance segments are in the ratio 1:1, then

b) Relative Speed

Consider two bodies moving at speeds u and v.

When they are moving in the same direction,

the relative speed between the two bodies is the difference of their speeds, i.e.u – v

if u > v or v – u if v > u.

When they are moving in the opposite directions,

the relative speed between the two bodies is the sum of their speeds, i.e. u + v

Types of problems in Time Speed Distance

Type I: Unit Conversion Time And Distance Problems (They Are The Most Basic Type)

This type is very easy to solve. You will get speed in one unit (e.g kilometer per hour or km/h) and you have to convert into another unit (e.g metres per second or m/s).

Question 1: A bus moves at a speed of 81 km/h. What is the speed of the bus in metres per second?

Solution:
In question, you can see that the speed is given in km/h.
You can write, X = 81 km/h
Therefore, speed in m/s (according to our formula) = X x 5/18
= (81 x 5/18)
= 22.5 m/s

Type II: Average Speed When Travelling To A Place And Returning

In this type of questions, a person will move from one place to another at certain speed. Then he will return to the starting place at a different speed. You may be asked to find average speed, distance, etc.

To solve such problems, you have to remember the formula given below.

Let a person move from one place to another at speed X and return to the starting place at a different speed Y.
Then, average speed for the whole journey = 2XY/(X+Y)

Question 2: Ram walked from his home to the bank at the rate of 20 km/h and returned back at the rate of 5 km/h. If he took 4 hours and 30 minutes for the whole journey, find the distance of the bank from his home.

Solution:
Part 1: Finding Average Speed
From the question, you know the following:
Speed of Ram from home to bank = X = 20 km/h
Speed of Ram from bank to home = Y = 5 km/h
Also, you know that the average speed for the whole journey = 2XY/X+Y
= (2 x 20 x 5) / (20+5)
= 200 / 25 = 8 kmph.

Part 2: Finding Distance
From the question, you know that the time taken for the whole journey = 4 hrs and 30 min
4 hrs and 30 min can be written as 4 ½ hours or 9/2 hours
In part 1 of the solution, you have found that the average speed = 8 kmph
You know the familiar formula that Speed = Distance / Time
Therefore, total distance travelled by Ram = Average Speed X Time Taken
= 8 x 9/2 = 36 Km

Note: But, this distance is the total distance travelled by Ram from his home to bank plus distance he travelled from bank to home. Therefore, to calculate the distance from home to bank, you have to divide the above value by 2.
Therefore, distance from home to bank = 36/2 = 18 Km

Type III: Problems Based On Changing Time And Changing Speed

This type is based on a simple fact that if a person increases his speed he will reach his destination faster and if he decreases his speed he will reach his destination slower.

Though this type looks tough at first, you can solve this type easily if you carefully read and understand the below example.

Question 3: If a cyclist rides at a speed of 4 km/h, he reaches the office by 5 minutes late. However, if he rides at a speed of 5 km/h, he reaches 4 minutes earlier. Find the distance covered by him to reach office?

Solution:
Assume that the distance travelled by cyclist (from home to office) to be D km.

Case I: Cyclist’s speed is 4 km/h
Time taken to cover D km at 4 km/h = Distance covered by cyclist / Speed of cyclist= D/4 hours
Note: Above equation is based on the simple formula: Speed = Distance / Time

Case II: Cyclist’s speed is 5 km/h
Time taken to cover D km at 5 km/h = Distance covered by cyclist / Speed of cyclist = D/5 hours.
Time difference between case I and II
You know that when cyclist travels at 4 km/h, he reaches 5 minutes LATE but when he travels at 5 kmph, he reaches 4 minutes EARLIER.
Therefore, difference in time taken between case I and II = (5 minutes + 4 minutes)
= 9 minutes or 3/20 hours

But we know that the time taken in case I is D/4 and that in case II is D/5. Therefore, above equation becomes,
D/4 – D/5 = 3/20
(5D-4D) / 20 = 3/20
D = 3 km

Type IV: Time And Speed Problems On Trains

This type is very popular in all government exams. In this type, starting time, speed, etc., of trains will be given. You will asked to find the time of their crossing.

Here is an example question.

Question 4: A and B are 2 stations 390 km apart. A train start from A at 10am and travels towards B at 65kmph. Another train start from B at 11am and travels towards A at 35kmph. At what time do they meet ? 

Solution:Till 11 am train A covers distance of 65 km. 
So remaining distance (390−65)=325km.
Relative speed =100km/hr 
So to cover 100km in 1 hour
Therefore to cover 325km in x hr x =325/100 
=3.25 hrs
Hence the train meet at  11 am + 3.25 hrs  =2.25 pm.

Type V: Problems On Bus With Stoppages

Though the bus you are travels appears to travel fast, it will reach its destination very slow if it stops at many places. Type V deals with such cases. Below example will help you to understand clearly.

Question 5: A bus without any stoppage can travel a certain distance at an average of 70km/h and with stoppages covers the same distance at an average speed of 50 km/h. How many minutes per hour does the bus stop?

Solution:
Assume that the total distance travelled by the bus to be D km.
Case I: Without stoppages
Without stoppages, the average speed of the bus is 70 km/h.
Time taken by bus without stoppages = Distance covered by the bus / Speed of the bus = D/70 hours …equation 1

Case II: With stoppages
With stoppages, the average speed of the bus is 50 km/h.
Time taken by the bus with stoppages = Distance covered by the bus / Speed of the bus = D/50 hours …equation 2

Calculation of Stopping Time
If we subtract the values of time taken by bus with and without stoppages, we can find the total stopping time. In other words, total stopping time equals the difference in values of equations 1 and 2
Therefore, we can write
Time taken by bus for stoppages (D/50 – D/70) hours = 7D – 5D / 350
= 2D/350
= D/175 hours. …. equation 3

Now, have a look at case II. In case II, bus with stoppages travels at 50 km/h to cover the distance D.
Therefore, journey time of bus with stoppages = Distance / Speed = D/50 hours
Also we found that the total stoppage time = D/175 hours (see equation 3)

In D/50 hours of journey, the bus stops for D/175 hours. The total stoppage time for 1 hour of journey can be calculated using the direct proportion table method as given below.

D/50 = D/175X
Or, X = 50/175 = 2/7 hours
= 2/7 x 60 minutes = 17.14 minutes

Alternate Shortcut Method:
You know that the speed of the bus is reduced by 20 km/h due to stoppages
In other words, the time that the bus takes without stopping to drive 20 km/h will be equal to stoppage time per hour.
Time taken to cover 20 km = (20 km / Actual speed of bus without stoppage )
= 20/70 hours = 2/7 hours
= 20/70 x 60 minutes
= 17.14 minutes