In MBA Exams Quantitative Ability section, Time, Speed and Distance & Time and Work are crucial topics. For the same, BYJU'S Exam Prep has brought together vital study notes for the aspirants preparing MBA Entrance exams.

**Time, Speed & Distance and Time & Work**

**Contents**

1.1 Basic Concepts

1.2 Unique Classification of Question types

1.2.1 Questions based on Proportionality

1.2.2 Meeting Point Questions

1.2.3 Relative Speed

1.3 Advanced Concepts

1.3.1 Constant Product Rule

1.3.2 Inverse Proportionality of Speed and Time (when Distance is constant)

**Time & Work**

**Contents**

2.1 Introduction

2.2 Unique Classification of Question Types

2.2.1) When efficiencies are equal

2.2.2) Efficiencies are equal within a group but more than one group exists

2.2.3) Inverse proportionality of efficiency and Time taken

2.2.4) Negative work

2.2.5) Questions where different people work partially i.e. leave or join job in between

2.2.6) Questions where different people work alternately (instead of working simultaneously)

**Time, Speed & Distance**

1.1 Basic Concepts

If you observe the Speedometer in a car, the numbers there indicate the Speed with which the car is travelling, at that particular instant of Time. Therefore, Speed can be defined as the amount of Distance covered in a given Time period.

The formula for Speed is:

From this we can deduce that

- Higher the Speed, the lesser the Time taken, when the Distance is constant i.e. Speed is inversely proportional to Time and vice versa if Distance is constant.
- In a given Time, with a higher Speed, you will cover a longer Distance i.e. Speed is directly proportional to Distance and vice versa when Time is constant.
- With a constant Speed, to cover a longer Distance, you will take more Time. i.e. Time is directly proportional to Distance and vice versa when the Speed is constant

**ILLUSTRATION:**

**A car is moving at a Speed of 20kmph for the first three hours and then at a Speed of 60 kmph for the next three**

**hours. Find the ratio of the Distance covered.**

Soln: As the Time taken is constant, the Distance covered will be directly proportional to the Speed. Here Speed is in ratio - 20:60 or 1:3. So the Distance will also be in ratio 1:3.

**Units **

Each of Speed, Distance and Time can be expressed in different units as in

Time (seconds(s), minutes (min), hours (hr))

Distance (meters (m), kilometers (km), miles, feet)

Speed (m/s, km/hr)

So if Distance (km) and Time (hr), then as Speed = Distance/Time, the units of Speed will be km/ hr

**ILLUSTRATION: **

**2) A train travels a Distance of 20 km in the first 2 hours, 20 kms in the next 3 hours and 60 kms in the last 5 hours. What is the average Speed of the train for the entire journey?**

**ILLUSTRATION**

**Amit and Aman have to travel from Delhi to Jaipur in their respective cars. Amit is driving at 60 kmph while Aman is driving at 90 kmph. Find the Time taken by Aman to reach Jaipur if Amit takes 9 hrs.**

As the Distance covered is constant in both the cases, the Time taken will be inversely proportional to the Speed. In the problem, Speed of Amit and Aman is in ratio 60: 90 or 2:3. So the ratio of the Time taken by Amit to that taken by Aman will be in the ratio 3:2. So if Amit takes 9 hrs, Aman will take 6 hrs.

**Ram and Shyam are standing at two ends of a room with a width of 30 m. They start walking towards each other along the width of the room with a Speed of 2 m/s and 1 m/s respectively. Find the total Distance traveled by Ram when he meets Shyam for the third Time.**

When Ram meets Shyam for the third Time, they together would have covered a Distance of 5d, i.e 5x30m = 150 m.

Ratio of Speed of Ram and Shyam = 2:1, so the total Distance traveled by them will also be in the ratio 2:1 as the Time taken is constant.

### 1.2.3 RELATIVE SPEED

“Relative” means “in comparison to”. We use the concept of relative Speed when we have two or more bodies moving with some Speeds. To make things simpler, we make one body stationary (Speed =zero) and take the Speed of the other body with respect to the stationary body, which is the sum of the Speeds if the bodies are moving in the opposite direction and the difference if they are moving in the same direction. This Speed of the moving body with respect to the stationary body is called Relative Speed.

**Relative Speed of 2 bodies**

**=Sum of their individual Speeds if they are moving in the opposite direction**

**=Difference of their individual Speeds if they are moving in the same direction**

So consider A and B starting at the same Time, travelling in opposite direction with Speed of 20 kmph and 30 kmph and they have to travel a Distance of 200 km. Now we can use the concept of relative Speed if we want to find the Time of their meeting.

Two bodies moving towards each other are A and B with Speeds 20 kmph and 30 kmph. Now make body A stationary and take the Speed of B in reference to A, so relative Speed of B = sum of their respective Speeds = 20 + 30 = 50 kmph.( as they are traveling in opposite directions)

**ILLUSTRATIONS:**

**The thief Bhagu Ram is spotted by the police man Pakad Singh from a Distance of 200m.Once they see each other they start running. What is the Distance that Bhagu Ram, running at 5 kmph would have covered before being caught by Pakad Singh, running at 7 kmph?****A bird is sitting on a train A moving at a Speed of 40 kmph. It sees another train B at a Distance of 200 ms with Speed of 60 kmph coming from the opposite direction on the same rail track. It flies with an average Speed of 10kmph and sits on another train. Again, it flies back to the first train immediately and then to the second train and so on. It does so before the two trains crash. What is the total Distance traveled by the bird?**

### SOME SPECIFIC CASES:

**CASE 1 :****Boats & Streams**

Let the Speed of boat in still water = x

Speed of stream = y

As we have previously discussed, in questions where two bodies with different Speeds are concerned, we will use the concept of relative Speed. Here we assume the stream to be the stationary body and we take the Speed of boat relative to the Speed of the stream.

Using the concept of relative Speed, we have the situation where the stream is stationary and only the boat is moving at a Speed determined by the direction of boat relative to the stream.

**Time & Work**

#### 2.1 Introduction

Work is defined as something which has an effect or outcome; often the one desired or expected;

The basic concept of Time and Work is similar to that across all Arithmetic topics, i.e. the concept of Proportionality.

#### Efficiency is inversely proportional to the Time taken when the amount of work done is constant.

We will use the concept of percentage of work done to solve most of the questions.

A few basic points one needs to know to use the percentage concept are:

When we say that someone has done a work à it means he has done 100 % of the work.

Hence, if A finishes a work in 4 days it means à In 4 days he will do 100% of the work

Hence, in one day he finishes 25 % (100/4) of the work.

Similarly in 3 days he finishes 75 % of the work.

Table of commonly used numbers:

Number of days | Percentage value |

2 | 50% |

3 | 33% |

4 | 25% |

5 | 20% |

6 | 16.67% |

7 | ≈14% |

8 | ≈12% |

9 | ≈11% |

We can also consider complete work as 1 unit. Then if A takes 4 days to finish a work, it means he can finish 1/4^{th} of the work in 1 day.

Let us look at some basic questions to illustrate this:

**ILLUSTRATION:**

**1) Rahim can finish a work in 10 days and Ram can finish the same work in 40 days. If Ram and Rahim both work together then what is the total number of days taken?**

We can solve this problem using two approaches.

**Approach 1: (using fractions)**

Ram can finish the work in 10 daysà in one day he will do 1/10^{th} of the work.

Rahim can finish the work in 40 days à in one day he will do 1/40^{th} of the work.

So in one day, both working together can finish = (1/10) + (1/40) = 5/40 = 1/8^{th} of the work. So to complete the work they will take 8 days.

**Approach 2: (using percentage: Recommended)**

Rahim can finish 100 % of work in 10 days à in one day he finishes 10% of the work. Ram can finish 100% of the work in 40 days à in one day he finishes 2.5 % of the work.

So working together, in a single day they can finish 12.5% of the work. So to complete 100% of the work, they will take 100/12.5 = 8 days.

**2) Ravi can do a job in 10 days. Raman can do the same job in 20 days. They together start doing the job but after 4 days Raman leaves. How many more days will be required by Ravi to complete this job alone?**

Ravi can finish a job in 10 days à in one day he can finish 10 % of the job.

Raman can finish the same job in 20 days à in one day he can finish 5 % of the job.

So working together, in a day they can do 10 + 5 = 15 % of the job.

### 2.2 UNIQUE CLASSIFICATION OF QUESTION TYPES

Time and work concepts and questions can be classified, for better understanding, as follows

- Efficiencies are equal
- Efficiencies are equal within a group but more than one group exists.
- Inverse proportionality of efficiency and Time taken
- Working against a work i.e. negative work.
- Questions where different people work partially i.e. leave or join in between
- Questions where different people work alternately (instead of working simultaneously)

### 2.2.1) When efficiencies are equal

*Approach 1: *

Using the concept of equal efficiencies:- When efficiencies are equal then the Time taken to do a certain work is always inversely proportional to the number of people employed.

*Approach 2:*

We can use the concept of Man- Days. Every work to be done has a quantity attached to it. That quantity is the product of number of men and the Time taken. So the quantity can be named as “MAN-HOURS” or “MAN – DAYS” or anything similar depending upon the question. For a work, this quantity will always be constant irrespective of any change in number of men or/and Time if we assume that the men are working with equal efficiency.

**ILLUSTRATION:**

**3) If 25 men can finish the work in 12 days, 30 men can finish the same work in how many days?**

*Approach 1.*

*Approach 2*

Given that 25 men can finish the work in 12 days.

So MAN-DAYS needed to complete the work = 25 * 12 = 300 MAN-DAYS.

*Even if we change the number of men working; for the completion of work we need 300 MAN-DAYS. When we have 30 men then number of days taken = *

### 2.2.5) Questions where different people work partially i.e. leave or join job in between

These are cases including

- A and B working on a project. B leaves after n days, how long does A take to complete the project?
- A and B working on a project. B leaves after n days, C joins in, so how long do A & C take to complete the project together?
- There are n taps in a tank. Initially tap 1 is open, after an hour, tap 2 is open and so on. Find the Time taken to fill the tank

The fastest way to solve these types of questions is to use the “percentage” concept.

Let us look at a few examples solved in this way

### 2.2.6) Questions where different people work alternately (instead of working simultaneously)

These are cases including

- A works for the first 2 days, B works for the next 2 days and C works till the finish. Find the Time taken
- Pipe A is open for 3 hours, Pipe B for 2 hours and Pipe C for 1.5 hours. Find the number of hours in which the cistern is filled

### TEST

Time, Speed & Distance and Time & Work 01

#### No. of Questions -15

#### Time- 30 minutes

1) Four people whose speeds are 1 m/s, 2 m/s, 4 m/s and 8 m/s start from the north, East, West and Southern most points of a circle simultaneously in the clockwise direction. If the fastest person takes T minutes to complete one single round, find when all of them would meet at a single point?

(a) After T minutes

(b) After 8T minutes

(c) They would never meet at the same point

(d) After 2T minutes

(e) After 4T minutes

2) Bhavana and Amita are cycling around a circular track of circumference 300 m. they start from the same point and cycle in opposite directions. Amita cycles at a speed of 22.5 m/min and she meets Bhavana every 7.5 minutes. At what frequency will they meet each other if they were cycling in the same direction?

a) 30 min b) 45 min c) 60 min d) 90 min e) 70 min

3) A man driving his bike at 24 kmph reaches his office 5 minutes late. Had he driven 25% faster on an average he would have reached 4 minutes earlier than the scheduled time. How far is his office?

a) 24 km b) 72 km c) 18 km d) 36 e) Data Insufficient

4) Efficiency of A is 33.33% more than that of B. B takes 24 days complete a work. How many days A will take to complete the work?

a) 20 b) 25 c) 18 d) 16 e) 14

5) 5 men or 8 women can reap a field in 12 days. Find the number of days (approximately) taken by 3 men and 4 women to reap the same field.

a) 10 b) 11 c) 12 d) 13 e) 14

6) One day Prakash started late for office by 1 hour, so he increased his normal speed by 5 km/hr so that he reaches on time. Find the normal time taken to reach his office if his office is at a distance of 60 km from his house.

a) 3 hrs b) 5 hrs c) 6 hrs d) 4 hrs e) 2 hrs

7) If Sajesh increases his speed from 12 km/hr to 15 km/hr while coming from Office to home, he reaches home one hour early. Determine the distance between his home and the office.

a) 40 kms b) 50 kms c) 60 kms d) 70 kms e) 80 kms

**Questions 8-9**

A starts from home for his office. He travels downhill, then on flat ground and then uphill to reach his office. It takes him 3 hrs to reach the office. On the way back home A takes 3 hrs 10 min to reach home along the same route. The speed downhill is 60 km/hr, on flat ground is 48 km/hr and uphill is 40 km/hr.

8) What is the distance between A’s home and his office?

a) 144 km b) 148 km c) 154 km d) 100 km e) Data insufficient

9) By what distance should his office be shifted so that the time taken to go to the office is same as time taken to reach home from the office?

a) 20 km b) 30 km c) 40 km d) 15 km e) Data insufficient.

10) If one pipe A can fill the tank in 10 hours then 4 pipes each of efficiency 20% as that of Pipe A can filled the same tank in how many hours?

a) 15 hrs b) 12 hrs c) 10 hrs d) 12.5 hrs e) 8 hrs

11) A and B are moving in a circular track in the same direction. They start simultaneously in a race which requires them to cover 15 rounds. Whenever A & B meet, it was found that the ratio of the number of rounds covered by them till then is 3:1. The time taken by B to complete the race if they meet every 5 minutes is?

a) 75 mins b) 100 min c) 150 min d) 50 min e) none of these

12) Three women and four men can complete a work in 4 days. Two men and five women can complete the same work in 5 days. Find out the money received by a man for his work if a woman is paid Rs. 60 for her work.

a) Rs. 120 b) Rs. 150 c) Rs. 130 d) Rs. 75 e) can’t be determined

13) A train travels at an average speed of 90 km/hr without any stoppages. However, its average speed decreases to 60km/hr on account of stoppages. On an average, how many minutes per hour does the train stop?

a) 12 minutes b) 18 minutes c) 24 minutes d) 20 minutes e) 30 minutes

14) The height of a certain flag pole is 30 feet. Grease is applied to the pole. A monkey attempts to climb the pole. It climbs 3 feet every second but slips down 2 ft in the next second. When will the monkey reach the top of the flag pole?

a) 28 secs b) 27 secs c) 60 secs d) 55 secs e) 54 secs

15) A starts towards Q from P at 9 am with a speed of 60 km/hr. B starts from Q to P a little later and travels at 80 km/hr. They meet when B has traveled 60 km. If both reach their destinations at the same time, then what is the distance between P and Q?

a) 120 km b) 210 km c) 140 km d) 150 e) none of these

#### Time, Speed & Distance and Time & Work 02

#### No. of Questions -15

Time - 30 mins

1) Amit takes 20 days to complete a certain work. Amit started the work and Suraj joined him 4 days before the work was completed. Find out the number of days for which Amit worked alone if Suraj’s efficiency is 25% more than that of Amit’s.

a) 10 days b) 8 days c) 12 days d) 11 days

2) The ratio of number of days taken by B is to C to complete a work is 2: 3. The ratio of efficiency of A is to C is 5:3. A takes 4 days less than C, when A and C complete the work individually. A, B and C started the work and B & C left after 2 days. The number of days taken by A to complete the remaining work is:

a) 1 day b) 2 days c) 3 days d) 5 days

3) Distance between Bombay and Pune is 210 km, A and B leave to Bombay from Pune simultaneously. A drives at 60 kmph and B drives at 80 kmph. B reaches Bombay and returns to Pune immediately and meets A on the way. find the distance from Bombay where they meet?

a) 50 km b) 40 km c) 20 km d) 30km

4) 20 men can complete a work in 20 days. After how many days should the number of men be increased by 50% so that the work is completed in 75% of the actual time?

a) 6 b) 4 c) 3 d) 5

5) 15 workers working 4 hours a day for 25 days can build a platform of width 120 meters, length 10 meter and height 14 meters. How many days will 12 workers working 5 hours a day will take if they have to build a platform of width 600 meters, length 14 metres and height 12 metres?

a) 150 b) 130 c) 125 d) 120

6) Vivek and Bharath go home daily after Office by Office Cab which has a speed of 40 kmph. Vivek takes 20% more time than Bharath to reach his home. If Bharath’s house is at a distance of 30 km from the office, then calculate the distance of Vivek’s house from the office.

a) 36 km b) 40 km c) 45 km d) 42 km

7) A,B,C and D participate in a cycling race on a circular track such that they are 100 m, 120 m, 110 m and 105 m away from the center of the circular track. Their speeds are 48 m/s, 40 m/s, 42 m/s and 44 m/s respectively. If their starting positions are collinear with the center of the track, which two players will have to change their positions so that both of them will complete two rounds of the circular track simultaneously?

a) A and C b) A and B c) C and D d) A and D e) either (b) or (c)

8) A tap can fill a tank in 12 hours, but because of a hole in the bottom of the tank, it fills the tank in 15 hours. Determine the time it will take to empty the tank if it is completely filled once and the tap is closed.

a) 30 hrs b) 60 hrs c) 20 hrs d) 25 hrs

9) Amit and Bala leave at 8 am everyday to meet each other at point X after 2 hours. On one day, Amit walks at 5/6^{th} of the usual speed while Bala starts one hour late. Bala thus increases his speed by 25%. Now Amit takes ½ hour more than usual to meet Bala and they meet ½ km away from the point X. Find out the speeds of Amit and Bala and the total distance travelled by them?

a) 5 kmph, 5 kmph, 20 km b) 6 kmph, 4 kmph, 22 km c) 6 kmph, 4 kmph, 20 km d) 4 kmph, 6 kmph, 20 km

10) A takes 4 days less than B and 2 days more than C to do a job. A and B together can do the job in the same time as C. Determine the ratio of number of days taken by A and B to complete the job individually.

a) 2: 3 b) (1 + √3): (3 + √3) c) 1: √2 d) (1 + √3): (2 + √6)

11) Peter, Sana and Gavin are visiting the Adams family who are staying 200km away. Each of their walking speeds is 10 kmph. Initially, Peter and Gavin travel in a car at the rate of 50kmph and Sana walks the distance. After a while, Gavin gets off the car as he feels nauseated and walks the rest of the distance to the house. Peter goes back in the car to fetch Sana and they all reach the house at the same time. What was the entire time involved in traveling?

a) 10 hours b) 7 hours c) 8 ½ hours d) 8 hours

12) At the construction site of a new mall, concrete was to be delivered from the factory in 8 hours. The material has to be delivered with 30 3 tonne trucks, which were used for 2 hours, and then nine 5 tonne trucks were added 2 hours later to complete the job in time. How many hours will it take for one three tonne truck alone and one five tonne truck alone respectively to complete the job?

a) 330 hr , 198 hr b)240 hr , 200 hr c) 210 hr, 320 hr d) 198 hr, 240 hr

13) Afsan is deciding which car to rent for a day for a class trip, between a Pinnova and a Bilto. The rate/km is in a ratio of 3:2, the seating capacity is in a ratio of 5:2. The speeds are in the ratio of 7:4. Find out the ratio of the maximum cost incurred that day for the two car types, given that there is no wastage of capacity or time?

a) 60:28 b) 56:30 c) 105:16 d) 140:12

14) Amit and Francis are on detention and they need to write 65 pages together, Amit writing for an hour extra than Francis. Francis can write 2 pages/hour more than Amit and therefore, he did 5 pages more than Amit. What is the speed per hour of Francis?

a) 6 b) 5 c) 7 d) 12

15) The time taken by 4 men to complete a job is double the time taken by 5 children to complete the same job. Each man is twice as fast as a woman. How long will 12 men, 10 children and 8 women take to complete a job, given that a child would finish the job in 20 days.

a) 2 days b) 2 ^{1}/_{8} days c) 4 days d) 1 day** **

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