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This Chapter is as much useful as other chapters in the **section of quantitative aptitude** where conceptual clarity of relationship between working efficiency and time is very important to understand.

The problems on **time ****and ****work **are based on the calculation of time required by a given number of Work-Force (which may include men, women and machine) to complete the given or assigned job. As we know that the work is directly related with time. However, sometimes we are required to find the work done in a **given ****time ****period**.

__Thus, it can be said that the problems on Time and Work fall in two categories:__

- To find the time required to complete the assigned job.
- To find the work done in given time period.

This chapter plays a crucial role in determining the selection of an aspirant to succeed in MBA entrance exam. Every year 3 - 4 problems are generally asked in **CAT **or many other entrance exams like **FMS**, **XAT**, **MAT**, **IIFT**, etc. Infact a normal question paper seems to be incomplete without having the problems based on work and time.

Basically, concept of **capacity **or **efficiency **is used in this chapter. Suppose a man can complete a work in 4 days then we can say that each day he does one fourth of the work or 25% work each day. Thus it’s clear that his capacity is 25% per day. The time can be calculated in days, hours, months etc.

**For ****instance**, if a man can do an amount of work in 8 days, then 1/8^{th} of work is done in one day. Similarly, if 1/8^{th} work is done in a day, the complete work can be completed in 8 days.

If a man can complete a work in 'n' days then his one day's work = 1/n and this one day's work in terms of percentage is called **capacity**. Also, if a man can complete 1/n work in one day, then he can complete the whole work in n days. Here, we assume that total work is one (unit), the number of days required to complete the given work would be equivalent to the reciprocal of the day's work.

A man can complete his work in n days, then his one day's work = 1/n, his percentage capacity = (1/n) x 100.

Basically for smarter and faster calculation you have to be very quick and smart with your percentage calculations.

__These type of problems can be solved through two methods:__

1. **Unitary ****method**

2. **Percentage ****capacity**

Unitary method is obsolete due to its difficult calculations.

Points to Note:

Moremenlessdays and converselymoredayslessmen.Moremenmorework and converselymoreworkmoremen.Moredaysmorework and converselymoreworkmoredays.

**➢ TYPE - 1****(a)** If M_{1} men can do W_{1} work in D_{1} days and M_{2} men can do W_{2} work in D_{2} days, then

**(b)** If we include working hours T_{1} and T_{2} for two groups, then

**(c)** If M_{1} men of E_{1} efficiency can do W_{1} work in D_{1} days with T_{1} working hours/day and M_{2} men of E_{2} efficiency can do W_{2} work in D_{2} days with T_{2} working hours/day, then

**Example 1. A contractor adopts a work to complete it in 100 days. He employed 100 workers. But after 50 days it was found that only 1/3rd of the work has completed, How many more worker he should employ to complete the work in time.****(A) **100**(B) **125

(C)

(D)

Question is 100 men in 50 days can do 1/3rd of the work. How many more men will be employed so that they can complete 2/3rd of the work in 50 days.

So the Question becomes

Hence, x = 100

**Example 2. ****If 25 men can do a piece of work in 36 days working 10 hours a day, then how many men are required to complete the work working 6 hours a day in 20 days?****(A) **65**(B)** 75

(C)

(D)

Ans. (B)

M

25 x 36 x 10 = M

⇒ M

By percentage change graphic, when time is decreased by 2/3 ( i.e., 66.66%), number of men is increased by 2 times (i.e., 200%)

Question 1:A single reservoir supplies the petrol to the whole city, while the reservoir is fed by a single pipeline filling the reservoir with the stream of uniform volume. When the reservoir is full and if 40,000 litres of petrol is used daily, the supply fails in 90 days. If 32,000 litres of petrol is used daily, it fails in 60 days. How much petrol can be used daily without the supply ever failing?

Question 2:A contractor employed a certain number of workers to finish construction of a road in a certain scheduled time. Sometime later, when a part of work had been completed, he realised that the work would get delayed by three-fourth of the scheduled time, so he at once doubled the number of workers and thus he managed to finish the road on the scheduled time. How much work had been completed, before increasing the number of workers?

If A can do a piece of work in x days and B can do it in y days then A and B working together will do the same work in

xy/(x+y)days.

**Example. ****A and B can do a piece of work in 12 days, B and C in 15 days C and A in 20**** ****days. How long would each take separately to do the same work?****(A) **A = 20 days, B = 30 days, C = 50 days **(B) **A = 30 days, B = 20 days, C = 60 days **(C) **A = 20 days, B = 20 days, C = 50 days **(D)** A = 30 days, B = 40 days, C = 30 days **Ans. (B) ****Solution.**

A+ B can do in 12 days.

B + C can do in 15 days.

A+ C can do in 20 days. **By the theorem: ****We see that 2(A+ B + C) can do the work in **

∴ A+ B + C can do the work in 5 x 2 =10 days. (less men more days)

Now,

• A can do the work in

= [As, A =(A+ B + C) - (B + C)]

• B can do the work in

=[As, B = (A+ B + C) - (A+ C)]

• C can do the work in

=

Question 3:Railneer is packaged in a water bottling plant, with the help of two machines M_{1} and M_{2}. M_{1} and M_{2} produce 400 and 600 bottles per minute respectively. One day's production can be processed by M_{1} operating alone for 9 hours, by M_{2} operating alone for 6 hours or by both M_{1} and M_{2} operating simultaneously for 3 hours and 36 minutes. If one day's production in processed by M_{1} operating alone for 1/3 of the time and M_{1} and M_{2} simultaneously operating for 2/3 of the time, then in how many hours total production of one day will be completed?

**➢ TYPE - 3**

If A, B and C do a work in x, y and z days respectively then all three working together can finish the work in

**Example. A can complete a work in 10 days B in 20 days and C in 25 days. If they work together in how many days they can complete the work. ****(A) ****(B) ****(C) ****(D) ****Ans. (A)****Solution.**

Question 4:A and B can do a piece of work in 10 days, B and C can do in 15 days and C and A in 20 days. How long A,B and C together will take to do the work.

**➢ ** **TYPE - 4**

If A and B together can do a work in x days and A alone can do it in y days then B alone can do it in

xy/(y-x)days.

**Example. ****The rate at which tap M fills a tank is 60% more than that of tap N. If both the taps are opened simultaneously, they take 50 hours to fill the tank. The time taken by N alone to fill the tank is (in hours)****(A) **90 hours.**(B)** 110 hours. **(C)** 130 hours. **(D) **150 hours. **Ans.** (C)**Solution.**

If the rate at which Tap N fills the tank is 10 units per hour, the rate of Tap M would be 16 units per hour.

Hence,

⇒ the capacity of the tank would be 26 x 50 =1300.

⇒ Time taken by Tap N alone would be 1300 /10 =130 hours

Question 5:25 men and 15 women can complete a piece of work in 12 days. All of them start working together and after working for 8 days the women stopped working. 25 men completed the remaining work in 6 days. How many days will it take for completing the entire job if only 15 women are put on job.

**➢ TYPE-5**

Efficiency is inversely proportional to the number of days (D) taken to complete the work.

Means,

**E ∝ 1/D or E = K/D**,

∴ ED = constant

Similarly, **E _{1}D_{1 = }E_{2}D_{2} **

These product methods are limited to the constant work, if the amount of work gets changed, then it does not work, then we have to take help from the **unitary ****method**.

When more than one man work on a particular work, the rate of work is calculated as the strength of workers working in a particular time. So, the amount of work is defined in terms of men days or man-hours or man-days-hours.

**Example. ****A takes 16 days to finish a job alone, while B takes 8 days to finish the same job, What is the ratio of their efficiency and who is less efficient.****(A) **1 : 2**(B) **1 : 3**(C)** 2 : 3**(D)** 3 : 4**Ans.** (A)**Solution.**

Since A takes more time than B to finish the same job hence A is less efficient or efficiency of A = 100/16 = 6.25 % and efficiency of B = 100/8 = 12.5 % ratio of efficiency of

Hence, B is twice efficient as A.

Question 6:A is twice as good a workman as B and is therefore able to finish a piece of work in 30 days less than B. In how many days they can complete the whole work; working together?

NOTE:

∴ Efficiency × time = constant work

Whole work is always considered as 1, in terms of fraction and 100%, in terms of percentage.

⇒In general, the number of days or hours = (100 / Efficiency)

Question 7:A can complete a work in 10 days, B in 12 days and C in 15 days. All of them began the work together, but A had to leave the work after 2 days of the start and B 3 days before the completion of the work. How long did the work last?

Question 8:Arun and Satyam can complete a work individually in 12 days and 15 days respectively. Arun does work only on Monday, Wednesday and Friday while Satyam does the work on Tuesday, Thursday and Saturday. Sunday is always off. But Arun and Satyam both work with half of their efficiencies on Friday and Saturday respectively. If Arun started the work on 1st January which falls on Monday followed by Satyam on the next day and so on (i.e., they work collectively on alternate days), then on which day work will be completed?

Question 9:Boston, Churchill and David are three workers, employed by a contractor. They completed the whole work in 10 days. Initially all of them worked together, but the last 60% of the work was completed by only Churchill and David together. Boston worked with Churchill and David only for initial two days then he left the work due to his poor health. Also Churchill takes 20% less time to finish the work alone then that of David working alone. If they were paid ₹ 3000 for the entire work, then what is the share of least efficient person?

Question 10:In the ancient city of Portheus, the emperor has installed an overhead tank that is filled by two pumps X and Y. X can fill the tank in 12 hours while Y can fill the tank in 15 hours. There is a pipe Z which can empty the tank in 10 hours. Both the pumps are opened simultaneously. The supervisor of the tank, before going out on a work, asks his assistant to open Z when the tank is exactly 40% filled so that the tank is exactly filled up by the time he is back. If he starts X and Y at exactly 11:00 AM and he comes back at A : B. Then find the value of A + B.

Question 11:Three water pipes, A,B and C are all used to fill a container. These pipes can fill the container individually in 6 minutes, 12 minutes and 18 minutes, respectively. All the three pipes were opened simultaneously. However, it was observed that pipes A and B were supplying water at 2/3rd of their normal rate. Pipe C supplied water at half of its normal rate for first 3 minutes, after which it supplied water at its normal rate. What fraction of the tank is empty after 2 minutes?

Question 12:A contract is to be completed in 72 days and 104 men are set to work, each working 8 hours a days. After 30 days, only 1/5th of the work is finished. How many additional men need to be employed so that the work may be completed on time. (If each man is now working 9 hours per day)?

Question 13:Two pipes can fill a cistern in 14 and 16 hours respectively. The pipes are opened simultaneously and it is found that due to leakage in the bottom of the cistern, it takes 32 minutes extra for the cistern to be filled up. When the cistern is full, in what time will the leak empty it?

Question 14:A and B can do a piece of work in 10 days, B and C can do in 15 days and C and A in 20 days. How long A,B and C together will take to do the work.

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