Time and work deals with the relation between time and work in mathematical terms.

For example, if I write 500 words a day, then how many days would it take for me to write 1500 words or how many words will I write in 4 days?

**Rule 1-**

**If someone can complete a task in X units of time, then the work done by that person in one unit of time is 1/X.**

Continuing the above example, if I can write an essay of 5000 words in 10 days, then the work done by me each day would be 5000/10 = 500 words.

**Rule 2-**

**Similarly, if someone does 1/X work in one unit of time, then the units of time required to complete a task would be X.**

Thus, if I write 500 words a day, I need 5 days to write a 2500 word newspaper article.

**It is important to note that work is always taken as a whole or as a one.**

**Illustration 1**

If A and B are doing a work together and A takes 6 hours to complete the work and B takes 10 hours to complete the work, then how many hours will it take for them to complete the work together?

Apply Rule 1 to calculate the work done by them together in one hour.

Work done by A in an hour = 1/6 work

Work done by B in an hour = 1/10 work

**(Work done by A in an hour + Work done by B in an hour = Work done by A and B in an hour)**

Work done by A and B in an hour = 1/10 +1/6 = 6+10/60 = 16/60 = 4/15 work

Then apply Rule 2 to take out the total time required to complete the work.

Work done by A and B in an hour = 4/15 work

(Here 1/X = 4/15, thus, X = 15/4)

Total time required to complete the work = 15/4 = 3.75 hours or 3 hours 45 minutes

**Illustration 2**

If A is twice as good as B and both of them complete a work in 15 days then how many days would it take

- i) For A to complete the work alone and ii) For B to complete the work alone?

First take out the ratio in which A and B complete the work.

As A is twice as fast as B then the ratio of the work done by them in one unit of time is 2:1.

Thus, A:B = 2:1.

Now we know that they both completed the work in 15 days.

Thus, the work done by them in one day = 1/15 work.

Now we know the ratio in which they do their work as well as the work done by them in one day.

Thus, work done by A in one day = 2/3 X 1/15 = 2/45 work and work done by B in one day = 1/3 X 1/15 = 1/45.

Applying Rule 2:

Total time needed by A = 45/2 = 22.5 days and total time needed by B is 45 days.

**Illustration 3**

**Negative work**

A alone can build a wall in 12 days. B alone can build it in 36 days. C can break the same wall in 18 days. How much time will it take to build the wall?

Wall build in a day = Wall build by A + Wall build by B – Wall demolished by C

Wall build in a day = 1/12 +1/36 – 1/18 = 3+1-2/36 = 2/36 = 1/18

Thus, number of days required to build the wall = 18 days

**Using percentages to solve the same question**

Wall to be build = 100%

Percentage of wall build by A in a day = 100%/12 = 8.33%

Percentage of wall build by B in a day = 100%/36 = 2.77%

Percentage of wall demolished by C in a day = 100%/18 = 5.55%

Wall build in a day = 8.33% + 2.77% – 5.55% = 5.55%

Number of days required to build the wall = 100/5.55% = 18 days

**Illustration 4**

A factory needed to complete a work in 60 days. It employed 30 workmen. However, after 30 days only 25% of the work was completed. How many more workmen are needed to complete the work on time?

30 workmen – 30 days – ¼ work

30 workmen – 1 day – 1/(4X30) work

1 workman – 1 day – 1/(4X30X30) work

Days left – 30 days

1 workman – 30 days – 30/(4X30X30) work

1 workman – 30 days – 1/120 work

Work left = ¾ work

Work done by 1 workman in 30 days = 1/120 work

Keeping the days constant, work done by 120 workmen in 45 days – 1 work

Work done by 120X3/4 workmen in 45 days – 3/4work

3/4th work will be completed in 45 days by 90 workmen.

Thus, the factory needs to employ 60 more workmen to complete the work on time.

**Faster way to solve this problem**

30 workmen – 30 days – ¼ work

Work left = ¾ work

Days remaining = 30 days

To complete 1/4th work in 30 days – 30 workmen

To complete 3/4th work in 30 days – 30 X 3 = 90 workmen

Workmen already employed – 30

Thus, the factory needs to employ 60 more workmen to complete the work on time.