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### Theory:

In the previous topics, we have learnt how to the different types of variations and the methods of solving them.

In this session, we shall learn the new concept of time and work.
Time and work is a problem which deals with a person or a group of people and the time taken by them to complete a piece of work.
Usually, total work is taken as $$1 \ unit$$.

The solution for the time and work problem can be determined using the unitary method.

Unitary method:

Let us remember the formulas to solve the time and work problem.

1. If $$A$$ can complete a piece of work in $$x$$ days, then $$A$$'s one day work is $$\frac{1}{x}$$.

2. If $$A$$ and $$B$$ can do the same work in $$x$$ and $$y$$ days, respectively then their one day's work together is $$\frac{1}{x} + \frac{1}{y}$$ $$= \frac{x+y}{xy}$$

3. Also, $$A$$ and $$B$$ together can complete the work in $$\frac{xy}{x+y}$$ days.
Example:
If Reeta can do a work in $$10$$ days and Sheenu can do the same work in $$15$$ days, together how long will it take them to complete the same work?

Solution:

Given:

Reeta can do the work in $$10$$ days.

Sheenu can do the work in $$15$$ days.
We know the formula that $$A$$ and $$B$$ together can complete the work in $$\frac{xy}{x+y}$$ days.
Here, $$x = 10$$ and $$y = 15$$

Substituting the values in the above formula, we have:

The time taken by Reeta and Sheenu to complete the work together $$= \frac{10 \times 15}{10 + 15}$$

$$= \frac{150}{25}$$

$$= 6$$ days

Therefore, the time taken by Reeta and Sheenu to complete the work together is $$6$$ days.
Also, remember these formulas.
1. Work done per day $$= 1/$$Time taken to do complete work

2. Work done $$=$$ Number of days worked $$\times$$ Work done per day

3. Remaining work $$= 1 -$$ Work done

4. Time taken to do remaining work $$=$$ Remaining work $$\times$$ Time taken to do full work