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# Formulas For Work And Time

## Formulas for work and time

**Work: When the certain amount of job is assigned to an individual known as Work**

**Time: Time is defined as the required duration to complete the task. **

## Work and Time Formula's

**Definition**

**Work:**In terms of mathematics work is defined as the amount of job assigned or the amount of job actually done.**Time:**Time is defined as the number of days or hours required to complete the task

## Formula's for Work and Time

**Work from Days: **

If A can do a piece of work in n days, then A’s one day work = \frac{1}{n}

**Days from work:**

If A’s one day work = \frac{1}{n}, then A can finish the work in n days

**Work Done by A and B**

A and B can do a piece of work in ‘a’ days and ‘b’ days respectively.

When working together they will take \frac{ab}{a+b} days to finish the work

In one day, they will finish ( \frac{a+b}{ab})^{th} part of work.

**Ratio:**

If A is thrice as good a workman as B, then:

Ratio of work done by A and B = 3: 1.

Ratio of times taken by A and B to finish a work = 1: 3

**Efficiency:**

Efficiency is inversely proportional to the

Time taken when the amount of work done is constant.

Efficiency α = \frac{1}{Time Taken}

## Basic Rules for Work and Time

**Rule 1: **If A completes a piece of work in x days. And B can completes same piece of work in y days .

Then,

One day work of A = \frac{1}{x} One day work of B = \frac{1}{y}

Work done by A + B = \frac{1}{x} + \frac{1}{y} = \frac{x+y}{xy}

Total time = \frac{xy}{x + y}

**Rule 2: **If A completes a piece of work in x days. B** **completes same piece of work in y days .C completes same piece of work in z days

Then,

One day work of A = \frac{1}{x}

One day work of B = \frac{1}{y}

One day work of C = \frac{1}{z}

Work done by A + B + C = \frac{1}{x} + \frac{1}{y} + \frac{1}{z} = \frac{yz+xz+xy}{xyz}

Total time = \frac{xyz}{xy + yz + zx}.

**Rule 3****: **If M_{1} men can complete a work W_{1} in D_{1} days and M_{2} men can complete a work W_{2} in D_{2} days then**, ** \frac{M_{1}D_{1}}{W_{1}} = \frac{M_{2}D_{2}}{W_{2}} _{ . }

If Time required by Both M_{1} and M_{2} is T_{1} and T_{2} respectively, then relation is \frac{M_{1}D_{1}T_{1}}{W_{1}} = \frac{M_{2}D_{2}T_{2}}{W_{2}}

**Rule 4: ** If A alone can complete a certain work in ‘x’ days and A and B together can do the same amount of work in ‘y’ days,

Work done by b =1/y – 1/x = x-y/xy

Then B alone can do the same work in xy/(x-y)

**Rule 5: **If A and B can do work in ‘x’ days.

If B and C can do work in ‘y’ days.

If C and A can do work in ‘z’ days.

Work done by A,B and C = ½ *(1/x+1/y+1/z)

Total time taken when A, B, and C work together xyz/(xy + yz + zx)

**Rule 6: **Work of one day = \frac{Total work}{Total number of working days}

Total work = one day work × Total number of working days

Remaining work = 1 – work done

Work done by A = A’s one day work × Total number of working days of A

**Rule 7:**If A can finish \frac{m}{n} part of the work in D days.

Then total time taken to finish the work by A = \frac{D}{\frac{m}{n}} = \frac{n}{m} × D days

**Rule 8:**

If A can do a work in ‘x’ days

B can do the same work in ‘y’ days

When they started working together, B left the work ‘m’ days before completion then total time taken to complete the work = (y+m)x/x+y

**Rule 9: **A and B finish work in a days.

They work together for ‘b days and then A or B left the work.

B or A finished the rest of the work in ‘d’ days.

Total time taken by A or B alone to complete the work = \frac{ad}{a – b} or \frac{bd}{a-b}

**Work and Time Formulas & Three Universal Rules**

- If M
_{1}persons can do W_{1}work in D_{1}days and M_{2}persons can do W_{2}works in D_{2 }days then the formula \frac{M_{1}D_{1}}{W_{1}} = \frac{M_{2}D_{2}}{W_{2}} .It can be written as M_{1}D_{1}W_{2 }= M_{2}D_{2}W_{1}. - If the persons work T1 and T2 hours per day respectively \frac{M_{1}D_{1}T_{1}}{W_{1}} = \frac{M_{2}D_{2}T_{2}}{W_{2}}
- It can be written as, M
_{1}D_{1}T_{1 }W_{2}= M_{2}D_{2}T_{2}W_{1} - If the persons has efficiency of E
_{1}and E_{2}respectively then \frac{M_{1}D_{1}T_{1}E_{1}}{W_{1}} = \frac{M_{2}D_{2}T_{2}E_{2}}{W_{2}} Therefore, M_{1}D_{1}T_{1}E_{1}W_{2}= M_{2}D_{2}T_{2}E_{2}W_{1}.

In all the above formula,

M = Number of workers

D = Number of days

T = Time required

W = Units of work

E = Efficiency of workers

**Read Also** – **Tips & tricks to solve work and time questions **

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