# Formulas For Work And Time

## Work and Time Formulas

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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. ## Introduction to Work and Time

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 3If M1 men can complete a work W1 in D1 days and M2 men can complete a work W2 in D2 days then, $\frac{M_{1}D_{1}}{W_{1}}$ = $\frac{M_{2}D_{2}}{W_{2}}$  .

If Time required by Both M1 and M2 is T1 and T2 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 =$\frac{1}{y} – \frac{1}{x} = \frac{x-y}{xy}$

Then B alone can do the same work in $\frac{xy}{(x-y)}$ days

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 = $\left ( \frac{1}{2}\right )\left ( \frac{1}{x}+\frac{1}{y}+\frac{1}{z} \right )$

Total time taken when A, B, and C work together $\frac{2xyz}{ ( xy+yz+zx )}$

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 M1 persons can do W1 work in D1 days and M2 persons can do W2 works in D2 days then the formula $\frac{M_{1}D_{1}}{W_{1}}$ =   $\frac{M_{2}D_{2}}{W_{2}}$ .It can be written as M1D1W2 = M2D2W1.
• 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, M1D1 T1 W2 = M2D2 T2W1
• If the persons has efficiency of E1 and E2 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, M1D1T1E1W2 = M2D2T2E2W1.

In all the above formula,

M = Number of workers

D = Number of days
T = Time required
W = Units of work
E = Efficiency of workers

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