# Formulas For Work And Time

## Work and Time Formulas

### 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.

### Formulas for Work and Time

1.      Work from Days:
If A can do a piece of work in n days, then A’s one day work = 1/n
2.      Days from work:
If A’s one day work = 1/n, then A can finish the work in n days
3.      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 (ab/a+b) days to finish the work
In one day, they will finish (a+b/ab)th  part of work.
4.      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
5.      Efficiency:
Efficiency is inversely proportional to the Time taken when the amount of work done is constant.
Efficiency α 1/time taken

### Basic Rules & Formulas for Work and Time

·         Rule 1:

If A completes a piece of work in x days
B completes same piece of work in y days
Then,
One day work of A = 1/x
One day work of B = 1/y
Work done by A + B = 1/x + 1/y = x+y/xy
Total time = 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 = 1/x
One day work of B = 1/y
One day work of C = 1/z
Work done by A + B + C = 1/x + 1/y + 1/z= x+y+z/xyz
Total time = xy/xy + yz + zx

·         Rule 3:

If M1 men can complete a work W1 in D1 days and M2 men can complete a work W2 in D2 days then,
M1D1/W1 = M2D2/W2
If Time required by Both M1 and M2 is T1 and T2 respectively, then relation is
M1D1T1/W1 = M2D2T2/W2

·         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, 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.
Total time taken when A, B, and C work together

2xyz/xy + yz + zx

·         Rule 6:

Work of one day = 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 m/n part of the work in D days. Then total time taken to finish the work by
A = D/(m/n) = 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 = ad/a-b or bd/a-b

Work and Time Formulas & Three Universal Rules

1.      If M1 persons can do W1 work in D1 days and M2 persons can do W2 works in D2 days then the formula

M1D1/W1 = M2D2/W2

It can be written as M1D1W2 = M2D2W1

2.      If the persons work T1 and T2 hours per day respectively

M1D1T1/W1 = M2D2 T2/W2

It can be written as, M1D1 T1W2 = M2D2 T2W1

3.      If the persons has efficiency of E1 and E2 respectively then

M1D1T1E1/ W1 = M2D2T2E2/ W2

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 