MetroEye02:Work-Time Formula

General Formula for Work-Time relationship

The inverse of time it takes working together equals the sum of the inverse of the time it takes each individually.

If A takes a hours and B takes b hours to finish a task, then A and B together takes t hours which can be calculated from the following formula:

\(\dfrac{1}{t} = \dfrac{1}{a} + \dfrac{1}{b}\)

MetroEye01:Rate of Work Problem

General Formula for work-person-rate problem

\(\dfrac{W_1}{M_1\cdot H_1}\) = \(\dfrac{W_2}{M_2\cdot H_2}\)

where W is the work to be done
M is number of persons required
H is the time required.

Progression03: Sum of Odd and Even Numbers

Sum of first n odd numbers

S = \( n^2\)

Number of terms(n), if nth term of odd-number series is \( t_n\) 

n =  \(\dfrac{t_n+1}{2}\)

Sum of first n even numbers

S = n(n+1)

Number of terms(n), if nth term of even-number series is \( t_n\) 

n =  \(\dfrac{t_n}{2}\)

Progression02:Sum of first n natural numbers

Sum of first n natural numbers 1,2,3,4,....... n 

S = \(\dfrac{n(n+1)}{2}\)

nth term/number of terms if sum of n terms is given

n = \(\dfrac{\sqrt{1+8S} -1}{2}\)

Progression01: Sum and nth term of a series

Any series a, a+d, a+2d, a+3d, ..........

where a --> First term
          d --> difference between the terms
          n --> number of terms

nth term of the series 
First term -> a
Second term -> a+1d
Third term -> a+2d
.
.
.
nth term -> a+(n-1)d

n'th term of the series = a + (n-1)d

Sum of the first n terms

S = a+a+d+a+2d+a+3d+.........+a+(n-1)d
   = na+d+2d+3d+.....(n-1)d
   = na+d(1+2+3+......+(n-1))
   = na+d((n-1)(n-1+1)/2)
   = n(a+(n-1)d/2)

Sum of the first n terms = \(\dfrac{n}{2}\left(2a+\left(n-1\right)d\right)\)

Number of terms in the series

Number of terms = \(\dfrac{\left(LastTerm-FirstTerm\right)}{d}+1\)

Fraction01: Sum of Simple Fractions

For all simple unit fractions, the following is valid