List of Important Formulas for Competitive Exams

Shortcut Formulas/Formulas for Competitive exams like CTET, SSC and others.

1.  (a+b)² = a² + b² + 2ab

2.  (a-b)²=a² + b² - 2ab 

3.   a² - b²= (a - b)(a + b)

4.  a³ - b³= (a-b)(a² + ab + b²)

5.  a³ + b³= (a+b)(a² - ab + b²)

6.  (a + b)² = (a - b)² +4ab

7.  (a - b)² = (a + b)² -4ab

8.  a³ + b³ + c³ - 3abc =(a + b + c)(a² + b² + c² - ab - bc - ca)

9.   If (a + b + c) = 0, then a³+b³+c³=3abc
  
10. (a+b+c)³ = (a + b + c)³= a³ + b³ + c³ + 3(a + b)(b + c)(c + a)

11. (a + b + c )²  = a² + b² + c² + 2ab + 2bc + 2ca

12. x² + x(a+b) + ab = (x + a)(x + b)


13. a²(b+c) + b²(c+a)+c²(a+b) + 3abc = (a+b+c)(ab + bc + ca)

14. a²(b-c) + b²(c-a)+c²(a-b) + 3abc = -(a - b)(b - c)(c - a)
                      
                                (Number System)

15. For list of shortcut formulas for time, speed and distance click here.

16. Sum of first n natural numbers is {(n)(n+1)/2 }

17. Sum of first n even natural numbers is {(n)(n+1)}

18. Sum of first n odd natural numbers is n²

19. Sum of first even number upto n is {(n/2)(n/2+1)}

20. Sum of cubes of first natural numbers is {(n)(n+1)/2 }² 

21. Sum of squares of first n natural numbers is {(n)(n + 1)(2n +1)/6}

22. a³-b³ = (a-b)³ + 3ab(a-b)

23.  a³+b³ = (a+b)³ - 3ab(a+b)

24. Average of n multiples of x is x(1+n)/2

25. Average of first n natural numbers even numbers is (n+1)

26. Average of first n natural odd number is n.
                       
                                  (Profit and Loss formulas)

27. Profit Percentage =  {(S.P. - C.P)/C.P.}x100

28. Loss Percentage  =  {(C.P. - S.P.)/C.P.}x100

29.To find the Cost Price when Selling Price (S.P.) and Percentage profit (x) is 
     given
              Cost Price =   {S.P/(100 + x)} x 100

30.To find the Selling Price when the cost price (C.P.) and percentage profit (x) is 

     given by
                   Selling Price = {(100 + x)(C.P.)/100}                                                               

31.To find the Cost Price when Selling Price (S.P.) and Percentage loss (x) is given

                  Cost Price =   {S.P/(100 - x)} x 100

32.To find the Selling Price when the cost price (C.P.) and percentage loss (x) is   

     given by

                  Selling Price = {(100 - x)(C.P.)/100}

33. If an article is sold at a profit of x% which is further sold at a profit of y% then the final Selling price is 

                  Final Selling Price = Initial Cost Price{(100+x)/100}{(100+y)/100)  

34. If two successive profits x and y are made on the same article then the net 
    profit is 
                  (x + y + xy/100)  

.
35. If a false weight is used instead of the true weight then, the profit 
      percentage  is 
                         {(True Weight - false weight)/False Weight} x 100
  
                              (Time and Work formulas)

36.  If A can do a piece of work in x days and B can do the work in y days, then working together they can complete the work in  (xy)/(x+y) days.

37.  If A can do a piece of work in x days and (A and B) can do the same work in y days, then B can complete the work in  (xy)/(x-y) days.