1) “By one more than previous (number)”

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• To express decimal equivalent of vulgar fraction when denominator is
ending in 9
e.g. 6/19 = 0.315789473684210526…
• To find the square of numbers ending in 5.
e.g. (85)2 = 7225.

2) “(Subtract) All from 9, last from 10”

• To multiply 2 numbers when one of them is nearer to 10 or powers of 10.
e.g. 4368-5632
X 9999-0001
43675632
3) “Vertically and crosswise.”

• To multiply 2 numbers.
e.g. 123
X 321
39483
• To divide one number by another number: e.g.38982/73
___ 534|0___

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73|383938|12 Ans: 534 and remainder 0

4) “Transpose and use”

• To divide one number by another which is nearer to 10 or powers of 10.
e.g. Divide 1234 by 112

1 1 2 | 1 2: 3 4
-1-2| -1:-2
| :-1-2
1 1: 0 2 Ans: 11 and remainder 02

5) “When there is similarity the entity is zero”

• (X+7)(X+9) = (X+3)(X+21)
Here 7x9 = 3x21 Therefore X = 0

6) “When one is in ratio the other is zero”

• To solve 2 simultaneous equations.
e.g. 6X + 7Y = 8
19X + 14Y = 16 Here 7:14 :: 8:16 Therefore X = 0 and Y = 8/7

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7) “By addition and subtraction”

• To solve 2 simultaneous equations.
e.g. 45X – 23Y = 113

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23X – 45Y = 91
By addition we get: 68X – 68Y = 204 i.e. X - Y = 3
By subtraction we get: 22X + 22Y = 22 i.e. X + Y = 1
Again by addition and subtraction we get: X = 2 & Y = - 1

8) “By completion and non-completion”

• To solve quadratic, cuboid type equations.
e.g. X3 - 6 X2 + 11X – 6 = 0
Complete the cube: X3 - 6 X2 + 12X – 8 – X + 2 = 0
(X – 2)3 = (X – 2) Therefore X – 2 = 0, -1 or 1 i.e. X = 2, 1 or 3.

9) “By differential calculus”

• To find roots of quadratic equation.
e.g. X2 – 7X + 12 = 0
Taking first differential equal to √ (b2 – 4ac) we get:
2X – 7 = +√ ((-7)2 – 4x1x12) i.e. X = 3 or 4.

10) “Whatever is less”

• To find the cube of numbers which are nearer to 10 or powers of 10.
e.g. (104)3 = 112 48 64. (Subtract -4 twice from 104 to get 112 Then multiply -4 and -12 to get 48 and then write the cube of 4 i.e. 64.

11) “Individual and whole”

• To solve algebraic equation.
e.g. (X+7)4+(X+5)4=706 Expressing it as sum of two forth powers we get:
(X+7)4+(X+5)4 =54+34 Therefore X+7=+5 and X+5=+3

12) “(Multiply) the remainders by last digit”

• 3, 2, 6, 4, 5, and 1 are the remainders when multiplied by 7 the last digits of the quotients will be: 1, 4, 2, 8, 5 and 7 respectively.
Therefore 1/7 = 0.142857

13“Last and twice the penultimate”

• To multiply by 12. e.g. 412 x 12 = 4944

• To solve the algebraic equations of the following form:
1/AB + 1/AC = 1/AD + 1/BC where A,B,C&D are in A.P. Then D+2C = 0.
e.g. 1/((X+2)(X+3)) + 1/((X+2)(X+4)) = 1/((X+2)(X+5)) + 1/((X+3)(X+4))
Here (X+5)+2(X+4) = 0 Therefore X = -13
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14) “By one less than previous”

• To multiply by series of 9 say 99, 999 etc. when the number of digits in the multiplicand is less than or equal the number of digits in multiplier. e.g. 8x9=72. 11x99=1089. 231x999=230769

15) ”Sum of co-efficients”

• X3+6 X2 +11X+6 = (X+1)(X+2)(X+3)
1+6+11+6 = (1+1)(1+2)(1+3)
24 = 2x3x4

16) “First differential is equal to sum of the factors”

• Consider: X2 + 3X + 2 = (X+1)(X+2)
Here first differential is 2X + 3 = sum of factors: X+1 and X+2.