Fractions

Classification of Fractions

1. Proper Fraction :

   A Proper Fraction is one whose numerator is less than its denominator. eg: 1/4, 5/8

2. Improper Fraction :

   An Improper Fraction is one whose numerator is equal to or greater than its denominator. eg: 6/4, 5/5

3. Mixed Fraction :

   A Mixed fraction is a quantity consisting of two parts, one a whole number and other a proper fraction. eg: 3¹/8, 8³/5 A Mixed Fraction can always be expressed as an improper fraction.

Simplification

 

VBODMAS

'VBODMAS' (Vinculum - Bracket - Of - Division - Multiplication - Addition - Subtraction) rule should be applied for solving problems involving one or more mathematical operations like multiplication, division, addition, subtraction etc.such problems are solved i n the order of vinculum, bracket, of, division, multiplication, addition, subtraction.

Simplification Using Identities

1. a x (b x c) = a x b + a x c
2. (a + b)² = a² + 2ab + b²
3. (a - b)² = a² - 2ab + b²
4. (a + b)² = (a - b)² + 4ab
5. (a - b)² = (a + b)² + 4ab 
6. (a - b) (a + b) = a² - b² 
7. (a + b)³ = a³ + 3ab (a + b) + b³ 
8. (a - b)³ = a³ - 3ab (a - b) - b³ 
9. a³ + b³ = (a + b) (a² - ab + b²)
10. a³ - b³ = (a - b) (a² + ab + b²)
11. a³ - b³ / a² + ab + b² = a - b 
12. a³ + b³ / a² - ab + b² = a + b 
13. a² - ab + b² / a³ + b³ = 1 / a + b 
14. a² + ab + b² / a³ - b³ = 1 / a - b
15. (a + b)² + (a - b)² = 2 (a² + b²)
16. (a + b)² - (a - b)² = 4ab

Basic Arithmetic Operations

 

Addition

1. Positive no + Positive no = Positive no , Eg: 6 + 5 = 11
2. Negative no + Negative no = Negative no , Eg: -4 + -7 = -11
3. Negative no = Positive no = Difference of the nos and sign of greater no , Eg: +8 + -10 = -2

Multiplication

1. Positive no x Positive no = Positive no , 6 x 3 =18
2. Negative no x Negative no  = Positive no , -2 x -6 = 12
3. Positive no  x Negative no = Negative no , 5 x -6 = -30
4. Negative no  x Positive no = Negative no , -8 x 2 = -16

Division

1. Positive no ÷ Positive no = Positive no , 18 ÷ 3 = 6
2. Negative no ÷ Negative no  = Positive no , -12 ÷ -6 = 2
3. Positive no  ÷ Negative no = Negative no , 30 ÷ -6 = -5
4. Negative no  ÷ Positive no = Negative no , -16 ÷ 2 = -8