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Basic Important Formulas

These are the basic formulae:

(a + b)² = a² + b² + 2ab

(a - b)² = a² + b² - 2ab

(a + b)² - (a - b)² = 4ab

(a + b)² + (a - b)² = 2(a² + b²)

(a² - b²) = (a + b)(a - b)

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

(a³ + b³) = (a + b)(a² - ab + b²)

(a³ - b³) = (a - b)(a² + ab + b²)

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

If a + b + c = 0 then a³ + b³ + c³ = 3abc

Terminating Decimals: when the decimal part is terminating , it is known as terminating decimals. For example:1/2, 1/4, 10/8.

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Recurring Decimals: The decimal fractions, in which one or more decimal digits are repeated again and again, are called recurring decimal fractions. E.g., 2/3 = 0.6666 = 0.6̄

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Pure and Mixed Recurring Decimal: In a recurring decimal, if all the figures are repeated after the decimal point, then it is called a pure recurring decimal and if some figures are not repeated, while some of them are repeated, then it is called a mixed recurring decimal.

For example:0.44444.....(pure recurring),
0.255555.....(Mixed recurring)

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Convert Pure Recurring Decimal into p/q form
Put as many 9's in the denominator as the number of digits in the period of decimal number, in numerator put number as it is or subtract digit (before decimal) from the number.
e.g.
(i) 0.999 = 9/9 = 1
(ii) 0.727272 = 72/99
(iii) 5.717171 = 566/99.

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Convert Mixed Recurring Decimal into p/q form
In numerator, put the number and then subtract it from the formed by the non - recurring digits.

In denominator, put as many 9's as there are digits in the period of the recurring decimal followed by as many zeros as there is number of digits which are not recurring after the decimal point.
e.g.
(i) 0.2454545 = 243/990
(ii) 0.8727272 = 864/990
(iii) 9.072222 = 8165/900

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In the denominator place as many nine as there are repeating digits and after nine put as many zeroes as the number of non-repeating digits.

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Proper fraction: In the case of proper fraction Numerator is Less than denominator. for example 2/3, 5/43, 9/32

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Improper fraction: In the case of improper fraction Numerator is greater than denominator. for example 12/3, 55/43, 19/32

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Mixed fraction: mixed fraction is the combination of whole number and fraction, basically it is improper type fraction.