Fractions: A fraction is an expression that indicates the quotient of two quantities.
Type of Fractions
Common Fraction:A common fraction also called Vulgar fraction is a fraction in which both numerator and denominator are integers.
Decimal Fraction:A decimal fraction is a fraction in which denominator is an integer power of ten. The term decimals are commonly used to refer decimal fractions.
Conversion of a Decimal into Common Fraction
Put 1 in the denominator under the decimal point and annex with it as many zeros as is the number of digits after the decimal point. Now, remove the decimal point and reduce the fraction to its lowest terms.
Basic Arithmetic operations  Decimal Fractions
Addition and Subtraction of Decimal Fractions
To add/subtract decimals,
(i) Write down the numbers, one under the other, with the decimal points lie in one column.
(ii) Now the numbers can be added normally (remember to put the decimal point in the answer).
Multiplication of a Decimal Fraction by a Power of 10
Shift the decimal point to the right by as many places as the power of 10.
Multiplication of Decimal Fractions
To multiply decimals,
(i) Multiply the given numbers considering them without decimal point.
(ii) In the product obtained, place the decimal point by starting at the right and moving a number of places equal to the sum of the decimal places in the given numbers.
Division of Decimal Fraction by a Counting Number
To divide a decimal by a counting number,
(i) Divide the given numbers considering them without decimal point.
(ii) In the quotient obtained, place the decimal point by starting at the right and moving a number of places equal to the sum of the decimal places in the given numbers.
Division of Decimal Fraction/Counting Number by a Decimal Fraction
To divide a decimal/counting number by a decimal,
(i) Multiply both the dividend and the divisor by a suitable power of 10 to make divisor a whole number.
(ii) Now divide the numbers normally.
Recurring Decimal
If a figure or a set of figures is repeated continuously in a decimal fraction, it is called a recurring decimal (also known as repeating decimal and circulating decimal).
A repeating decimal is denoted by placing a horizontal line above the repeated numerals. Another way to denote a recurring decimal is placing dots over the first and last digits.
Type of Recurring Decimals
Recurring Decimals can be classified into two categories  Pure Recurring Decimals and Mixed Recurring Decimals
Pure Recurring Decimals: Pure Recurring Decimal is a decimal fraction in which all the figures after the decimal point are repeated.
Mixed Recurring Decimals: Mixed Recurring Decimal is a decimal fraction in which some figures are not repeated and some figures are repeated.
Conversion of Recurring Decimals into Vulgar Fractions
To convert a pure recurring decimal into a vulgar fraction,
(1) Remove the number left to the decimal point, if any.
(2) Write the repeated figures only once in the numerator without the decimal point.
(3) Write as many nines in the denominator as the number of repeating figures.
(4) Add the number removed in step 1(if any) with the fraction obtained in the above steps.
To convert a mixed recurring decimal into a vulgar fraction,
(1) Remove the number left to the decimal point, if any.
(2) Numerator is the difference between the number formed by all the digits (taking repeated digits only once) and that formed by the digits which are not repeated.
(3) Denominator is the number formed by taking as many nines as the number of repeating figures followed by as many zeros as the number of nonrepeating digits.
(4) Add the number removed in step 1(if any) with the fraction obtained in the above steps.
Comparison of Fractions
Fractions with same denominators.
if fractions have same denominator. So just compare the numerators. Bigger the numerator, bigger the number.
Fractions with same numerators.
if fractions have same numerator. So just compare the denominators. Bigger the denominator, smaller the number.
Fractions with different numerators and denominators.
To compare such fractions, find out LCM of the denominators.then convert each of the given fractions into an equivalent fraction with LCM as the denominator.And compare the numerators. Bigger the numerator, bigger the number.
