Fractions

A digit which can be represented in p/q form, where q ≠0, is called a fraction. Here, p is called the numerator and q is called the denominator.

Simple Fraction:

The fraction which has denominator other then power of 10 is called simple fraction

Types of Simple fraction:

1. Proper Fraction :When the numerator of a fraction is less than its denominator, then fraction is called proper fraction. Eg 1/2 , 2/5 etc.

2. Improper Fraction :When the numerator of a fraction is greater than its denominator, then fraction is called improper fraction. Eg. 17/13,18/14 etc.

3. Compound Fraction :A fraction, in which numerator or denominator or both are in fraction, then it is called compound fraction. Eg  

4. Inverse Fraction: if we inverse the numerator and the denominator of a fraction, then the resultant fraction will be the inverse fraction of the original fraction. 
Eg given fraction =3/8 then its inverse fraction =8/3

 5. Mixed Fraction: The fraction, which is the combination of integer and fraction, is called mixed fraction. Eg 

6. Continuous Fraction: It has no certain definition but only say that a fraction contains additional fractions in its denominators, is called continuous fraction.eg 

Note: lf simplify a continuous fraction, start from bottom and work upwards

Comparison of Simple Fractions

• Cross Multiplication Method
• By Changing Fractions in Decimal Form
• By Equating Denominators of Given Fractions
• By Equating Numerators of Given Fractions

1. Cross Multiplication Method

By Changing Fractions in Decimal Form

3.By Equating Denominators of Given Fractions

4. By Equating Numerators of Given Fractions

• Arrange the given fractions in increasing order
•             4/5 ,6/7  ,5/6     
• If in the given fractions, the difference between numerator and denominator are same, then fraction having larger numerator is the largest and fraction having smaller numerator is the smallest.

So 4/5<5/6<6/7

• Which of the following fractions is largest and lowest.
•            2/5  ,5/11 ,8/17 ,11/23
• If in the given fractions, the numerators are increasing by a definite value and the denominator is also increasing by a definite value but the value of denominator is greater than numerator, then the fraction having smaller numerator will be the smallest fraction and the fraction having larger numerator will be the largest fraction.

So 11/23 is largest and 2/5 is lowest

• If any number is divided by a/b instead of multiply by a/b then the obtained number will be x greater than the original value.

    The given number= abx/b²-a²

Decimal Fraction

The fractions whose denominator (bottom number) is 10 or higher powers of 10, i.e., 100, 1000, 10,000 etc., are called decimal fractions.

Note: We can also write decimal fractions with a decimal point (without a denominator), that makes easier to solve math calculations like addition and multiplication on fractions.

Operation on decimal number:

1.    6202.5+620.25+62.025+6.2025+0.62025  =6891.59775
2.    5.064+3.98+0.7036+7.6+.3+2  =19.6476
3.    31.004-17.2386  =13.7654
4.    13-5.1967  =7.8033
5.    5172.49+378.352+?=9318.678  =3767.836
6.    2.61×1.3  =3.393
7.    0.4×.04×.004×40  =0.00256
8.    0.63÷9  =0.07
9.    0.0204÷13  =0.00156
10.    2.5÷0.0005  =5000

Recurring Decimal Fraction:

The decimal fraction, in which one or more decimal digits are repeated again and again, is called recurring decimal fraction. To represent these fractions, a line is drawn on the digits which are repeated.

1. Pure Recurring Decimal Fraction: When all the digits in a decimal fraction are repeated after the decimal point, then the decimal fraction is called as pure recurring decimal fraction.

2. Mixed Recurring Decimal Fraction: A decimal fraction in which some digits are repeated and some are not repeated after decimal is called as mixed recurring decimal fraction.

1. Convert a Pure Recurring Decimal into 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.

Examples
(a)  0.=3/9=1/3

(b)  0.=7/9=7/9

(c)  1.=1+3/9=1+1/3=4/3

(g)  5.=5+36/99=5+4/11=59/11

2. Convert a Mixed Recurring Decimal into 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 non-repeating digits.
(4) Add the number removed in step 1(if any) with the fraction obtained in the above steps.

Examples

(a)  =(5429−54)/9900 =5375/9900=215/396

(b)  =(16−1)/90 =15/90=1/6

(c)  =(583−58)/900 =525/900=7/12

(d)  =1+(318−3)/990 =1+315/990=1+7/22=29/22