Fractionभिन्न:- (from Latin fractus, "broken") It is a part of a whole.

A fraction describes how many parts of a certain size there are, for example, one-half, eight-fifths, three-quarters.

Part of a fraction        

Numerators and denominators

Numerator अंश- The top number of fraction

Denominator हर - The bottom number of fraction                  

The numerator represents a number of equal parts, and the denominator, which cannot be zero, indicates how many of those parts make up a unit or a whole.

Ex- 3/4, the numerator, 3, tells us that the fraction represents 3 equal parts, and the denominator, 4, tells us that 4 parts make up a whole.

Note-   Other uses for fractions are to represent ratio and division.

Thus the fraction 3/4 is also used to represent the ratio 3:4 (the ratio of the part to the whole) and the division 3 ÷ 4 (three divided by four).

Note:- (a) The denominator cannot be zero. (b) A fractions can be positive or negative

TYPES OF FRACTIONS(7)

There are seven types of fractions are-

समान भिन्न - Like fractions

असमान भिन्न - Unlike fractions

एकांक भिन्न-Unite fractions        

सम भिन्न- Proper Fraction

विषम भिन्न- Improper Fraction 

मिश्र भिन्न-Mixed Fraction           

तुल्य भिन्न-Equal Fraction  

1. Like fractions :- Those fractions whose denominators are equal are called like fractions.     

Ex-  ,  ,  ,

2. Unlike fractions:-Those fractions whose denominators are unequal are called unlike fractions.                     

Ex-  ,  ,  ,

3. Unit fractions:- Those fractions whose numerators are one are called unit fractions.

Ex-  ,   ,  ,

Unit fractions can also be expressed using negative exponents.

Ex- 2⁻¹, which represents 1/2,

and 2⁻², which represents 1/(2²) or 1/4.

4. Proper fraction:- Those Fractions whose numerators are less than the denominators are called proper                                                 fractions. (Numerator < denominator)           

Ex- ,  ,   ,   ,   ,  ,

Note:-  The value of a proper fraction is always less than 1.

5. Improper fraction:- Fractions with the numerator either equal to or greater than the denominator are called improper fraction.

(Numerator ≥ denominator)

Ex-  ,  ,  ,  ,  ,  ,
Notes:- (i) Every natural number can be written as a fraction in which 1 is it's denominator.

Ex-      2 = , 3 = , 4 =  etc.

(ii) Every natural number is an improper fraction.

(iii) The value of an improper fraction is always equal to or greater than 1.

6. Mixed fraction:- A combination of a proper fraction and a whole number is called a mixed fraction.

Ex- 

Note: A mixed number is formed with a whole number and a fraction.

Property 1:

* A mixed fraction may always be converted into an improper fraction.

7. Equal/Equivalent Fractions:- Those fractions whose value are same are called equal fractions.

Ex-   and   

Converting a Mixed Fraction into an Improper Fraction

Converting an Improper Fraction into a Mixed Fraction

Divide the Numerator by the denominators that the quotient will be the whole number and remainder will be the numerator, while denominator will remain the same. 

How to find the equivalent fractions?

To find the equivalent fraction of proper and improper fraction, we have the multiply both the numerator and denominator with the same number.

Example

Reciprocal of a fraction

If we have two non-zero numbers whose product is one then these numbers must be the reciprocals of each other.

To find the reciprocal of any fraction, we just need to flip the numerator with the denominator.

Multiplication of Fractions

1. How to multiply a fraction with a whole number?

a. If we have to multiply the proper or improper fraction with the whole number then we simply multiply the numerator with that whole number and the denominator will remain the same.

Example

b. If we have to multiply the mixed fraction with the whole number then first convert it in the form of improper fraction then multiply as above.

Example

c. Fraction as an operator “of”.

If it is written that find the 1/2 of 24 then what does ‘of’ means here?

Here ‘of’ represents the multiplication.

2. How to multiply a fraction with another fraction?

If we have to multiply the proper or improper fraction with another fraction then we simply multiply the numerator of both the fractions and the denominator of both the fractions separately and write them as the new fraction.

Example

Value of the products of the fractions

Generally when we multiply two numbers then we got the result which is greater than the numbers.

5 × 6 = 30, where, 30 > 5 and 30 > 6

But in case of a fraction, it is not always like that.

a. The product of two proper fractions

If we multiply two proper fractions then their product will be less than the given fractions.

Example

b. The product of two improper fractions

If we multiply two improper fractions then their product will be greater than the given fractions.

Example

c. The product of one proper and one improper fraction

If we multiply proper fraction with the improper fraction then the product will be less than the improper fraction and greater than the proper fraction.

Example

Division of Fractions

1. How to divide a whole number by a Fraction?

a. If we have to divide the whole number with the proper or improper fraction then we will multiply that whole number with the reciprocal of the given fraction.

Example

b. If we have to divide the whole number with the mixed fraction then we will convert it into improper fraction then multiply it’s reciprocal with the whole number.

Example

2. How to divide a Fraction with a whole number?

To divide the fraction with a whole number, we have to take the reciprocal of the whole number then divide it with the whole number as usual

Example

3. How to divide a fraction with another Fraction?

To divide a fraction with another fraction, we have to multiply the first fraction with the reciprocal of the second fraction.

Example

Note-

* An improper fraction can be always being converted into a mixed fraction.

* The value of Proper fraction is between 0 to 1.

* Improper fraction is 1 or greater than 1. Mixed fraction is greater than 1.

Additive inverse of a fraction- If the sum of two fractions is zero then they are called additive inverse of fraction of each other.

* To get additive inverse change the sign.

Ex-      25 + (- 25) = 0

Multiplicative inverse of a fraction- If the product of two fractions is 1then they are called multiplicative inverse of fraction of each other.

* To get multiplicative inverse write reciprocal.

Ex-      25 ×  52 = 1

Ratios:- A ratio is a relationship between two or more numbers that can be expressed as a fraction.                                                  

* A number of items are grouped and compared in a ratio, specifying numerically the relationship between each group.

* Ratios are expressed as "group 1 to group 2 ... to group n". For example, if a car lot had 12 vehicles, of which 2 are white, 6 are red, and 4 are yellow

Then the ratio of red to white to yellow cars is 6 to 2 to 4. The ratio of yellow cars to white cars is 4 to 2 and may be expressed as 4:2 or 2:1.

A ratio is often converted to a fraction when it is expressed as a ratio to the whole. In the above example, the ratio of yellow cars to all the cars on the lot is 4:12 or 1:3. We can convert these ratios to a fraction and say that 4/12 of the cars or ⅓ of the cars in the lot are yellow. Therefore, if a person randomly chose one car on the lot, then there is a one in three chance or probability that it would be yellow.