**Operations with fractions :**

We can not imagine the subject Math without the term "Fraction". Because, we solve many problems with it. So, we must be aware of the operations which we often do in fractions.

Let us explore the following operations with fractions.

(i) Addition

(ii) Subtraction

(iii) Multiplication

(iv) Division

(v) Reciprocal

**Example 1 : **

Simplify : 2/5 + 3/5

**Solution : **

Here, for both the fractions, we have the same denominator, we have to take only one denominator and add the numerators.

Then, we get

2/5 + 3/5 = (2+3) / 5 = 5/5 = 1

**Example 2 :**

Simplify : 7/5 - 3/5

**Solution : **

Here, for both the fractions, we have the same denominator, we have to take only one denominator and subtract the numerators.

Then, we get

7/5 - 3/5 = (7-3) / 5 = 4/5

Here, we explain two methods to add two fractions with different denominators.

1) Cross multiplication method

2) L.C.M method

**Cross multiplication method :**

If the denominators of the fractions are co-prime or relatively prime, we have to apply this method.

Fro example, let us consider the two fractions 1/8, 1/3.

In the above two fractions, denominators are 8 and 3.

For 8 and 3, there is no common divisor other than 1. So 8 and 3 are co-prime.

Here we have to apply cross-multiplication method to add the two fractions 1/8 and 1/3 as given below.

**L.C.M method :**

If the denominators of the fractions are not co-prime (there is a common divisor other than 1), we have to apply this method.

Fro example, let us consider the two fractions 5/12, 1/20.

In the above two fractions, denominators are 12 and 20.

For 12 and 20, if there is at least one common divisor other than 1, then 12 and 20 are not co-prime.

For 12 & 20, we have the following common divisors other than 1.

**2 & 4**

So 12 and 20 are not co-prime.

In the next step, we have to find the L.C.M (Least common multiple) of 12 and 20.

12 = 2² x 3

20 = 2² x 5

When we decompose 12 and 20 in to prime numbers, we find 2, 3 and 5 as prime factors for 12 and 20.

To get L.C.M of 12 and 20, we have to take 2, 3 and 5 with maximum powers found above.

So, L.C.M of 12 and 20 = 2² x 3 x 5

= 4 x 3 x 5

= 60

Now we have to make the denominators of both the fractions to be 60 and add the two fractions 5/12 and 1/20 as given below.

**Note : **

We have to do the same process for subtraction of two fractions with different denominators.

Let us look at next stuff on "Fractions in mathematics"

To multiply a proper or improper fraction with the whole number,first, we have to multiply the whole number with the numerator of the fraction, keeping the denominator same.

For example,

2 x 3/5 = 6/5

3 x 7/11 = 21/11

To multiply a mixed fraction by a whole number, first convert the mixed fraction to an improper fraction and then multiply.

For example,

4 x 3 4/7 = 4 x 25/7 = 100/7 = 14 2/7

Let us look at next stuff on "Fractions in mathematics"

The picture given below clearly illustrates, how to convert improper fractions in to mixed numbers

To multiply a proper or improper fraction by another proper or improper fraction, we have to multiply the numerators and denominators.

For example,

2/3 x 4/5 = 8/15

1/3 x 7/11 = 7/33

If the product of two non-zero numbers is equal to one then each number is called the reciprocal of the other.

So the reciprocal of 3/5 is 5/3.

**Note: **

Reciprocal of 1 is 1 itself. 0 does not have a reciprocal.

To divide a whole number by any fraction, multiply that whole number by the reciprocal of that fraction.

For example,

6 ÷ 2/5 = 6 x 5/2 = 30/2 = 15

While dividing a whole number by a mixed fraction, first convert the mixed fraction into improper fraction and then solve it.

6 ÷ 3 4/5 = 6 ÷ 19/5 = 6 x 5/19 = 30/19 = 1 11/19

To divide a fraction by another fraction, multiply the first fraction by the reciprocal of the second fraction.

For example,

1/5 ÷ 3/7 = 1/5 x 7/3 = 7/15

After having gone through the stuff given above, we hope that the students would have understood "Operations with fractions".

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