# Operations on rational numbers

Rational numbers are numbers that can be expressed in fractional form. The ideal form of rational numbers is ‘p/q’ where q is any non-zero number. Here p is the numerator and q the denominator. For example, 2/5, 7/21, 12/12 are rational numbers as the denominator is not equal to zero.

Operations on rational numbers can be defined as the arithmetic operations which are applicable to rational numbers. Arithmetic operations such as addition, subtraction, multiplication and division are applicable to rational numbers. However, there are different methods to use arithmetic operations on rational numbers.

## Methods of using operations on rational numbers

1. Addition of rational numbers having the same denominator.
2. Addition of rational numbers having different denominators.
3. Subtraction of rational numbers having the same denominator.
4. Subtraction of rational numbers having different denominator
5. Multiplication of rational numbers
6. Division of rational numbers

## Operations on rational numbers

1. Addition and subtraction of rational numbers
The addition and subtraction of rational numbers follow the same procedure.
1. Having same denominators
For adding rational numbers having the same denominator, proceed the same way as for real numbers. Since the denominators are the same, add the numerators while not changing the denominator. For example, consider two rational numbers 2/13 and 5/13
2/13 + 5/13 = (2 + 5)/13
2/13 + 5/13 = 7/13
Here, we added 2 and 5 directly because both had the same denominator and the sum of 2 and 5 was placed as the numerator of the common denominator 13.
Similarly, for subtraction, we subtract the numerators and write the difference as the numerator while keeping the denominator same. For example, for rational numbers 2/13 and 5/13
2/13 – 5/13 = (2 – 5)/13
2/13 – 5/13 = -3/13
2. Having different denominators
For addition of rational numbers of different denominators, make the denominators same by taking their LCM and proceed the same way as in step i. For example, let two rational numbers be 2/3 and 4/5. LCM of 3 and 5 is 15. Therefore,
2/3 + 4/5 = ([2 . 5) + (4 .3)]/15
2/3 + 4/5 = (10 + 12)/15
2/3 + 4/5 = 22/15
Similarly, for subtraction
2/3 - 4/5 = ([2 . 5) - (4 .3)]/15
2/3 - 4/5 = (10 - 12)/15
2/3 + 4/5 = -2/15

2. Multiplication of rational numbers
The procedure of multiplication of rational numbers having the same and different denominators is the same and unaffected by the nature of the denominator. For example, for multiplication of 1/3 and 2/3, the final product is equal to the ratio of the product of numerator to the product of denominator.
1/3 . 2/3 = (1 . 2)/(3 . 3)
1/3 . 2/3 = 2/9
Similarly, for different denominators as in ½ and ¾
½ . ¾ = (1 . 3)/(2 . 4)
½ . ¾ = 3/8

3. Division of rational numbers
For dividing two rational numbers steps as below are followed:
1. The dividend is considered as it is.
2. The sign of division is changed into multiplication.
3. Finally, the divisor value is taken as its reciprocal to divide given rational numbers
For example, the division of 2/3 and 5/3 is as follows
2/3 ÷ 5/3 = 2/3 . 3/5
therefore, 2/3 ÷ 5/3 = 2/5
Which is the required answer.

The vertices of adjacent sides of a rectangle always join to make a right angle which indicates that its opposite sides are parallel. This also means that the opposite sides are equal in measure. A quadrilateral with equal and parallel opposite sides is said to be a parallelogram. Hence, a rectangle is also a parallelogram with vertex angles 90°.

## Diagonals of a rectangle

A rectangle has two diagonals of equal length. These diagonals bisect each other i.e. divide each other into two equal parts. For example,

Here, in rectangle ABCD, AD and BC are diagonals and O is the point of intersection of the diagonals such that

AO = OD

BO = OC

Length of diagonals: The length of each diagonal can be calculated by a simple formula as below

d = √(l² + b²)

where ‘d’ is the length of diagonal. This formula is derived from the Pythagoras theorem as

we know

(hypotenuse)² = (perpendicular)² + (base)²

In rectangle any rectangle ABCD,

Using Pythagoras theorem,

(AD)² = (AC)² + (CD)²

AD =√[(AC)² + (CD)²]

For AD = d; AC = l: CD = b

d = √(l² + b²)

## Properties of a rectangle

1. It is a 4-sided closed 2-D shape with 4 vertices or corners.
2. All the vertex angles are equal i.e. 90° each.
3. Opposite angles are equal which makes it a parallelogram.
4. Its opposite sides are equal and parallel.
5. The sum of its interior angles is 360°.
6. The diagonals of a rectangle are of equal length.
7. The diagonals bisect each other.
8. If the length and breadth of a rectangle are equal and diagonals bisect each other at 90° it becomes a square.

## Perimeter of a rectangle

The perimeter of a rectangle is the total measure of its boundary. To calculate the perimeter of a rectangle, add all its sides or use the formula

Perimeter of rectangle = 2 . (length + breadth)

Perimeter of rectangle ABCD = 2. (AB + AC)

where length is AB and CD and breadth is AC and BD.

## Area of a rectangle

The area of rectangle is defined as the space occupied by a rectangle in a plane. It is calculated as the product of its length and breadth and measured in square units.

Area of rectangle = length . breadth

Area of rectangle ABCD = AB . AC

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