# Class-7 Rational Numbers

Introduction to Rational Number

Subtraction of Rational Number

Multiplication of Rational Number

## Introduction to Rational Numbers

Rational number can be defined as a ratio or fraction (^{P}⁄_{Q}) form where numerator and denominator are integer and denominator (Q) should not be equal to zero.

**Example:** ^{5}⁄_{8} is an example of rational number as 5 and 8 are integers and 8 ≠ 0

In the same way ^{-11}⁄_{2}, ^{9}⁄_{-2}, ^{-43}⁄_{-7} are also rational number.

## Positive Rational Number

When both the numerator and denominator are positive or negative integer then they are said to be Positive Rational Number. Let's see some examples.

**Examples** ^{-25}⁄_{-13} (can be written as ^{25}⁄_{13}), ^{17}⁄_{19} etc.

## Negative Rational Number

When one of the numerator or denominator is negative integer then they are said to be Negative Rational Number. Let's see some examples.

**Example:** ^{-25}⁄_{27}, ^{15}⁄_{-17} etc.

- If both the numerator and denominator of a rational number is either positive or negative, then multiplying −1 to numerator or denominator will convert the rational number to negative.
- If any of the numerator or denominator of a rational number is negative, then multiplying −1 to the negative part of the numerators or denominator will convert the rational number to positive.

**Example:** ^{3}⁄_{5} = ^{(3 × (-1))}⁄_{5} = ^{-3}⁄_{5}

^{-6}⁄_{15} = ^{((-6)×(-1))}⁄_{15} = ^{6}⁄_{15}

## Properties of Rational Number

1. Every integer is a rational number but every rational number is not an integer.

E.g. ^{7}⁄_{9} is a rational number and 9 ≠ 0 but ^{7}⁄_{9} not an integer.

E.g. ^{-7}⁄_{9} is a rational number and 9 ≠ 0 but ^{-7}⁄_{9} not an integer.

2. Every natural number and whole number is also an integer and so a rational number.

3. zero (0) is also a rational number but it is neither positive nor negative.

## Comparison of Rational Number

Before comparing the rational numbers, we must remember the following points:

1. Every positive rational number is greater than 0 and every negative number.

^{9}⁄_{5} > 0, ^{9}⁄_{5} > ^{-3}⁄_{2}

2. Every negative rational number is less than 0 and every positive number.

^{-3}⁄_{3} < 0, ^{-5}⁄_{3} < ^{132}⁄_{3}

3. Zero is greater than every negative number and smaller than every positive number.

^{4}⁄_{18} > 0 > ^{-4}⁄_{18}

**Example 1.** Compare ^{5}⁄_{7} and ^{-2}⁄_{5}

**Solution.** As we know Every positive rational number is greater than every negative number.

^{5}⁄_{7} > ^{-2}⁄_{5}.

**Example 2.** Compare ^{-1}⁄_{2} and ^{-5}⁄_{4}

**Solution.** Here both are negative rational number having different denominators. So, we have to make both the denominators same.

^{-1}⁄_{2} = ^{(-1×2)}⁄_{(2×2)} = ^{-2}⁄_{4}

Then compare the rational number of same denominators.

^{-2}⁄_{4} > ^{-5}⁄_{4}

^{-1}⁄_{2} > ^{-5}⁄_{4}

## Addition of Rational Number

**When the denominators are equal:**

- By keeping the denominators same simply add the numerators.
- Simplify the result if possible.

**Example 1.** Add ^{2}⁄_{5} and ^{3}⁄_{5}

**Solution.** ^{2}⁄_{5} + ^{3}⁄_{5}

= ^{(2+3)}⁄_{5}

= ^{5}⁄_{5}

= 1

**Example 2.** Add ^{5}⁄_{9} and ^{-4}⁄_{9}

**Solution.** ^{5}⁄_{9} + ^{-4}⁄_{9}

= ^{{5+(-4)}}⁄_{9}

= ^{(5-4)}⁄_{9}

= ^{1}⁄_{9}

**When the denominators are unequal:**

- Find out LCM of the denominators of the given rational numbers.
- Convert the given rational numbers to have LCM as the common denominator.
- Add the newly converted rational numbers by following the process of equal denominators

**Example 1.** Add ^{-4}⁄_{5} and ^{7}⁄_{2}

**Solution.** ^{-4}⁄_{5} + ^{7}⁄_{2}

LCM of 5 and 2 is 10.

^{-4}⁄_{5} = ^{-4×2}⁄_{5×2} = ^{-8}⁄_{10}

^{7}⁄_{2} = ^{7×5}⁄_{2×5} = ^{35}⁄_{10}

Now, add ^{-8}⁄_{10} and ^{35}⁄_{10}.

^{-8}⁄_{10} + ^{35}⁄_{10}

^{(-8+35)}⁄_{10} = ^{27}⁄_{10}

## Subtraction of Rational Number

**When the denominators are equal:**

- By keeping the denominators same simply subtract the numerators.
- Simplify the result if possible.

**Example 1.** Subtract ^{2}⁄_{5} from ^{3}⁄_{5}

**Solution.** ^{3}⁄_{5} − ^{2}⁄_{5}

= ^{(3-2)}⁄_{5}

= ^{1}⁄_{5}

**Example 2.** Subtract ^{-2}⁄_{7} from ^{3}⁄_{7}

**Solution.** ^{3}⁄_{7} − ^{-2}⁄_{7}

= ^{{3-(-2)}}⁄_{7}

= ^{(3+2)}⁄_{7}

= ^{5}⁄_{7}

**When the denominators are unequal:**

- Find out LCM of the denominators of the given rational numbers.
- Convert the given rational numbers to have LCM as the common denominator.
- Subtract the newly converted rational numbers by following the process of equal denominators

**Example 1.** Subtract ^{4}⁄_{5} from ^{7}⁄_{2}

**Solution.** ^{7}⁄_{2} − ^{4}⁄_{5}

LCM of 5 and 2 is 10.

^{7}⁄_{2} = ^{7×5}⁄_{2×5} = ^{35}⁄_{10}

^{4}⁄_{5} = ^{4×2}⁄_{5×2} = ^{8}⁄_{10}

Now, subtract ^{35}⁄_{10} from ^{8}⁄_{10}.

^{35}⁄_{10} − ^{8}⁄_{10}

^{(35-8)}⁄_{10} = ^{27}⁄_{10}

## Multiplication of Rational Number

- Multiply the numerators of given rational numbers and the product becomes numerator
- multiply the denominators of given rational numbers and the product becomes denominator of the result
- Simplify the result if possible

^{a}⁄

_{b}and

^{c}⁄

_{d}are two rational numbers.

^{a}⁄

_{b}x

^{c}⁄

_{d}=

^{(axc)}⁄

_{(bxd)}

**Example 1.** Multiply ^{4}⁄_{5} and ^{2}⁄_{7}.

**Solution.** ^{4}⁄_{5} × ^{2}⁄_{7}

= ^{(4×2)}⁄_{(5×7)}

= ^{8}⁄_{35}

**Example 2.** Multiply ^{-5}⁄_{9} and ^{3}⁄_{5}.

**Solution.** ^{-5}⁄_{9} × ^{3}⁄_{5}

= ^{(-5×3)}⁄_{(9×5)}

= ^{-15}⁄_{45}

= ^{-1}⁄_{3}

## Reciprocal of Rational Number

When the product of two rational number is 1 then each one is called the reciprocal of other. In other words ^{a}⁄_{b} rational number reciprocal is ^{b}⁄_{a}.

Let's see some examples.

**Example 1.** Find the reciprocal of ^{4}⁄_{5}.

**Solution.** Reciprocal of ^{4}⁄_{5} = ^{5}⁄_{4}

**Example 2.** Find the reciprocal of ^{7}⁄_{-9}.

**Solution.** Reciprocal of ^{7}⁄_{-9} = ^{-9}⁄_{7}

## Division of Rational Number

If ^{P}⁄_{Q} and ^{R}⁄_{S} are two rational number to be divided, then multiply ^{P}⁄_{Q} with reciprocal of ^{R}⁄_{S} i.e. ^{S}⁄_{R}.

Let's see some examples.

**Example 1.** Divide ^{4}⁄_{5} by ^{2}⁄_{5}.

**Solution.** ^{4}⁄_{5} ÷ ^{2}⁄_{5}

Reciprocal of ^{2}⁄_{5} = ^{5}⁄_{2}

^{4}⁄_{5} × ^{5}⁄_{2} = 2

**Example 2.** Divide ^{3}⁄_{7} by ^{-5}⁄_{14}.

**Solution.** ^{3}⁄_{7} ÷ ^{-5}⁄_{14}

Reciprocal of ^{-5}⁄_{14} = ^{14}⁄_{-5} = ^{-14}⁄_{5}

^{3}⁄_{7} × ^{-14}⁄_{5} = ^{-6}⁄_{5}

## Class-7 Rational Numbers Test

## Class-7 Rational Numbers Worksheet

Rational Numbers Worksheet - 1

Rational Numbers Worksheet - 2

Rational Numbers Worksheet - 3

Rational Numbers Worksheet - 4

## Answer Sheet

**Rational-Numbers-Answer**Download the pdf

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