Class-8 Rational Numbers

Introduction to Rational Number

Facts of Rational Numbers

Addition of Rational Numbers

Addition Properties of Rational numbers

Subtraction of Rational Number

Subtraction Properties of Rational Number

Multiplication of Rational Numbers

Multiplication Properties of Rational Number

Division of Rational Number

Division Properties of Rational Number

Rational Number Between any Two Given Rational Numbers

Rational Numbers Test

Rational Number Worksheet

Answer Sheet

Introduction to Rational Numbers

In class 7, we learnt following things about Rational numbers.

Define rational number
Number line representation of rational numbers
Comparison of rational numbers
Operation like addition, subtraction, multiplication, and division

Now, in class 8 we will study the recap of all operation, properties of all operation and find rational number between any two given rational numbers.

Facts of Rational Numbers

Numbers in form of ^a⁄_b where 'a' and 'b' are integer. Here, 'b' is not equal to zero, known as rational numbers.

some examples are ⁵⁄₉, ^-6⁄₁₁, 9, -3, 0, ³⁄_-11 etc.

Every integer is a rational number, but every rational number is not an integer. It is because, every integer 'i' can be written as ⁱ⁄₁ which is in fraction form. For example, ^-3⁄₈ is a rational number but not an integer.

Every rational number is not a fraction, but every fraction is a rational number. For example, ^-5⁄₉ is a rational number but not a fraction.

The rational number ^-a⁄_b is same as −^a⁄_b.

Every rational number should be written with positive integer as denominator.

Two rational numbers ^a⁄_b = ^c⁄_d, if a × d = b × c.

Rational number is positive if both numerator and denominator are positive integers, or both are negative integers. If one is positive integer and other is negative integer then rational number is said to be negative.

All finite decimal numbers are rational numbers. Let's see some examples

0.5 = ¹⁄₂, 0.26 = ¹³⁄₅₀ and 0.625 = ⁵⁄₈.

Addition of Rational Numbers

When we add two rational numbers, first convert each rational number with positive denominator.

Rational Numbers with same denominator

If ^a⁄_b and ^c⁄_d are two rational numbers, then (^a⁄_b) + (^c⁄_b) = ^(a+c)⁄_b.

Example 1. Add ⁶⁄₉ and ⁸⁄_-9.

Solution. ⁶⁄₉ + ⁸⁄_-9

Here, first convert ⁸⁄_-9 into positive denominator.

⁸⁄_-9 = ^{8×(-1)}⁄_{-9×(-1)}

= ^-8⁄₉

Then, ⁶⁄₉ + (-⁸⁄₉) = ⁶⁄₉ − ⁸⁄₉ = ^(6-8)⁄₉ = ^-2⁄₉.

Rational Numbers With Different Denominator

Here we first find LCM of denominators of two rational numbers. Then we covert the denominator of each rational number to have same value as LCM. At last, add the rational numbers following the process of same denominator.

Example 1. Add ^-5⁄₇ + ⁷⁄₃.

Solution.LCM of 7 and 3 = 21

^-5⁄₇ = ^(-5×3)⁄_(7×3) = ^-15⁄₂₁

⁷⁄₃ = ^(7×7)⁄_(3×7) = ⁴⁹⁄₂₁

Then, ^-15⁄₂₁ + ⁴⁹⁄₂₁ = ^(-15+49)⁄₂₁ = ³⁴⁄₂₁.

Addition Properties of Rational numbers

Closure property
Commutative
Associative
Special properties

Closure Property

When two rational numbers are added, then sum of two rational numbers always a rational number. Here, if we take ^a⁄_b and ^c⁄_d are two rational numbers, then (^a⁄_b + ^c⁄_d) is a rational number. Let's see some example

Example 1. Add ¹⁄₂ and ²⁄₅.

Solution. ¹⁄₂ + ²⁄₅ = ⁽⁵⁺⁴⁾⁄₁₀ = ⁹⁄₁₀

So, ⁹⁄₁₀ is a rational number.

Example 2. Add ^-5⁄₉ and ^-2⁄₃

Solution. ^-5⁄₉ + ^-2⁄₃ = ^{-5+(-6)}⁄₉ = ^-11⁄₉

Example 3. Add ^-6⁄₅ and ¹⁄₄.

Solution. ^-6⁄₅ + ¹⁄₄ = ^(-6×4+1×5)⁄₂₀ = ^(-24+5)⁄₂₀ = ^-19⁄₂₀

Commutative

Addition of rational numbers are commutative that is ^a⁄_b + ^c⁄_d = ^c⁄_d + ^a⁄_b, here, b and d are not equal to 0. Let's see some examples.

Example 1. ^-5⁄₇ + ³⁄₅ = ³⁄₅ + ^-5⁄₇

Solution. We have to prove LHS = RHS

LHS = ^-5⁄₇ + ³⁄₅ = ^(-25+21)⁄₃₅ = ^-4⁄₃₅

RHS = ³⁄₅ + ^-5⁄₇ = ^{21+(-25)}⁄₃₅ = ^-4⁄₃₅

LHS = RHS. So, it is following commutative properties.

Associative

Addition of rational number is associative, that is (^a⁄_b + ^c⁄_d) + ^e⁄_f = ^a⁄_b + (^c⁄_d + ^e⁄_f). Here b, d, f are not equal to 0. Let's see some examples.

Example 1. (¹⁄₂ + ²⁄₃) + ^-3⁄₅ = ¹⁄₂ + (²⁄₃ + ^-3⁄₅).

Solution. We have to prove LHS = RHS

LHS = (¹⁄₂ + ²⁄₃) + ^-3⁄₅ = ⁽³⁺⁴⁾⁄₆ + (^-3⁄₅) = ⁷⁄₆ - ³⁄₅ = ^(35−18)⁄₃₀ = ¹⁷⁄₃₀

RHS = ¹⁄₂ + (²⁄₃ + ^-3⁄₅) = ¹⁄₂ + ^{10+(-9)}⁄₁₅ = ¹⁄₂ + ¹⁄₁₅ = ⁽¹⁵⁺²⁾⁄₃₀ = ¹⁷⁄₃₀

LHS = RHS, hence it is proved that it follows associative property.

Special Properties of Rational Numbers

The number zero is known as additive identity of rational number, that is, ^a⁄_b + 0 = ^a⁄_b = 0 + ^a⁄_b.

Additive inverse of rational number can be calculated by changing its sign, that is additive inverse of ^a⁄_b is (^-a⁄_b) and additive inverse of ^-a⁄_b is ^a⁄_b. Let's see some examples.

Example 1. Additive inverse of ^-3⁄₅ = −(^-3⁄₅) = ³⁄₅.

Example 2. Additive inverse of ^-4⁄_-9 = −(^-4⁄_-9) = ^-4⁄₉.

Subtraction of Rational Number

Subtraction of rational numbers can be done by adding additive inverse of subtrahend with minuend. That is, ^a⁄_b − ^c⁄_d = ^a⁄_b + (^-c⁄_d).here ^-c⁄_d is additive inverse of ^c⁄_d.

Example 1. Subtract ⁵⁄₉ from ^-3⁄₇.

Solution. ^-3⁄₇ − ⁵⁄₉

        = ^-3⁄₇ + additive inverse of ⁵⁄₉

        = ^-3⁄₇ + (^-5⁄₉)

        = ^{(-3×9−5×7)}⁄₆₃

        = ^{-27+(-35}⁄₆₃

        = ^-62⁄₆₃

Example 2. subtract ^-3⁄₂₅ from 3¹⁄₅.

Solution. 3¹⁄₅ − (^-3⁄₂₅)

        = ¹⁶⁄₅ + additive inverse of (^-3⁄₂₅)

        = ¹⁶⁄₅ + ³⁄₂₅

        = ⁽⁸⁰⁺³⁾⁄₂₅

        = ⁸³⁄₂₅

        = 3⁸⁄₂₅

Subtraction Properties of Rational Number

Closure Property
Not Commutative
Not Associative

Closure Property

Like addition, difference of two rational number is rational number. Let's see some examples.

Example 1. ⁶⁄₉ − ³⁄₄ = ^(24-27)⁄₃₆ = ^-3⁄₃₆ = ^-1⁄₁₂
^-1⁄₁₂ is a rational number.

Example 2. ^-7⁄₉ − ⁴⁄₃ = ^(-7-12)⁄₉ = ^-19⁄₉ = −2¹⁄₉
−2¹⁄₉ is a rational number.

Not Commutative

If ^a⁄_b and ^c⁄_d are two rational number, then, (^a⁄_b − ^c⁄_d ) is not equal to (^c⁄_d − ^a⁄_b).
Let's see some example.

Example 1. Prove ¹⁄₃ − ⁵⁄₆ ≠ ⁵⁄₆ − ¹⁄₃.

Solution. LHS = ¹⁄₃ − ⁵⁄₆ = ^(2-5)⁄₆ = ^-1⁄₂

RHS = ⁵⁄₆ − ¹⁄₃ = ^(5-2)⁄₆ = ¹⁄₂

LHS ≠ RHS i.e. ^-1⁄₂ ≠ ¹⁄₂

Hence it is proved that ¹⁄₃ − ⁵⁄₆ ≠ ⁵⁄₆ − ¹⁄₃.

Not Associative

If ^a⁄_b, ^c⁄_d and ^e⁄_f are three rational numbers, then (^a⁄_b − ^c⁄_d) − ^e⁄_f ≠ ^a⁄_b − (^c⁄_d − ^e⁄_f).
Let's see some example.

Example 1. prove (^-3⁄₅ − ²⁄₇) − (^-1⁄₅) ≠ ^-3⁄₅ − (²⁄₇ − ^-1⁄₅).

Solution. LHS = (^-3⁄₅ − ²⁄₇) − (^-1⁄₅)

        = ^(-21-10)⁄₃₅ + ¹⁄₅

        = ^(-31)⁄₃₅ + ¹⁄₅

        = ^(-31+7)⁄₃₅ = ^-24⁄₃₅

RHS = ^-3⁄₅ − (²⁄₇ − ^-1⁄₅)

        = ^-3⁄₅ − (²⁄₇ + ¹⁄₅)

        = ^-3⁄₅ − ⁽¹⁰⁺⁷⁾⁄₃₅

        = ^-3⁄₅ − ¹⁷⁄₃₅

        = ^(-21-17)⁄₃₅ = ^-38⁄₃₅

^-24⁄₃₅ ≠ ^-38⁄₃₅ i.e. LHS ≠ RHS

Hence, it is proved that subtraction does not follow associative property.

Multiplication of Rational Numbers

To get the product of two rational number, numerators are multiplied, and denominators are multiplied. Let's see some examples.

Example 1. Multiply ^-3⁄₇ and ⁸⁄₉

Solution. ^-3⁄₇ × ⁸⁄₉

= ^(-3×8)⁄_7×9

= ^-8⁄₂₁

Example 2. Multiply 3¹⁄₇ and -4³⁄₄

Solution. 3¹⁄₇ × (-4³⁄₄)

        = ²²⁄₇ × (^-19⁄₄)

        = ^{22×(-19)}⁄_7×4

        = ^-418⁄₂₈

        = -14²⁶⁄₂₈

Multiplication Properties of Rational Number

Closure Property
Commutative Property
Associative Property
Distributive Properties
Multiplicative Identity
Multiplicative Inverse

Closure Property

Product of two rational number always a rational number. If ^a⁄_b and ^c⁄_d are two rational number ,then ^a⁄_b × ^c⁄_d is also rational numbers. Let's see some examples.

Example 1. ²⁄₅ × ⁷⁄₉ = ^2×7⁄_5×9 = ¹⁴⁄₄₅
¹⁴⁄₄₅ is a rational number.

Example 2. ³⁄₄ × ¹⁄₆ = ^3×1⁄_4×6 = ³⁄₂₄ = ¹⁄₈
¹⁄₈ is a rational number.

Example 3. ⁴⁄₇ × ⁵⁄₈ = ^4×5⁄_7×8 = ⁵⁄₁₄
⁵⁄₁₄ is a rational number.

Commutative Property

If ^a⁄_b and ^c⁄_d are two rational numbers, then ^a⁄_b × ^c⁄_d = ^c⁄_d × ^a⁄_b. Let's see some examples.

Example 1. Prove ²⁄₃ × (^-4⁄₅) = (^-4⁄₅) × ²⁄₃.

Solution. LHS = ²⁄₃ × (^-4⁄₅) = ^{2×(-4)}⁄_(3×5) = ^-8⁄₁₅

RHS = ^-4⁄₅ × ²⁄₃ = ^(-4×2)⁄_5×3 = ^-8⁄₁₅

LHS = RHS i.e. ²⁄₃ × (^-4⁄₅) = (^-4⁄₅) × ²⁄₃

Hence, it is proved that multiplication of rational number is commutative.

Associative Property

Like addition, multiplication have associative property. If ^a⁄_b, ^c⁄_d and ^e⁄_f are three rational numbers, then (^a⁄_b × ^c⁄_d) × ^e⁄_f = ^a⁄_b × (^c⁄_d × ^e⁄_f). Let's see some example.

Example 1. Prove (⁵⁄₇ × ^-3⁄₂) × ^-1⁄₇ = ⁵⁄₇ × (^-3⁄₂ × ^-1⁄₇)

Solution. LHS = (⁵⁄₇ × ^-3⁄₂) × ^-1⁄₇ = ^-15⁄₁₄ × ^-1⁄₇ = ¹⁵⁄₉₈

RHS = ⁵⁄₇ × (^-3⁄₂ × ^-1⁄₇) = ⁵⁄₇ × ³⁄₁₄ = ¹⁵⁄₉₈

LHS = RHS i.e. (⁵⁄₇ × ^-3⁄₂) × ^-1⁄₇ = ⁵⁄₇ × (^-3⁄₂ × ^-1⁄₇)

Hence, multiplication of fraction is associative.

Distributive Property

When two rational numbers are added or subtracted and multiply with another rational number, then it should follow distributive laws.

If ^a⁄_b, ^c⁄_d and ^e⁄_f are three rational numbers, then ^a⁄_b × (^c⁄_d + ^e⁄_f) = ^a⁄_b × ^c⁄_d + ^a⁄_b × ^e⁄_f and

^a⁄_b × (^c⁄_d − ^e⁄_f ) = ^a⁄_b × ^c⁄_d − ^a⁄_b × ^e⁄_f.

Multiplicative Identity

The number 1 is multiplicative identity means ^a⁄_b × 1 = ^a⁄_b = 1 × ^a⁄_b.

Multiplicative Inverse

When the numerator and denominator of non-zero rational number are interchange called multiplicative inverse or reciprocal. Multiplicative inverse of ^a⁄_b is ^b⁄_a.

product of rational number with its reciprocal always 1.

Zero has no multiplicative inverse.

Example 1. Find multiplicative invers of ⁷⁄₉ × ¹⁰⁄₇.

Solution. First, we multiply ⁷⁄₉ and ¹⁰⁄₇, then convert into its reciprocal

        ⁷⁄₉ × ¹⁰⁄₇

        = ^(7×10)⁄_(9×7)

        = ⁷⁰⁄₆₃ = ¹⁰⁄₉

Multiplicative invers of ¹⁰⁄₉ = ⁹⁄₁₀

Division of Rational Number

When two rational numbers are divided, first we find out reciprocal of the divisor, then it is multiplied with the dividend. In other words, if ^a⁄_b and ^c⁄_d are rational numbers ,then ^a⁄_b ÷ ^c⁄_d = ^a⁄_b × ^d⁄_c (reciprocal of ^c⁄_d) = ^a×d⁄_b×c.
Let's see some examples.

Example 1. 2¹⁄₅ ÷ (^-1⁄₃).

Solution. 2¹⁄₅ ÷ (^-1⁄₃)

        = ¹¹⁄₅ ÷ (^-1⁄₃)

Reciprocal of ^-1⁄₃ is equal to -3.

        = ¹¹⁄₅ × (-3)

        = ^-33⁄₅

Example 2. Divide the sum of ¹⁄₇ and ¹⁄₂ by the product of ⁹⁄₂ and ³⁄₇

Solution. (¹⁄₇ + ¹⁄₂) ÷ (⁹⁄₂ × ³⁄₇)

         = ⁽²⁺⁷⁾⁄₁₄ ÷ ^(9×3)⁄_(2×7)

         = ⁹⁄₁₄ ÷ ²⁷⁄₁₄

Reciprocal of ²⁷⁄₁₄ is ¹⁴⁄₂₇

         = ⁹⁄₁₄ × ¹⁴⁄₂₇

         = ^(9×14)⁄_(14×27)

         = ⁹⁄₂₇

         = ¹⁄₃

Division Properties of Rational Number

Closure Property
Division is Non-commutative
Division is Non-associative

Closure Property

When two non-zero rational numbers are divided their result remain rational number. If ^a⁄_b and ^c⁄_d are non-zero rational number, then ^a⁄_b ÷ ^c⁄_d is rational number. If we do not consider 0 as a rational number, then rational number closed under division.

Division is Non-commutative

If ^a⁄_b and ^c⁄_d are on-zero rational number, ^a⁄_b ÷ ^c⁄_d ≠ ^c⁄_d ÷ ^a⁄_b

Division is Non-associative

If ^a⁄_b, ^c⁄_d and ^e⁄_f are non-zero rational numbers, ^a⁄_b ÷ (^c⁄_d ÷ ^e⁄_f) ≠ (^a⁄_b ÷ ^c⁄_d) ÷ ^e⁄_f.

Rational Number Between any Two Given Rational Numbers

In order to know rational number between two rational numbers, we have to calculate the mean of two given rational numbers. If ^a⁄_b and ^c⁄_d are two rational numbers. Then mean of ^a⁄_b and ^c⁄_d is written as ^{(^a⁄_b+^c⁄_d)}⁄₂.

Example 1. Find a rational number between ^-4⁄₅ and ⁵⁄₆.

Solution. First, we calculate mean of (^-4⁄₅ + ⁵⁄₆) ÷ 2

= ^(-24+25)⁄₃₀ ÷ 2

= ¹⁄₃₀ × ¹⁄₂ = ¹⁄₆₀

So, ¹⁄₆₀ is a rational number between ^-4⁄₅ and ⁵⁄₆.