Class-7 Rational Numbers

Introduction to Rational Number

Positive Rational Number

Negative Rational Number

Properties of Rational Number

Comparison of Rational Number

Addition of Rational Number

Subtraction of Rational Number

Multiplication of Rational Number

Reciprocal of Rational Number

Division of Rational Number

Rational Numbers Test

Rational Number Worksheet

Answer Sheet

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.

Let's see some examples.

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

Let's see some examples.

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.

Let's see some examples.

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

Let's see some examples.

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

Let's consider ^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