Rational Numbers — for Class 8

What Are Rational Numbers?

A rational number is any number that can be written as $\frac{p}{q}$ , where $p$ and $q$ are integers and $q \neq 0$ . That’s the complete definition — and it’s more powerful than it looks.

Every integer is rational. Every fraction is rational. Even decimals like $0.75$ are rational because $0.75 = \frac{3}{4}$ . The only numbers that aren’t rational are things like $\sqrt{2}$ and $\pi$ , which can never be expressed as a neat fraction.

In Class 8, rational numbers are where your number system truly grows up. We move from “fractions are positive things between 0 and 1” to a full number line stretching infinitely in both directions. The rules you’ll learn here — especially properties like commutativity and distributivity — are the same ones JEE students rely on years later.

Key Terms and Definitions

Rational Number: A number of the form $\frac{p}{q}$ where $p, q \in \mathbb{Z}$ and $q \neq 0$ .

Examples: $\frac{3}{4}$ , $-\frac{7}{2}$ , $0$ , $5$ (which is $\frac{5}{1}$ ), $-3$ (which is $\frac{-3}{1}$ )

Standard Form: A rational number $\frac{p}{q}$ is in standard form when $q > 0$ and $\gcd(p, q) = 1$ (no common factors other than 1).

Example: $\frac{-6}{4}$ → standard form is $\frac{-3}{2}$

Equivalent Rational Numbers: Numbers that represent the same value. Multiply or divide both numerator and denominator by the same non-zero integer.

\frac{2}{3} = \frac{4}{6} = \frac{-4}{-6} = \frac{10}{15}

Additive Inverse: The additive inverse of $\frac{p}{q}$ is $\frac{-p}{q}$ . Their sum is zero.

Multiplicative Inverse (Reciprocal): The multiplicative inverse of $\frac{p}{q}$ (where $p \neq 0$ ) is $\frac{q}{p}$ . Their product is 1.

Properties of Rational Numbers

This is the section most students underestimate. These properties are heavily tested in MCQs — a single property violation means a number system collapses.

Closure Property

A set is closed under an operation if performing that operation on two members always gives another member of the same set.

Operation	Closed?	Example
Addition	✓ Yes	$\frac{1}{2} + \frac{1}{3} = \frac{5}{6}$ ✓
Subtraction	✓ Yes	$\frac{3}{4} - \frac{5}{4} = \frac{-2}{4} = \frac{-1}{2}$ ✓
Multiplication	✓ Yes	$\frac{2}{3} \times \frac{3}{5} = \frac{6}{15} = \frac{2}{5}$ ✓
Division	✗ No	Division by zero is excluded

Students often say “rational numbers are closed under division.” They’re not — because $\frac{0}{1} \div \frac{0}{1}$ is undefined. The correct statement: rational numbers are closed under division by non-zero rational numbers.

Commutative Property

The order doesn’t matter.

\frac{a}{b} + \frac{c}{d} = \frac{c}{d} + \frac{a}{b} \quad \checkmark

\frac{a}{b} \times \frac{c}{d} = \frac{c}{d} \times \frac{a}{b} \quad \checkmark

Subtraction and division are NOT commutative: $\frac{3}{4} - \frac{1}{2} \neq \frac{1}{2} - \frac{3}{4}$

Associative Property

How we group numbers doesn’t matter (for addition and multiplication).

\left(\frac{a}{b} + \frac{c}{d}\right) + \frac{e}{f} = \frac{a}{b} + \left(\frac{c}{d} + \frac{e}{f}\right) \quad \checkmark

Again, subtraction and division are NOT associative.

Distributive Property

This is the most-used property in actual calculations:

a \times (b + c) = a \times b + a \times c

\frac{p}{q} \times \left(\frac{r}{s} + \frac{t}{u}\right) = \frac{p}{q} \times \frac{r}{s} + \frac{p}{q} \times \frac{t}{u}

Role of Zero and One

Additive identity: $0$ — because $\frac{a}{b} + 0 = \frac{a}{b}$
Multiplicative identity: $1$ — because $\frac{a}{b} \times 1 = \frac{a}{b}$
Additive inverse of $\frac{a}{b}$ : $\frac{-a}{b}$
Multiplicative inverse of $\frac{a}{b}$ (when $a \neq 0$ ): $\frac{b}{a}$

Operations on Rational Numbers

Addition and Subtraction

Same denominator: Add/subtract numerators directly.

\frac{5}{7} + \frac{-3}{7} = \frac{5 + (-3)}{7} = \frac{2}{7}

Different denominators: Find LCM of denominators first.

\frac{3}{4} + \frac{-5}{6}

LCM of 4 and 6 = 12

= \frac{3 \times 3}{12} + \frac{-5 \times 2}{12} = \frac{9}{12} + \frac{-10}{12} = \frac{-1}{12}

Multiplication

Multiply numerators together, denominators together, then simplify.

\frac{3}{5} \times \frac{-7}{9} = \frac{3 \times (-7)}{5 \times 9} = \frac{-21}{45} = \frac{-7}{15}

Cross-cancel before multiplying — it keeps numbers small. Here, 3 and 9 share a factor of 3: $\frac{\cancel{3}}{5} \times \frac{-7}{\cancel{9}^3} = \frac{-7}{15}$ . Same answer, less arithmetic.

Division

Dividing by a rational number = multiplying by its reciprocal.

\frac{p}{q} \div \frac{r}{s} = \frac{p}{q} \times \frac{s}{r}

\frac{-4}{5} \div \frac{3}{10} = \frac{-4}{5} \times \frac{10}{3} = \frac{-40}{15} = \frac{-8}{3}

Rational Numbers on the Number Line

To place $\frac{3}{5}$ on the number line:

Identify it lies between 0 and 1 (since $0 < \frac{3}{5} < 1$ )
Divide the segment from 0 to 1 into 5 equal parts
Count 3 parts from 0

For negative rationals like $\frac{-3}{5}$ : mirror the process on the left side of 0.

Rational Numbers Between Two Rationals

Between any two rational numbers, infinitely many rational numbers exist. This is called the dense property.

Method: To find $n$ rational numbers between $\frac{a}{b}$ and $\frac{c}{d}$ , convert both to equivalent fractions with denominator $b \times d \times (n+1)$ or simply find the mean repeatedly.

Mean method: The rational number $\frac{1}{2}\left(\frac{a}{b} + \frac{c}{d}\right)$ lies between them.

Find 3 rationals between $\frac{1}{4}$ and $\frac{1}{2}$ :

Convert: $\frac{1}{4} = \frac{5}{20}$ and $\frac{1}{2} = \frac{10}{20}$

So $\frac{6}{20}, \frac{7}{20}, \frac{8}{20}$ all work. Simplified: $\frac{3}{10}, \frac{7}{20}, \frac{2}{5}$ .

Solved Examples

Example 1 — Easy (CBSE Level)

Find the additive inverse of $\frac{-7}{12}$ .

The additive inverse of $\frac{p}{q}$ is $\frac{-p}{q}$ .

Additive inverse of $\frac{-7}{12}$ = $\frac{-(-7)}{12} = \frac{7}{12}$

Check: $\frac{-7}{12} + \frac{7}{12} = \frac{0}{12} = 0$ ✓

Example 2 — Easy (CBSE Level)

Simplify: $\frac{5}{6} - \frac{-3}{8}$

Subtracting a negative = adding a positive:

\frac{5}{6} - \frac{-3}{8} = \frac{5}{6} + \frac{3}{8}

LCM(6, 8) = 24

= \frac{20}{24} + \frac{9}{24} = \frac{29}{24}

Example 3 — Medium (CBSE Level)

Using distributive property, find: $\frac{3}{7} \times \frac{4}{5} + \frac{3}{7} \times \frac{-2}{5}$

We notice $\frac{3}{7}$ is a common factor — pull it out.

= \frac{3}{7} \times \left(\frac{4}{5} + \frac{-2}{5}\right) = \frac{3}{7} \times \frac{2}{5} = \frac{6}{35}

This is where distributive property saves you time — instead of two multiplications, just one.

Example 4 — Medium (CBSE Level)

Find 5 rational numbers between $\frac{-3}{5}$ and $\frac{-1}{5}$ .

Both fractions have denominator 5. We need 5 numbers between them, but there are only 2 integers between $-3$ and $-1$ (just $-2$ ). So multiply numerator and denominator by 6 (or more):

\frac{-3}{5} = \frac{-18}{30}, \quad \frac{-1}{5} = \frac{-6}{30}

Numbers between $\frac{-18}{30}$ and $\frac{-6}{30}$ :

\frac{-17}{30}, \frac{-15}{30}, \frac{-13}{30}, \frac{-10}{30}, \frac{-7}{30}

Example 5 — Hard (CBSE Class 8 / Competition Level)

Verify: $\frac{-5}{6} \times \left[\frac{3}{4} + \frac{-7}{12}\right] = \frac{-5}{6} \times \frac{3}{4} + \frac{-5}{6} \times \frac{-7}{12}$

LHS:

\frac{3}{4} + \frac{-7}{12} = \frac{9}{12} + \frac{-7}{12} = \frac{2}{12} = \frac{1}{6}

\frac{-5}{6} \times \frac{1}{6} = \frac{-5}{36}

RHS:

\frac{-5}{6} \times \frac{3}{4} = \frac{-15}{24} = \frac{-5}{8}

\frac{-5}{6} \times \frac{-7}{12} = \frac{35}{72}

\frac{-5}{8} + \frac{35}{72} = \frac{-45}{72} + \frac{35}{72} = \frac{-10}{72} = \frac{-5}{36}

LHS = RHS = $\frac{-5}{36}$ ✓ Distributive property verified.

Exam-Specific Tips

CBSE Class 8 Pattern: Questions on rational numbers typically appear in the 1-mark (MCQ/fill-in-the-blank) and 3-mark sections. Chapter 1 (Rational Numbers) is a guaranteed 10-12 marks in most CBSE papers. Standard form, properties, and finding rationals between two numbers are the most repeated question types.

For 1-mark questions: Memorise all 8 properties in a table. Standard form questions are straightforward — reduce the fraction and ensure the denominator is positive.

For 3-mark questions: “Find $n$ rational numbers between…” is almost always asked. Practice the multiplication trick (multiply numerator and denominator by $n+1$ or higher).

For 4-mark questions: Verification of distributive property comes with full working marks. Show every step — examiners award marks for process, not just the final answer.

For CBSE marking, always convert your final answer to standard form. Writing $\frac{-6}{8}$ instead of $\frac{-3}{4}$ can cost you a half-mark in strict evaluation.

Common Mistakes to Avoid

Mistake 1 — Sign errors in subtraction

$\frac{3}{5} - \frac{-2}{5}$ is NOT $\frac{1}{5}$ . Subtracting a negative means adding: the answer is $\frac{5}{5} = 1$ .

Mistake 2 — Wrong LCM

LCM(4, 6) is 12, not 24. Always find the least common multiple. Using a larger common denominator isn’t wrong, but it creates bigger numbers and more simplification work — and more chances for arithmetic errors under exam pressure.

Mistake 3 — Forgetting standard form

$\frac{6}{-4}$ is not in standard form. The denominator must be positive: write it as $\frac{-6}{4} = \frac{-3}{2}$ .

Mistake 4 — Confusing additive and multiplicative inverse

Additive inverse of $\frac{3}{7}$ is $\frac{-3}{7}$ (flip the sign). Multiplicative inverse of $\frac{3}{7}$ is $\frac{7}{3}$ (flip the fraction). These are completely different operations. Mixing them up on a 1-mark MCQ is a very common error.

Mistake 5 — Claiming zero has a multiplicative inverse

Zero has no multiplicative inverse. $0 = \frac{0}{1}$ and the “reciprocal” would be $\frac{1}{0}$ , which is undefined. This is why we write " $q \neq 0$ " in the definition of rational numbers.

Practice Questions

Q1. Write the rational number $\frac{-36}{-48}$ in standard form.

Both numerator and denominator are negative, so: $\frac{-36}{-48} = \frac{36}{48}$

GCD(36, 48) = 12

$\frac{36}{48} = \frac{3}{4}$ ✓

Q2. Is $\frac{-3}{-7}$ the same as $\frac{3}{7}$ ? Justify.

Yes. $\frac{-3}{-7} = \frac{3}{7}$ because both numerator and denominator are negative — dividing a negative by a negative gives a positive. The standard form of both is $\frac{3}{7}$ .

Q3. Find: $\frac{-4}{9} + \frac{7}{12}$

LCM(9, 12) = 36

$\frac{-4}{9} = \frac{-16}{36}$ , $\frac{7}{12} = \frac{21}{36}$

$\frac{-16}{36} + \frac{21}{36} = \frac{5}{36}$

Q4. Multiply: $\frac{-13}{5} \times \frac{15}{-26}$

$\frac{-13}{5} \times \frac{15}{-26}$

Cross-cancel: 13 and 26 share factor 13; 5 and 15 share factor 5.

$= \frac{-\cancel{13}}{\cancel{5}} \times \frac{\cancel{15}^3}{-2\cancel{6}^2} = \frac{-1 \times 3}{1 \times (-2)} = \frac{-3}{-2} = \frac{3}{2}$

Q5. Find 4 rational numbers between $\frac{2}{3}$ and $\frac{3}{4}$ .

Multiply both by 5 (we need 4 numbers, so $\times 5$ gives room):

$\frac{2}{3} = \frac{40}{60}$ , $\frac{3}{4} = \frac{45}{60}$

Four rationals: $\frac{41}{60}, \frac{42}{60}, \frac{43}{60}, \frac{44}{60}$

Simplified: $\frac{41}{60}, \frac{7}{10}, \frac{43}{60}, \frac{22}{30}$

Q6. Using distributive property, simplify: $\frac{4}{9} \times \frac{-7}{6} + \frac{4}{9} \times \frac{1}{6}$

$= \frac{4}{9} \times \left(\frac{-7}{6} + \frac{1}{6}\right) = \frac{4}{9} \times \frac{-6}{6} = \frac{4}{9} \times (-1) = \frac{-4}{9}$

Q7. The product of two rational numbers is $\frac{-15}{28}$ . If one of them is $\frac{-5}{7}$ , find the other.

Other number = Product ÷ Known number

$= \frac{-15}{28} \div \frac{-5}{7} = \frac{-15}{28} \times \frac{7}{-5}$

$= \frac{-15 \times 7}{28 \times (-5)} = \frac{-105}{-140} = \frac{105}{140} = \frac{3}{4}$

Q8. Verify that $\frac{-2}{3}$ is the additive inverse of $\frac{2}{3}$ and $\frac{3}{2}$ is the multiplicative inverse of $\frac{2}{3}$ .

Additive inverse check: $\frac{2}{3} + \frac{-2}{3} = \frac{2-2}{3} = \frac{0}{3} = 0$ ✓

Multiplicative inverse check: $\frac{2}{3} \times \frac{3}{2} = \frac{6}{6} = 1$ ✓

Both verified.

FAQs

What is the difference between a rational number and a fraction?

Every fraction $\frac{p}{q}$ with positive $p$ and $q$ is a rational number, but not every rational number is a “fraction” in the elementary sense. Rational numbers include negatives like $\frac{-3}{4}$ and integers like $5 = \frac{5}{1}$ . So rational numbers are a broader category.

Is 0 a rational number?

Yes. $0 = \frac{0}{1}$ , which fits the form $\frac{p}{q}$ with $p = 0$ and $q = 1 \neq 0$ . Zero is rational. However, zero has no multiplicative inverse.

Are all integers rational numbers?

Yes. Any integer $n$ can be written as $\frac{n}{1}$ , satisfying the definition. So $\mathbb{Z} \subset \mathbb{Q}$ (integers are a subset of rationals).

How many rational numbers exist between two given rational numbers?

Infinitely many. Between any two distinct rational numbers, you can always find another one (take their average). This repeats infinitely — the rational numbers are dense on the number line.

Which properties do rational numbers satisfy that natural numbers do not?

The big ones: rational numbers have additive inverses (negatives exist) and multiplicative inverses for all non-zero elements. Natural numbers have neither. This makes the rational numbers a field in higher mathematics.

Why must the denominator be non-zero?

Division by zero is undefined. If $q = 0$ , then $\frac{p}{0}$ has no consistent meaning — you’d need a number that when multiplied by 0 gives $p$ , but anything times 0 is 0, so no such number exists.

How do you compare two negative rational numbers?

On the number line, a number further left is smaller. So $\frac{-5}{6} < \frac{-1}{6}$ even though 5 > 1. To compare $\frac{-3}{4}$ and $\frac{-5}{7}$ : convert to same denominator. $\frac{-3}{4} = \frac{-21}{28}$ and $\frac{-5}{7} = \frac{-20}{28}$ . Since $-21 < -20$ , we get $\frac{-3}{4} < \frac{-5}{7}$ .

What comes after rational numbers in the number system?

Irrational numbers — numbers like $\sqrt{2}$ , $\pi$ , and $e$ that cannot be expressed as $\frac{p}{q}$ . Together, rational and irrational numbers form the real numbers. You’ll meet irrationals formally in Class 9.