CBSE Class 8 Maths — Rational Numbers

Chapter Overview & Weightage

Rational Numbers is Chapter 1 of CBSE Class 8 Maths. It carries approximately 5–8 marks in the annual exam and forms the foundational number theory for Class 9 and beyond (real numbers, number line, irrational numbers).

Board exams test: (1) properties of rational numbers (closure, commutativity, associativity), (2) finding rational numbers between two given rationals, (3) standard form and comparison. The properties question often appears as fill-in-the-blank or match-the-column for 2–3 marks.

Topic	Typical marks
Properties of rational numbers	2–3 marks
Operations on rational numbers	2–3 marks
Rational numbers between two rationals	1–2 marks
Word problems	2 marks

Key Concepts You Must Know

Rational number: A number expressible as $\frac{p}{q}$ where $p, q$ are integers and $q \neq 0$ .
Standard form: A rational number is in standard form when the denominator is positive and GCD(p, q) = 1. Example: $\frac{-3}{4}$ is standard; $\frac{-6}{-8}$ is not.
Equivalent rational numbers: $\frac{p}{q} = \frac{mp}{mq}$ for any non-zero integer $m$ .
Additive inverse: The additive inverse of $\frac{p}{q}$ is $\frac{-p}{q}$ (their sum = 0).
Multiplicative inverse (reciprocal): The reciprocal of $\frac{p}{q}$ is $\frac{q}{p}$ (their product = 1).
Density property: Between any two rational numbers, there are infinitely many rational numbers.

Properties of Rational Numbers

Property	Addition	Multiplication
Closure	✓	✓
Commutativity	✓: $a+b = b+a$	✓: $a \times b = b \times a$
Associativity	✓: $(a+b)+c = a+(b+c)$	✓: $(ab)c = a(bc)$
Identity element	0 ( $a+0 = a$ )	1 ( $a \times 1 = a$ )
Inverse element	$-a$	$1/a$ (for $a \neq 0$ )
Distributivity	$a(b+c) = ab + ac$ — yes

Note: Subtraction and division of rational numbers are NOT commutative or associative.

Important Formulas

Method 1 (mean): $\frac{a+b}{2}$ is always between $a$ and $b$ .

Method 2 (common denominator): Convert to common denominator, then pick numerators between them.

To find $n$ rationals between $a$ and $b$ : Find $\frac{a(n+1) + b}{n+1}$ — or more simply, take $n$ equally spaced values using $d = \frac{b-a}{n+1}$ .

\frac{p}{q} + \frac{r}{s} = \frac{ps + qr}{qs}

\frac{p}{q} \times \frac{r}{s} = \frac{pr}{qs}

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

Solved Previous Year Questions

PYQ 1 — Properties (2 marks)

Q: Name the property used: $\frac{-3}{5} + \left(\frac{2}{7} + \frac{-5}{8}\right) = \left(\frac{-3}{5} + \frac{2}{7}\right) + \frac{-5}{8}$

Answer: Associativity of addition for rational numbers.

PYQ 2 — Finding rational numbers between two rationals (3 marks)

Q: Find 3 rational numbers between $\frac{1}{4}$ and $\frac{1}{2}$ .

Solution:

Convert to common denominator 8: $\frac{1}{4} = \frac{2}{8}$ and $\frac{1}{2} = \frac{4}{8}$ .

We need numbers between $\frac{2}{8}$ and $\frac{4}{8}$ . The only option is $\frac{3}{8}$ here — only one number. So convert to denominator 16:

$\frac{1}{4} = \frac{4}{16}$ and $\frac{1}{2} = \frac{8}{16}$ .

Three numbers between them: $\frac{5}{16}$ , $\frac{6}{16} = \frac{3}{8}$ , $\frac{7}{16}$ .

Answer: $\frac{5}{16}, \frac{3}{8}, \frac{7}{16}$ (any 3 valid values between $\frac{1}{4}$ and $\frac{1}{2}$ ).

PYQ 3 — Word problem (2 marks)

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

Solution:

Let the other number = $x$ .

\frac{-5}{7} \times x = \frac{-15}{28}

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

The other number is $\mathbf{\frac{3}{4}}$ .

Difficulty Distribution

Level	Marks	Type
Easy	1 mark	Fill in blank with property name, additive inverse
Medium	2–3 marks	Operations, finding rationals between two
Hard	4–5 marks	Word problem, combining multiple operations

Expert Strategy

For finding rational numbers between two rationals, always escalate the denominator if you don’t have enough room. Going from $\frac{1}{4}$ and $\frac{1}{2}$ to denominator 8 gives only 1 number in between ( $\frac{3}{8}$ ). Going to 16 gives 3. Going to 100 gives many. There are always infinitely many — you just need to look at a finer scale.

For property questions, remember the landmark: rational numbers are NOT closed under division (dividing by 0 is undefined) — don’t write “closure holds for division.”

For operations, always reduce to standard form at the end. If your answer is $\frac{6}{8}$ , simplify to $\frac{3}{4}$ .

Common Traps

Trap 1: Rational numbers are closed under division. They are NOT — division by zero is undefined. The correct statement is: rational numbers are closed under addition, subtraction, and multiplication; division is closed only when the divisor is non-zero.

Trap 2: Adding fractions without common denominators. $\frac{1}{3} + \frac{1}{4} \neq \frac{2}{7}$ . Always find the LCM first.

Trap 3: Forgetting to simplify the final answer. Examiners expect standard form. An answer like $\frac{12}{16}$ will lose marks in CBSE if not simplified to $\frac{3}{4}$ .

Trap 4: Confusing additive inverse with multiplicative inverse. Additive inverse of $\frac{2}{3}$ is $\frac{-2}{3}$ . Multiplicative inverse (reciprocal) is $\frac{3}{2}$ . These are different and both appear in exam questions.