Class 8 Rational Numbers Worksheet PDF: Properties and Operations

Rational numbers worksheet for Class 8 with 24 comprehensive problems. Learn all 8 properties, operations (addition, subtraction, multiplication, division), number line representation, and real-world applications.

Understanding Rational Numbers

A rational number is any number that can be expressed in the form p/q, where p and q are integers and q ≠ 0.

Examples: 3/4, -5/7, 2 (can be written as 2/1), 0 (can be written as 0/1), -3, 0.5 (= 1/2)

Important Properties

Closure Property: The sum, difference, and product of two rational numbers is always a rational number. Example: 1/2 + 1/3 = 5/6 (rational)

Commutative Property:

• Addition: a + b = b + a
• Multiplication: a × b = b × a

Associative Property:

• Addition: (a + b) + c = a + (b + c)
• Multiplication: (a × b) × c = a × (b × c)

Additive Identity: a + 0 = a (Zero is the additive identity)

Multiplicative Identity: a × 1 = a (One is the multiplicative identity)

Additive Inverse: For every rational number a/b, there exists -(a/b) such that a/b + (-(a/b)) = 0

Multiplicative Inverse (Reciprocal): For every non-zero rational number a/b, there exists b/a such that a/b × b/a = 1

Distributive Property: a × (b + c) = (a × b) + (a × c)

Operations on Rational Numbers

Addition/Subtraction: Find LCM of denominators, convert to like fractions, then add/subtract numerators

Multiplication: Multiply numerators and multiply denominators: (a/b) × (c/d) = (a×c)/(b×d)

Division: Multiply by reciprocal: (a/b) ÷ (c/d) = (a/b) × (d/c)

Rational Numbers on Number Line

Every rational number can be represented on a number line. Between any two rational numbers, there are infinitely many rational numbers.

Solved Example

Problem: Find: -2/3 + 5/6 - 1/2

Solution:

Step 1: Find LCM of denominators (3, 6, 2)LCM(3, 6, 2) = 6

Step 2: Convert to equivalent fractions with denominator 6-2/3 = (-2 × 2)/(3 × 2) = -4/65/6 = 5/6 (already has denominator 6)-1/2 = (-1 × 3)/(2 × 3) = -3/6

Step 3: Add/Subtract numerators (keep same denominator)-4/6 + 5/6 - 3/6 = (-4 + 5 - 3)/6 = -2/6

Step 4: Simplify to lowest terms-2/6 = -1/3

Key Points: Always find LCM of denominators first when adding/subtracting. Convert each fraction to equivalent fraction with LCM as denominator. Add or subtract only the numerators, keep the denominator same. Always simplify to lowest terms.

Sample Practice Problems

Write three rational numbers between 1 and 2.

Find: 1/4 + 1/4

What is the additive inverse of -3/7?

Find the multiplicative inverse (reciprocal) of 5/8.

Find: 3/5 + (-2/3)

Simplify: -5/6 - 7/9

Find: (-2/5) × (3/7) × (-5/6)

Word Problem: Rani ate 1/4 of a pizza and Raj ate 1/3 of the same pizza. What fraction of the pizza did they eat together?

Divide: 5/6 ÷ (-2/3)

Find five rational numbers between -1/2 and -1/3.

Simplify: (2/3 + 1/6) - (3/4 - 1/2)

Real-life Problem: A water tank is 3/4 full. If 1/6 of the water is used, what fraction of the tank is still filled?

Simplify using distributive property: -3/5 × (7/2 - 2/3)

The sum of two rational numbers is -3/5. If one of them is 2/3, find the other.

Complex Problem: A rectangular field is 7/4 km long and 3/2 km wide. Find its area in square kilometers. If the cost of fencing is ₹150 per km, what is the total cost of fencing the field?

Verify the distributive property for: a = 1/2, b = -2/3, c = 3/4. Show that: a × (b + c) = (a × b) + (a × c)

Challenge Problem: Three friends Amit, Priya, and Kabir shared a cake. Amit ate 2/5 of the cake, Priya ate 1/4 of the remaining cake, and Kabir ate 1/2 of what was left after Priya. What fraction of the original cake is still remaining?

Scoring Guide

20-24 correct: Excellent! Outstanding! Move on to linear equations in one variable, algebraic expressions, and advanced fraction problems.

15-19 correct: Very Good! Great work! Practice more complex word problems and mixed operations. Focus on problems requiring multiple steps and property verification.

10-14 correct: Good Effort! Keep practicing! Memorize all properties with examples. Practice finding LCM for addition/subtraction and converting to reciprocals for division. Do 15 problems daily.

0-9 correct: Keep Trying! Review the concept section carefully. Start with addition of simple fractions with same denominators, then different denominators. Master one operation before moving to the next.

Tips for Improvement

Master LCM: Quick LCM finding is essential for adding/subtracting rational numbers

Remember reciprocals: Division means multiply by reciprocal (flip the second fraction)

Simplify always: Always reduce your answer to the lowest terms (divide by HCF)

Sign rules: Negative × Negative = Positive; Negative × Positive = Negative

Property cards: Create flashcards for all 8 properties with examples

Number line practice: Draw number lines and mark fractions to visualize better

Word problems: Identify the operation needed (together = add, left = subtract, of = multiply, per/each = divide)

Common Mistakes to Avoid

Adding/subtracting numerators AND denominators (WRONG: 1/2 + 1/3 ≠ 2/5)

Forgetting to find LCM before adding/subtracting unlike fractions

Not simplifying the final answer to lowest terms

Confusing additive inverse with multiplicative inverse (reciprocal)

Wrong signs: forgetting that -(-a) = +a

In division, multiplying by the same fraction instead of its reciprocal

In word problems: "of" means multiply, not add!

Thinking 0 has a multiplicative inverse (it doesn't - can't divide by 0!)

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