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Chapter 8 · NCERT 2025-26

🎭 Fractions in Disguise

Decimals · Percentages · Rational Numbers · Operations · Properties

¾
0.75
75%
½ = 0.5
50%
−⅖
📐 Introduction: What Are Fractions in Disguise?

Have you ever noticed that ¾, 0.75, and 75% all represent the same quantity? Fractions are everywhere around us — they just wear different costumes! Sometimes they appear as decimals, sometimes as percentages, and sometimes as rational numbers written as p/q. This chapter reveals the hidden connections between all these forms.

In the NCERT Ganita textbook for Class 8 (2025-26), "Fractions in Disguise" explores how fractions transform into different avatars while keeping their value the same. Understanding this is crucial because it helps you solve problems in everyday life — from calculating discounts while shopping to understanding statistics in news reports.

🎭 The Big Idea

Fractions, decimals, percentages, and rational numbers are all different ways of expressing the same relationships. Learning to convert between them is a superpower for problem solving!

🌎 Real-Life Connection

When a shop offers "25% off," it means you pay ¾ of the price, which is the same as 0.75 times the price. Three disguises, one number!
📚 What You Will Learn
Topic Key Concepts
Fractions Revisited Proper, Improper, Mixed fractions; equivalent fractions
Decimals as Fractions Converting between decimals and fractions; terminating & recurring decimals
Percentages as Fractions Percent to fraction, fraction to percent, real-life applications
Rational Numbers Definition (p/q form), positive & negative rationals, representation
Operations Addition, subtraction, multiplication, division of rational numbers
Properties Closure, commutativity, associativity, identity, inverse, distributivity
Number Line Placing rational numbers on a number line, between any two rationals
💡 Think of It This Way: A fraction is like a person who can wear different outfits. The fraction ½ can dress up as 0.5 (decimal costume), 50% (percentage costume), or −(−1/2) (negative disguise). The person inside is always the same!
🎲 Types of Fractions: A Quick Review

Before exploring the disguises, let us recall the three main types of fractions:

✅ Proper Fraction

Numerator < Denominator
Examples: ½, ¾, 2/5, 7/10
Value is always less than 1.

🔵 Improper Fraction

Numerator ≥ Denominator
Examples: 5/3, 7/4, 9/2, 11/5
Value is always 1 or greater.

🔶 Mixed Fraction

Whole number + Proper fraction
Examples: 1½, 2¾, 3⅖
Another way to write an improper fraction.
🔄 Converting Between Types

Example 1: Convert the improper fraction 17/5 to a mixed fraction.

Step 1: Divide 17 by 5.
17 ÷ 5 = 3 remainder 2
Step 2: Write as mixed fraction: 3 2/5
(Quotient = whole part, Remainder = numerator, Divisor = denominator)

Example 2: Convert the mixed fraction 4⅗ to an improper fraction.

Step 1: Multiply the whole number by the denominator and add the numerator.
4 × 5 + 3 = 20 + 3 = 23
Step 2: Write over the same denominator: 23/5
🔀 Equivalent Fractions

Two fractions are equivalent if they represent the same value. You can create equivalent fractions by multiplying or dividing both numerator and denominator by the same non-zero number.

a/b = (a × n) / (b × n) for any non-zero n Fundamental property of equivalent fractions

Examples: ½ = 2/4 = 3/6 = 4/8 = 5/10 = 50/100. These are all the same fraction wearing different disguises!

💡 Key Insight: The simplest form of a fraction is when the numerator and denominator have no common factor other than 1. For example, 6/8 simplifies to ¾ (divide both by 2). Always simplify fractions to their lowest terms!
🔢 Decimals: Fractions in Decimal Disguise

A decimal is simply a fraction whose denominator is a power of 10 (10, 100, 1000, etc.). The decimal point separates the whole part from the fractional part.

🔄 Decimal to Fraction Conversion
Decimal As Fraction (unsimplified) Simplified Fraction Rule
0.5 5/10 ½ 1 decimal place → denominator 10
0.75 75/100 ¾ 2 decimal places → denominator 100
0.125 125/1000 ⅛ 3 decimal places → denominator 1000
2.4 24/10 12/5 Move decimal, simplify
3.25 325/100 13/4 Move decimal, simplify

Example 3: Convert 0.375 to a fraction in simplest form.

Step 1: Count the decimal places — three digits after the decimal point.
0.375 = 375/1000
Step 2: Simplify by finding the GCD of 375 and 1000.
GCD(375, 1000) = 125
Step 3: Divide both by 125.
375 ÷ 125 = 3,   1000 ÷ 125 = 8
Answer: 3/8
🔄 Fraction to Decimal Conversion

To convert a fraction to a decimal, simply divide the numerator by the denominator.

Example 4: Convert 7/8 to a decimal.

Step 1: Divide 7 by 8.
7 ÷ 8 = 0.875
Answer: 7/8 = 0.875 (a terminating decimal)
🔀 Terminating vs Recurring Decimals

✅ Terminating Decimal

The division ends with remainder 0. The decimal has a finite number of digits.
Examples: ¼ = 0.25, ⅜ = 0.375, 1/5 = 0.2
Rule: A fraction in lowest terms gives a terminating decimal only if the denominator has no prime factors other than 2 and 5.

🔄 Recurring (Repeating) Decimal

The division never ends; a block of digits repeats forever.
Examples: ⅓ = 0.333..., 1/6 = 0.1666..., 2/7 = 0.285714...
Rule: If the denominator has prime factors other than 2 and 5, the decimal will repeat.
💡 Quick Check: To tell if a fraction will terminate, check its denominator (in lowest terms). If it factors as 2a × 5b only, it terminates. Otherwise, it repeats. Example: 1/40 = 1/(23 × 5) terminates. 1/6 = 1/(2 × 3) repeats because of the factor 3.
📈 Percentages: Fractions in Percentage Disguise

The word percent comes from the Latin "per centum" meaning "per hundred." So a percentage is simply a fraction with denominator 100.

x% = x/100 Every percentage is a fraction with denominator 100
🔄 Converting Between Forms
Fraction Decimal Percentage How to Convert
½ 0.5 50% ½ × 100 = 50
¼ 0.25 25% ¼ × 100 = 25
¾ 0.75 75% ¾ × 100 = 75
⅕ 0.2 20% ⅕ × 100 = 20
⅔ 0.666... 66.67% ⅔ × 100 = 66.67
1½ 1.5 150% 3/2 × 100 = 150

Example 5: A student scored 72 out of 90 marks. What is the percentage?

Step 1: Write as a fraction: 72/90
Step 2: Simplify: 72/90 = 4/5
Step 3: Convert to percentage: 4/5 × 100 = 80%

Example 6: A shirt costs ₹800 and is on a 15% discount. What is the sale price?

Step 1: Discount amount = 15% of 800 = (15/100) × 800 = ₹120
Step 2: Sale price = 800 − 120 = ₹680
Shortcut: You pay (100% − 15%) = 85% of the price = 0.85 × 800 = ₹680
💡 Conversion Shortcuts:
• Fraction → Percentage: Multiply by 100
• Percentage → Fraction: Divide by 100, then simplify
• Decimal → Percentage: Move decimal 2 places right (multiply by 100)
• Percentage → Decimal: Move decimal 2 places left (divide by 100)
🔢 Rational Numbers: The Ultimate Disguise

A rational number is any number that can be expressed in the form p/q, where p and q are integers and q ≠ 0. The word "rational" comes from "ratio."

Rational Number = p/q, where p, q ∈ ℤ and q ≠ 0 Every fraction is a rational number, and so is every integer!
📚 Examples of Rational Numbers

🎲 Fractions

½, −¾, 5/7, −11/3
Already in p/q form!

🔢 Integers

5 = 5/1, −3 = −3/1, 0 = 0/1
Every integer is a rational number.

📈 Decimals (Terminating)

0.5 = 1/2, 2.75 = 11/4, −0.8 = −4/5
Terminating decimals are always rational.

🔄 Decimals (Recurring)

0.333... = 1/3, 0.1666... = 1/6
Repeating decimals are also rational.
🔴 Positive and Negative Rationals

A rational number is positive if both p and q have the same sign (both positive or both negative). It is negative if p and q have opposite signs.

Rational Number Standard Form Positive or Negative?
3/5 3/5 Positive (both positive)
−3/5 −3/5 Negative (opposite signs)
3/(−5) −3/5 Negative (opposite signs)
(−3)/(−5) 3/5 Positive (both negative = positive)
0/7 0 Neither positive nor negative
💡 Standard Form: A rational number is in standard form when (i) the denominator is positive, and (ii) the numerator and denominator have no common factor other than 1. For example, −6/8 in standard form is −3/4.
💡 Is π Rational? No! π = 3.14159... is a non-terminating, non-repeating decimal. It cannot be written as p/q with integer p and q, so π is irrational. Similarly, √2 = 1.41421... is irrational. But every fraction you encounter in this chapter IS rational!
⚙ Operations on Rational Numbers
➕ Addition of Rational Numbers

Same Denominator

a/b + c/b = (a + c)/b
Just add the numerators!
Example: 2/7 + 3/7 = 5/7

Different Denominators

Find the LCM, convert to equivalent fractions with the same denominator, then add.
Example: ⅓ + ¼ = 4/12 + 3/12 = 7/12

Example 7: Add −2/3 and 5/4.

Step 1: Find LCM of 3 and 4 → LCM = 12
Step 2: Convert to equivalent fractions.
−2/3 = −8/12,   5/4 = 15/12
Step 3: Add: −8/12 + 15/12 = 7/12
Answer: 7/12
➖ Subtraction of Rational Numbers

To subtract, add the additive inverse (negative) of the second number.

a/b − c/d = a/b + (−c/d) Subtraction = Addition of the negative

Example 8: Subtract 3/5 from 7/10.

Step 1: 7/10 − 3/5 = 7/10 − 6/10 (convert 3/5 to 6/10)
Step 2: = (7 − 6)/10 = 1/10
✕ Multiplication of Rational Numbers
(a/b) × (c/d) = (a × c) / (b × d) Multiply numerators together, multiply denominators together

Example 9: Multiply −3/7 by 2/5.

Step 1: (−3/7) × (2/5) = (−3 × 2) / (7 × 5)
Step 2: = −6/35
Answer: −6/35
➗ Division of Rational Numbers

To divide by a rational number, multiply by its reciprocal (multiplicative inverse).

(a/b) ÷ (c/d) = (a/b) × (d/c) Dividing = Multiplying by the reciprocal (flip the second fraction)

Example 10: Divide 4/9 by −2/3.

Step 1: Reciprocal of −2/3 is −3/2.
Step 2: (4/9) × (−3/2) = (4 × −3) / (9 × 2) = −12/18
Step 3: Simplify: −12/18 = −2/3
💡 KCF Rule for Division: Keep the first fraction, Change the operation to multiplication, Flip the second fraction. Keep → Change → Flip!
💡 Properties of Rational Numbers

Rational numbers follow several important mathematical properties. Understanding these helps you simplify calculations and verify answers.

Property Addition Multiplication
Closure a/b + c/d is rational ✅ (a/b) × (c/d) is rational ✅
Commutative a + b = b + a ✅ a × b = b × a ✅
Associative (a+b)+c = a+(b+c) ✅ (a×b)×c = a×(b×c) ✅
Identity a + 0 = a (additive identity = 0) a × 1 = a (multiplicative identity = 1)
Inverse a + (−a) = 0 (additive inverse) a × (1/a) = 1 (multiplicative inverse, a ≠ 0)
📚 Detailed Explanation of Each Property

🔒 Closure Property

When you add, subtract, or multiply two rational numbers, the result is always a rational number. For division, the result is rational as long as the divisor is not zero. Example: ½ + ⅓ = 5/6 (rational).

🔀 Commutative Property

You can swap the order: a + b = b + a and a × b = b × a. Example: ⅔ + ⅘ = ⅘ + ⅔. Note: Subtraction and division are NOT commutative! (5 − 3 ≠ 3 − 5)

🔗 Associative Property

Grouping does not matter: (a+b)+c = a+(b+c) and (a×b)×c = a×(b×c). Example: (½ + ⅓) + ¼ = ½ + (⅓ + ¼). Note: Subtraction and division are NOT associative!

🔭 Distributive Property

Multiplication distributes over addition: a × (b + c) = a×b + a×c. Example: ½ × (⅓ + ¼) = ½×⅓ + ½×¼ = 1/6 + 1/8 = 7/24.

🎯 Identity Elements

Additive identity is 0: any number + 0 = same number.
Multiplicative identity is 1: any number × 1 = same number.

🔄 Inverse Elements

Additive inverse of a/b is −a/b: their sum is 0.
Multiplicative inverse (reciprocal) of a/b is b/a: their product is 1.
Note: 0 has no multiplicative inverse (you cannot divide by 0)!
💡 Why Does This Matter? These properties let you rearrange and simplify expressions. For instance, to quickly compute 17 × 98, use the distributive property: 17 × (100 − 2) = 1700 − 34 = 1666. Much easier than multiplying directly!
📏 Rational Numbers on the Number Line

Every rational number has a unique position on the number line. To place a rational number on the number line:

  1. Identify the two consecutive integers between which the rational number lies.
  2. Divide the interval between those integers into as many equal parts as the denominator indicates.
  3. Count from the left integer as many parts as the numerator indicates.

Placing ¾ on the Number Line

0
¼
½
¾
1

The segment from 0 to 1 is divided into 4 equal parts. The 3rd mark represents ¾.

🔢 Negative Rational Numbers on the Number Line

Placing −⅔ on the Number Line

−1
−⅔
−⅓
0

The segment from −1 to 0 is divided into 3 equal parts. The 1st mark from −1 represents −⅔.

💡 Between Any Two Rational Numbers

Here is a beautiful fact: between any two rational numbers, there are infinitely many other rational numbers! This is called the dense property of rational numbers.

Example 11: Find three rational numbers between ¼ and ½.

Method: Convert to equivalent fractions with a larger common denominator.
¼ = 4/16,   ½ = 8/16
Three numbers between 4/16 and 8/16:
5/16, 6/16 (= 3/8), 7/16
Alternative Method: The average of two rationals is always between them.
Average of ¼ and ½ = (¼ + ½)/2 = (¾)/2 = 3/8
💡 Quick Method: To find a rational number between a/b and c/d, take their mean: (a/b + c/d) / 2. This always gives a number exactly halfway between them. Repeat the process to find as many as you want!
✏️ Worked Examples & Word Problems

Problem 1: A recipe needs ⅔ cup of sugar. If Meena wants to make 1½ times the recipe, how much sugar does she need?

Step 1: Convert 1½ to improper fraction: 3/2
Step 2: Multiply: ⅔ × 3/2 = (2 × 3)/(3 × 2) = 6/6 = 1 cup

Problem 2: Ravi spent ⅖ of his salary on rent and ¼ on food. What fraction of his salary is left?

Step 1: Total spent = ⅖ + ¼
LCM of 5 and 4 = 20
= 8/20 + 5/20 = 13/20
Step 2: Fraction left = 1 − 13/20 = 20/20 − 13/20 = 7/20

Problem 3: The temperature at 6 AM was −3½°C. By noon it rose by 8¾°C. What was the temperature at noon?

Step 1: Convert to improper fractions.
−3½ = −7/2,   8¾ = 35/4
Step 2: Add: −7/2 + 35/4 = −14/4 + 35/4 = 21/4
Step 3: Convert to mixed: 21/4 = 5¼°C

Problem 4: A tank is ⅗ full. If 120 litres of water are in the tank, what is its total capacity?

Step 1: Let total capacity = x litres.
⅗ of x = 120
Step 2: x = 120 ÷ (3/5) = 120 × (5/3) = 600/3 = 200 litres

Problem 5: Verify the distributive property: ½ × (⅔ + ¾).

LHS: ½ × (⅔ + ¾) = ½ × (8/12 + 9/12) = ½ × 17/12 = 17/24
RHS: ½ × ⅔ + ½ × ¾ = 2/6 + 3/8 = 8/24 + 9/24 = 17/24
Conclusion: LHS = RHS = 17/24 ✅ — Distributive property verified!
📊 Chapter Summary
📋 Key Conversion Formulas
Fraction → Decimal: Divide numerator by denominator ¾ = 3 ÷ 4 = 0.75
Decimal → Fraction: Count decimal places, put over power of 10, simplify 0.375 = 375/1000 = 3/8
Fraction → Percentage: Multiply by 100 ⅖ × 100 = 40%
Percentage → Fraction: Divide by 100, simplify 65% = 65/100 = 13/20
📚 Operations Summary
Operation Rule Example
Addition Same denominator: add numerators. Different: find LCM first. ⅓ + ⅙ = 2/6 + 1/6 = 3/6 = ½
Subtraction Add the additive inverse. ¾ − ½ = ¾ − 2/4 = ¼
Multiplication Multiply numerators, multiply denominators. ⅔ × ⅗ = 6/15 = ⅖
Division Multiply by the reciprocal. ⅔ ÷ ⅘ = ⅔ × &frac54; = 10/12 = 5/6
⚠️ Common Mistakes to Avoid

❌ Adding Across

WRONG: ⅓ + ¼ = 2/7.
RIGHT: Find LCM first! ⅓ + ¼ = 4/12 + 3/12 = 7/12.

❌ Forgetting the Reciprocal

When dividing fractions, always flip the second fraction and multiply. Do not just divide numerators and denominators separately.

❌ Sign Errors

Remember: (−) × (−) = (+) and (−) × (+) = (−). Be careful with signs in every step!

❌ Not Simplifying

Always simplify your final answer to lowest terms. Check if the numerator and denominator share any common factors.
💡 Final Revision Mantra:
• Fraction = Decimal = Percentage — Same value, different disguise
• Rational number = p/q where q ≠ 0
• Add/Subtract → Need common denominator
• Multiply → Straight across (top × top, bottom × bottom)
• Divide → Keep, Change, Flip (KCF)
• Number line → Infinitely many rationals between any two
🧠 Multiple Choice Questions (10 MCQs)

Test your understanding! Click on an option to check your answer. Track your score at the bottom.

  • Q1. The decimal 0.625 expressed as a fraction in simplest form is:
    • a) 625/100
    • b) 5/8
    • c) 3/5
    • d) 25/40
    ✅ Answer: (b) 5/8 — 0.625 = 625/1000 = 5/8 (divide by 125).
  • Q2. Which of the following is NOT a rational number?
    • a) −3/7
    • b) 0
    • c) √2
    • d) 5
    ✅ Answer: (c) √2 — √2 is irrational; it cannot be expressed as p/q with integers p and q.
  • Q3. 45% expressed as a fraction in simplest form is:
    • a) 45/100
    • b) 4/5
    • c) 9/20
    • d) 9/10
    ✅ Answer: (c) 9/20 — 45% = 45/100 = 9/20 (divide by 5).
  • Q4. The additive inverse of −5/9 is:
    • a) −9/5
    • b) 5/9
    • c) 9/5
    • d) −5/9
    ✅ Answer: (b) 5/9 — The additive inverse of a number is the number that gives 0 when added to it. (−5/9) + 5/9 = 0.
  • Q5. What is ⅔ ÷ &frac49;?
    • a) 8/27
    • b) 6/12
    • c) 3/2
    • d) 2/3
    ✅ Answer: (c) 3/2 — ⅔ ÷ &frac49; = ⅔ × 9/4 = 18/12 = 3/2.
  • Q6. The multiplicative identity for rational numbers is:
    • a) 1
    • b) 0
    • c) −1
    • d) 1/2
    ✅ Answer: (a) 1 — Any rational number multiplied by 1 gives the same number: a × 1 = a.
  • Q7. Which decimal is equivalent to ⅙?
    • a) 0.15
    • b) 0.1666...
    • c) 0.6
    • d) 0.16
    ✅ Answer: (b) 0.1666... — 1 ÷ 6 = 0.16666... (the 6 repeats forever). This is a recurring decimal.
  • Q8. A rational number between ⅓ and ½ is:
    • a) ¼
    • b) 2/3
    • c) 5/12
    • d) 1/6
    ✅ Answer: (c) 5/12 — ⅓ = 4/12 and ½ = 6/12. Since 4/12 < 5/12 < 6/12, the number 5/12 lies between them.
  • Q9. (−3/5) × (−10/9) equals:
    • a) −30/45
    • b) 2/3
    • c) −2/3
    • d) 30/45
    ✅ Answer: (b) 2/3 — (−3/5) × (−10/9) = 30/45 = 2/3. Negative × Negative = Positive.
  • Q10. If a shirt originally costs ₹600 and is sold at 20% discount, the selling price is:
    • a) ₹120
    • b) ₹500
    • c) ₹480
    • d) ₹580
    ✅ Answer: (c) ₹480 — Discount = 20% of 600 = 120. Selling price = 600 − 120 = ₹480.
📚 NCERT Questions & Answers
✍️ Fill in the Blanks
1. The decimal form of ⅜ is ___________.
0.375 — Divide 3 by 8: 3 ÷ 8 = 0.375.
2. The additive identity for rational numbers is ___________.
0 — Adding 0 to any rational number gives the same number.
3. 0.8 expressed as a fraction in simplest form is ___________.
4/5 — 0.8 = 8/10 = 4/5.
4. 125% expressed as a decimal is ___________.
1.25 — 125% = 125/100 = 1.25.
5. The reciprocal (multiplicative inverse) of −7/3 is ___________.
−3/7 — The reciprocal flips the fraction: (−7/3) × (−3/7) = 21/21 = 1.
✅ True or False
1. Every integer is a rational number.
True — Any integer n can be written as n/1, which is in the form p/q.
2. Subtraction of rational numbers is commutative.
False — a − b ≠ b − a in general. For example, 5 − 3 = 2, but 3 − 5 = −2.
3. 0.333... is a rational number.
True — 0.333... = ⅓, which is in the form p/q. All repeating decimals are rational.
4. The product of a rational number and its reciprocal is always 0.
False — The product of a rational number and its reciprocal is always 1 (not 0). For example, ⅔ × 3/2 = 1.
5. Between any two rational numbers, there are exactly 10 rational numbers.
False — Between any two rational numbers, there are infinitely many rational numbers (dense property).
✍️ Short Answer Questions
  • Q1. Express 0.45 as a fraction in simplest form.
    0.45 = 45/100 = 9/20 (dividing numerator and denominator by 5).
  • Q2. Find the value of (−4/7) + (2/3).
    LCM of 7 and 3 = 21.
    −4/7 = −12/21,   2/3 = 14/21
    −12/21 + 14/21 = 2/21
  • Q3. What fraction of an hour is 45 minutes? Express as a percentage too.
    45 minutes out of 60 minutes = 45/60 = ¾.
    As a percentage: ¾ × 100 = 75%.
  • Q4. Find the multiplicative inverse of −5/11.
    The multiplicative inverse (reciprocal) is −11/5.
    Verification: (−5/11) × (−11/5) = 55/55 = 1 ✅
  • Q5. Find three rational numbers between −1 and 0.
    −1 = −4/4 and 0 = 0/4.
    Three rational numbers: −3/4, −1/2 (−2/4), −1/4.
    Many other answers are possible!
📖 Long Answer Questions
Q1. Verify the distributive property of multiplication over addition for the rational numbers a = ½, b = ⅔, c = ¾.
We need to verify: a × (b + c) = a × b + a × c

LHS: a × (b + c) = ½ × (⅔ + ¾)
= ½ × (8/12 + 9/12) = ½ × 17/12 = 17/24

RHS: a × b + a × c = ½ × ⅔ + ½ × ¾
= 2/6 + 3/8 = ⅓ + 3/8
LCM of 3 and 8 = 24
= 8/24 + 9/24 = 17/24

Conclusion: LHS = RHS = 17/24 ✅
The distributive property is verified.
Q2. A farmer has a rectangular field of length 7½ metres and width 4⅔ metres. Find the area and the perimeter of the field.
Convert to improper fractions:
Length = 7½ = 15/2 m,   Width = 4⅔ = 14/3 m

Area = Length × Width = 15/2 × 14/3 = (15 × 14)/(2 × 3) = 210/6 = 35 m²

Perimeter = 2 × (Length + Width) = 2 × (15/2 + 14/3)
LCM of 2 and 3 = 6
= 2 × (45/6 + 28/6) = 2 × 73/6 = 146/6 = 24⅓ m
Q3. In a class of 50 students, 30% like cricket, ⅖ like football, and the rest like basketball. How many students like each sport? Express each as a fraction of the total.
Cricket: 30% of 50 = 30/100 × 50 = 15 students (fraction: 30/100 = 3/10)

Football: ⅖ of 50 = 2/5 × 50 = 20 students (fraction: 2/5 = 40/100 = 40%)

Basketball: 50 − 15 − 20 = 15 students
Fraction: 15/50 = 3/10 = 30%

Summary: Cricket = 3/10, Football = 2/5, Basketball = 3/10
Verification: 3/10 + 2/5 + 3/10 = 3/10 + 4/10 + 3/10 = 10/10 = 1 ✅
Q4. Find 5 rational numbers between −½ and ½. Represent them on the number line and arrange in ascending order.
Step 1: Convert to equivalent fractions with a larger denominator.
−½ = −6/12,   ½ = 6/12

Step 2: Choose 5 numbers between −6/12 and 6/12:
−5/12, −1/4 (−3/12), 0, 1/6 (2/12), 5/12

Ascending order: −5/12 < −1/4 < 0 < 1/6 < 5/12

On the number line: These points lie in order from left to right between −½ and ½. Mark the segment from −½ to ½ and divide it into 12 equal parts. Then place each rational number at the appropriate position.
Q5. Explain the closure property of rational numbers for all four operations with examples. For which operations does it fail?
Closure property means that performing an operation on two members of a set always produces a result that is also a member of the same set.

1. Addition ✅ (Closed):
½ + ⅓ = 5/6 (rational). The sum of any two rationals is always rational.

2. Subtraction ✅ (Closed):
¾ − ½ = ¼ (rational). The difference of any two rationals is always rational.

3. Multiplication ✅ (Closed):
⅔ × ⅘ = 8/15 (rational). The product of any two rationals is always rational.

4. Division ❌ (NOT always closed):
¾ ÷ ½ = 3/2 (rational) ✅
But ¾ ÷ 0 is undefined (not a rational number) ❌

Conclusion: Rational numbers are closed under addition, subtraction, and multiplication. They are closed under division only when the divisor is not zero.
🌟 Fun Facts & Did You Know?

🇮🇳 India & Zero

The concept of zero as a number was formalised by Brahmagupta in 628 CE. Without zero, we could not have 0 as the additive identity or represent rational numbers like 0/5 = 0!

🍴 Pizza Math

If you eat 2 slices of an 8-slice pizza, you have eaten ¼ = 0.25 = 25% of the pizza. That is the same fraction no matter how big or small the pizza is! Fractions describe proportions, not absolute sizes.

💰 Stock Markets

When the news says "the stock fell 3.5%," it means the stock lost 3.5/100 = 7/200 of its value. Every financial calculation uses fractions, decimals, and percentages interchangeably!

🔮 Ancient Egypt

Ancient Egyptians only used unit fractions (fractions with numerator 1), like ½, ⅓, ¼. They wrote ⅔ as ½ + 1/6. Their fraction system was much harder to use than ours!

🎮 Game Scores

When a game shows "Level 75% complete," it means you have finished ¾ of the level. Game developers use percentages (fractions in disguise!) to display progress bars and health bars.

🔢 Infinite Rationals

Between 0 and 1, there are infinitely many rational numbers. In fact, between any two rational numbers, no matter how close, there are infinitely many more! You can never "run out" of rational numbers.
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