Chapter 8 of 14

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CBSE Class 8 Mathematics★ 8-10 marks · CBSE Class 8 Maths (Ganita Prakash Part 2, Chapter 8)

Fractions in Disguise

'Fractions in Disguise' reveals fractions hiding in unfamiliar forms — decimals, percentages, repeating decimals, and algebraic expressions. Master converting between forms, recognising equivalent fractions, and operating on algebraic rational expressions. CBSE Class 8 Maths (Ganita Prakash Part 2) Chapter 8.

⏱ 22 min readMedium difficulty✓ Free · NCERT-aligned

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By the end of this chapter you'll be able to…

1Convert between decimals, fractions, percentages, ratios
2Convert repeating decimals to fractions
3Simplify algebraic rational expressions
4Perform operations on algebraic fractions
5Recognise equivalent forms of the same value

💡

Why this chapter matters

Master converting between fractions, decimals, percentages, ratios, and algebraic expressions. Recognise the unity behind all these forms.

Before you start — revise these

A 5-minute refresher here will save you 30 minutes of confusion below.

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Proportional Reasoning

Fractions in Disguise — Class 8 Mathematics (Ganita Prakash Part 2)

"Every decimal, every percentage, every repeating digit pattern — they are all fractions wearing different clothes."

1. About the Chapter

'Fractions in Disguise' is the first chapter of Ganita Prakash Part 2. It teaches you to recognise fractions in many different forms:

Terminating decimals (0.5, 0.25)
Repeating decimals (0.333..., 0.142857...)
Percentages (25%, 60%)
Ratios (3:4)
Algebraic rational expressions (x/(x+1))

All these are 'disguised fractions'. Mastering the conversions between forms is a critical skill.

2. Decimal Fractions

Decimal Place Value

A decimal extends the place-value system to the right of the decimal point:

Tenths (1/10)
Hundredths (1/100)
Thousandths (1/1000)
And so on...

Example: 0.345 = 3/10 + 4/100 + 5/1000 = 345/1000

Converting Decimal to Fraction

0.5 = 5/10 = 1/2
0.25 = 25/100 = 1/4
0.125 = 125/1000 = 1/8
0.875 = 875/1000 = 7/8

Converting Fraction to Decimal

By long division:

3/8 = 0.375 (terminates)
1/3 = 0.333... (repeats)
5/16 = 0.3125 (terminates)
2/11 = 0.1818... (repeats)

3. Repeating Decimals — The Hidden Fractions

Notation

A bar over digits indicates they repeat:

0.3̄ means 0.3333... (the 3 repeats forever)
0.27̄ means 0.272727... (the 27 repeats)
0.142857̄ means 0.142857142857... (the 142857 repeats)

Converting Repeating Decimal to Fraction

Method (1-digit repeating): Let x = repeating decimal. Multiply by 10. Subtract.

Example: Convert 0.3̄ to fraction.

Let x = 0.333...
10x = 3.333...
10x − x = 3 ⟹ 9x = 3 ⟹ x = 3/9 = 1/3
∴ 0.3̄ = 1/3 ✓

Example: Convert 0.6̄ to fraction.

Let x = 0.666...
10x = 6.666...
9x = 6 ⟹ x = 6/9 = 2/3

Method (2-digit repeating): Multiply by 100.

Example: Convert 0.45̄ to fraction.

Let x = 0.4545...
100x = 45.4545...
99x = 45 ⟹ x = 45/99 = 5/11

General Rule

For n-digit repeating block: multiply by 10ⁿ, then subtract.

4. Mixed Repeating Decimals

A decimal might have a non-repeating part AND a repeating part:

0.2333... = 0.23̄ (23 → 3 repeats; 2 doesn't)

Method: Let x = 0.23̄ = 0.23333...

10x = 2.3333...
100x = 23.3333...
100x − 10x = 21 ⟹ 90x = 21 ⟹ x = 21/90 = 7/30

5. Percentages as Fractions

Quick Conversions

50% = 1/2
25% = 1/4
75% = 3/4
20% = 1/5
10% = 1/10
33.33% = 1/3
12.5% = 1/8

Why Percent?

'Percent' means 'per 100'. So x% = x/100 — a fraction with denominator 100.

Common Percents and Fractions

%	Fraction	Decimal
1%	1/100	0.01
5%	1/20	0.05
10%	1/10	0.1
25%	1/4	0.25
33.33%	1/3	0.333...
50%	1/2	0.5
75%	3/4	0.75
100%	1	1.0

6. Ratios as Fractions

A ratio a : b can be written as a fraction a/b.

3 : 4 = 3/4
5 : 8 = 5/8
2 : 5 = 2/5

This explains why so many ratio problems can also be solved as fraction problems.

7. Algebraic Fractions (Rational Expressions)

Definition

An expression of the form P(x)/Q(x) where P and Q are polynomials and Q ≠ 0.

Examples

x/(x+1)
(3x+5)/(2x−7)
(x²−1)/(x+1)

Simplification

Reduce common factors in numerator and denominator.

Example: Simplify (x²−1)/(x+1)

Numerator = (x+1)(x−1) (difference of squares)
= (x+1)(x−1)/(x+1)
Cancel (x+1): = x − 1

Example: Simplify (x²+3x+2)/(x²+5x+6)

Numerator: x²+3x+2 = (x+1)(x+2)
Denominator: x²+5x+6 = (x+2)(x+3)
= (x+1)(x+2)/((x+2)(x+3))
Cancel (x+2): = (x+1)/(x+3)

8. Operations on Algebraic Fractions

Addition / Subtraction

Same as numeric fractions — take LCM of denominators.

Example: 1/x + 2/y

LCM of denominators = xy
= y/(xy) + 2x/(xy) = (y + 2x)/(xy)

Multiplication

Multiply numerators, multiply denominators.

Example: (x/2)(3/y) = 3x/(2y)

Division

Multiply by reciprocal.

Example: (x/3) ÷ (5/y) = (x/3)(y/5) = xy/15

9. Worked Examples

Example 1: Decimal to Fraction

Convert 0.125 to fraction.

0.125 = 125/1000 = 1/8 ✓

Example 2: Repeating Decimal to Fraction

Convert 0.7̄ to fraction.

Let x = 0.7777...
10x = 7.7777...
9x = 7 ⟹ x = 7/9

Example 3: Mixed Repeating

Convert 0.16̄ to fraction.

Let x = 0.16666...
10x = 1.6666...
100x = 16.6666...
100x − 10x = 90x = 15
x = 15/90 = 1/6

Example 4: % to Fraction

Convert 87.5% to fraction.

87.5% = 87.5/100 = 875/1000 = 7/8

Example 5: Simplify Algebraic Fraction

Simplify (x² − 9)/(x² − 6x + 9)

Numerator: x² − 9 = (x+3)(x−3)
Denominator: x² − 6x + 9 = (x−3)²
= (x+3)(x−3)/(x−3)² = (x+3)/(x−3)

Example 6: Add Algebraic Fractions

Compute 1/(x−1) + 1/(x+1).

LCM = (x−1)(x+1) = x² − 1
= (x+1)/(x²−1) + (x−1)/(x²−1)
= (x+1+x−1)/(x²−1) = 2x/(x²−1)

Example 7: Multiply

Compute (x+2)/(x−1) × (x²−1)/(x²+4x+4).

Numerator: (x+2)(x²−1) = (x+2)(x+1)(x−1)
Denominator: (x−1)(x²+4x+4) = (x−1)(x+2)²
= (x+2)(x+1)(x−1) / ((x−1)(x+2)²)
Cancel: = (x+1)/(x+2)

Example 8: Recognising Equivalent Forms

Which of these are equal? 3/4, 0.75, 75%, 3:4

3/4 = 0.75 ✓
75% = 75/100 = 3/4 ✓
3:4 = 3/4 ✓
All equal!

10. Common Mistakes

Converting repeating decimal wrong
- 0.7̄ = 7/9 (NOT 7/10 or 0.777)
Missing the non-repeating part
- 0.16̄ (where 6 repeats) ≠ 16/99
- Correct method: 100x − 10x
Cancelling without factoring
- (x+3)/(x+5) ≠ 3/5 (cannot cancel x)
- Only cancel COMMON FACTORS
Forgetting LCM
- 1/x + 1/y ≠ 1/(x+y)
- Correct: 1/x + 1/y = (x+y)/(xy)
Dividing by 0
- x/0 undefined for ANY x
- 0/0 indeterminate

11. Tips for Mastery

For Decimals

Memorise common decimal-fraction equivalents (1/2, 1/4, 1/8, 1/3, 1/6, 1/5, 1/10)

For Repeating Decimals

Memorise the technique: multiply by 10ⁿ, subtract original

For Algebraic Fractions

ALWAYS factorise BEFORE cancelling
Use identities (a² − b² = (a+b)(a−b), etc.) to factorise

For Operations

LCM for adding/subtracting
Multiply: just multiply numerators and denominators
Divide: multiply by reciprocal

12. Real-World Applications

Banking

Interest rates often given as percentages (5.5%) — convert to fraction (11/200) for exact calculation.

Science

Concentration: 0.05 g/mL = 5%
Probability: 0.25 = 1/4 = 25% chance

Cooking

1/3 cup ≈ 0.333 cup
3/4 teaspoon = 0.75 tsp

Construction

Measurements often in fractions of an inch: 3/8", 5/16", 7/16".

Sports

Cricket strike rate: 75% = 75 runs per 100 balls
Football possession: 60% = 3/5 of time

13. Conclusion

'Fractions in Disguise' opens your eyes to the unity of mathematical forms. A fraction, a decimal, a percentage, a ratio, an algebraic expression — they are all the same thing in different costumes.

Mastery of conversion between forms is a lifelong skill. You'll use it in:

Financial calculations
Scientific data interpretation
Algebra and calculus (later)
Everyday decision-making

In Class 9, you'll extend to rational numbers' decimal expansions more formally. In Class 11, you'll work with partial fractions for calculus. The foundation laid here will support all of that.

Key formulas & results

Everything you need to memorise, in one card. Screenshot this for revision.

Repeating decimal

0.x̄ = x/9 (single digit repeating)

0.3̄ = 3/9 = 1/3

2-digit repeating

0.xȳ = xy/99

0.45̄ = 45/99 = 5/11

n-digit repeating

Block/10ⁿ−1 (denominator has n 9s)

Mixed repeating

Use 100x − 10x technique

% to fraction

x% = x/100

Algebraic addition

a/b + c/d = (ad + bc)/(bd)

Algebraic multiplication

(a/b)(c/d) = (ac)/(bd)

Algebraic division

(a/b) ÷ (c/d) = (a/b)(d/c) = (ad)/(bc)

⚠️

Common mistakes & fixes

These are the exact errors that cost students marks in board exams. Read them once, save yourself the trouble.

WATCH OUT

✗ 0.3̄ = 3/10

✓ WRONG. 0.3̄ = 1/3 (use 10x − x = 3, so x = 3/9 = 1/3).

WATCH OUT

✗ Cancelling non-factors

✓ (x+3)/(x+5) cannot be simplified. Only CANCEL COMMON FACTORS, not individual terms.

WATCH OUT

✗ 1/x + 1/y = 1/(x+y)

✓ WRONG. 1/x + 1/y = (x+y)/(xy). Take LCM first.

WATCH OUT

✗ Dividing fractions wrong

✓ a/b ÷ c/d = (a/b)(d/c), NOT (ac)/(bd). Flip the second fraction.

WATCH OUT

✗ Forgetting to factorise first

✓ (x²−1)/(x+1) simplifies after factorising x²−1 = (x+1)(x−1). Without factoring, you can't cancel.

NCERT exercises (with solutions)

Every NCERT exercise from this chapter — what it covers and how many questions to expect.

Try These - Decimals

Convert between decimal and fraction

Questions

Try These - Repeating

Convert repeating decimals

Questions

Try These - Algebraic

Simplify algebraic fractions

Questions

Ex 8.1

Exercise 8.1

Decimals, repeating decimals, percentages

Practice problems

Try each one yourself before tapping "Show solution". Active recall > rereading.

Q1EASY· Conversion

Express 87.5% as a fraction in simplest form.

▶ Show solution

✦ Answer: 87.5% = 87.5/100 = 875/1000 = 7/8 (after dividing by HCF = 125).

Q2EASY· Decimal

Convert 0.5̄ (repeating) to fraction.

▶ Show solution

✦ Answer: Let x = 0.555... Then 10x = 5.555..., so 9x = 5, giving x = 5/9.

Q3MEDIUM· Repeating

Convert 0.272̄ (where 272 repeats) to a fraction in simplest form.

▶ Show solution

Step 1 — Set up.
  Let x = 0.272272272...

Step 2 — Multiply by 1000 (since 3 digits repeat).
  1000x = 272.272272...

Step 3 — Subtract.
  1000x − x = 272.272... − 0.272...
  999x = 272
  x = 272/999

Step 4 — Simplify.
  HCF(272, 999) = 1 (272 = 16×17; 999 = 27×37 — no common factor)
  Wait, let me verify: 272 = 16×17; 999 = 3³ × 37. No common factors.
  So x = 272/999 is in lowest form.

✦ Answer: 0.272̄ = 272/999.

Q4MEDIUM· Algebraic

Simplify: (x² − 4) / (x² + x − 6)

▶ Show solution

Step 1 — Factorise numerator.
  x² − 4 = (x + 2)(x − 2) [difference of squares]

Step 2 — Factorise denominator.
  x² + x − 6: find p, q with p+q = 1, p×q = −6.
  Try 3 and −2: 3 + (−2) = 1 ✓; 3 × (−2) = −6 ✓.
  So x² + x − 6 = (x + 3)(x − 2).

Step 3 — Substitute and cancel.
  (x+2)(x−2) / ((x+3)(x−2))
  Cancel (x−2): = (x+2)/(x+3)

Step 4 — State restriction.
  Result valid for x ≠ 2 (which made denominator zero before cancellation).

✦ Answer: (x² − 4)/(x² + x − 6) = (x+2)/(x+3), for x ≠ 2.

Q5HARD· Operations

Simplify: 1/(x−1) + 1/(x+1) − 2x/(x²−1)

▶ Show solution

Step 1 — Observe common denominator.
  x² − 1 = (x − 1)(x + 1) (difference of squares)
  LCD = (x − 1)(x + 1) = x² − 1

Step 2 — Convert each fraction.
  1/(x−1) = (x+1)/(x²−1)
  1/(x+1) = (x−1)/(x²−1)
  2x/(x²−1) is already over LCD

Step 3 — Combine.
  = [(x+1) + (x−1) − 2x] / (x² − 1)
  = (x + 1 + x − 1 − 2x) / (x² − 1)
  = (2x − 2x) / (x² − 1)
  = 0 / (x² − 1)
  = 0

Step 4 — Verify.
  The three fractions cancel exactly. The expression evaluates to 0 for all x ≠ ±1.

Step 5 — State restriction.
  Result holds for x ≠ 1 and x ≠ −1.

✦ Answer: The expression equals 0 (for x ≠ ±1).

5-minute revision

The whole chapter, distilled. Read this the night before the exam.

•Decimal → fraction: 0.345 = 345/1000
•Repeating decimal: 0.x̄ = x/9; 0.xȳ = xy/99
•n digits repeat → denominator has n 9s
•Mixed repeating: 100x − 10x technique
•Percentage = /100
•Ratio a:b = fraction a/b
•Algebraic fraction: P(x)/Q(x), Q ≠ 0
•ALWAYS factorise before cancelling
•Addition/subtraction: take LCM
•Multiplication: numerator × num; denominator × den
•Division: multiply by reciprocal
•Common equivalents: 1/2=0.5=50%; 1/4=0.25=25%; 1/8=0.125=12.5%; 1/3=0.333...=33.33%

CBSE marks blueprint

Where the marks come from in this chapter — so you can plan your prep.

Typical chapter weightage: 8-10 marks per chapter

Question type	Marks each	Typical count	What it tests
MCQ / Very Short	1	2	Quick conversion; fraction-decimal-percent
Short Answer	3	2	Repeating decimal conversion; simple algebraic simplification
Long Answer	5	1	Multi-step algebraic manipulation

Prep strategy

Memorise common equivalent forms (1/2 = 0.5 = 50%, etc.)
Master the 10x − x technique for repeating decimals
Practise factorising before cancelling
Practice LCM-based addition of algebraic fractions
Always state restrictions (x ≠ ...) for algebraic simplifications

Where this shows up in the real world

This chapter isn't just an exam topic — it lives in the world around you.

Financial calculations

Compound interest formulas involve algebraic fractions. EMI calculations use ratios of payments to principal.

Physics formulas

Many laws (Coulomb's, gravitational) have rational expressions. Lens equations involve 1/f = 1/u + 1/v.

Engineering

Mechanical advantage = output/input — a ratio. Electrical resistance in parallel: 1/R = 1/R₁ + 1/R₂.

Statistics

Probabilities are fractions between 0 and 1. Percentages, decimals, fractions all represent the same statistical concepts.

Exam strategy

Battle-tested tips from teachers and toppers for this chapter.

Convert ALL forms (decimal, %, ratio) to fractions first if asked to operate
Factorise before cancelling in algebraic fractions
State restrictions for valid simplifications
Use LCM for addition/subtraction of fractions
Verify by substituting a specific value (e.g., x = 0 or x = 1)

Going beyond the textbook

For olympiad aspirants and curious learners — topics that build on this chapter.

Continued fractions (Stern-Brocot tree)
Egyptian fractions: any rational as sum of distinct unit fractions
Partial fractions (Class 11)
Farey sequences
Decimal expansion patterns of 1/n for small n

Where else this chapter is tested

CBSE board isn't the only one — other exams test this chapter too.

CBSE Class 8 School ExamVery High

Class 8 OlympiadHigh

Class 9 Algebraic IdentitiesVery High

Class 10 Rational NumbersVery High

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Questions students ask

The real ones — pulled from the Q&A community and tutor sessions.

Let x = 0.999... Then 10x = 9.999... Subtract: 9x = 9, so x = 1. This is mathematically rigorous — 0.999... is EXACTLY equal to 1 (not just approximately). Many students find this counter-intuitive, but it's a true statement about how decimals work.

ONLY when the cancelled term is a FACTOR of both numerator and denominator. So you can cancel (x+1) from (x+1)(x+2) / (x+1)(x+3). But you CANNOT cancel x from (x+1)/(x+2) — the x is part of a larger term, not a factor on its own.

When you cancel a factor like (x − 2), you're dividing by (x − 2). Division by zero is undefined. So the simplified expression is valid only when the original (x − 2) ≠ 0, i.e., x ≠ 2. This restriction is part of the complete answer.

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Fractions in Disguise

By the end of this chapter you'll be able to…

Before you start — revise these

Fractions in Disguise — Class 8 Mathematics (Ganita Prakash Part 2)

1. About the Chapter

2. Decimal Fractions

Decimal Place Value

Converting Decimal to Fraction

Converting Fraction to Decimal

3. Repeating Decimals — The Hidden Fractions

Notation

Converting Repeating Decimal to Fraction

General Rule

4. Mixed Repeating Decimals

5. Percentages as Fractions

Quick Conversions

Why Percent?

Common Percents and Fractions

6. Ratios as Fractions

7. Algebraic Fractions (Rational Expressions)

Definition

Examples

Simplification

8. Operations on Algebraic Fractions

Addition / Subtraction

Multiplication

Division

9. Worked Examples

Example 1: Decimal to Fraction

Example 2: Repeating Decimal to Fraction

Example 3: Mixed Repeating

Example 4: % to Fraction

Example 5: Simplify Algebraic Fraction

Example 6: Add Algebraic Fractions

Example 7: Multiply

Example 8: Recognising Equivalent Forms

10. Common Mistakes

11. Tips for Mastery

For Decimals

For Repeating Decimals

For Algebraic Fractions

For Operations

12. Real-World Applications

Banking

Science

Cooking

Construction

Sports

13. Conclusion

Key formulas & results

Common mistakes & fixes

NCERT exercises (with solutions)

Practice problems

5-minute revision

CBSE marks blueprint

Where this shows up in the real world

Exam strategy

Going beyond the textbook

Where else this chapter is tested

AI helpers for this chapter

Generate 5 more questions

Explain at a different level

📝 My notes

Questions students ask

Why does 0.999... = 1?

When can I cancel in an algebraic fraction?

Why state 'x ≠ value' as restrictions?

Practice more, ask doubts, find a tutor

Related chapters

A Square and A Cube

Power Play

A Story of Numbers

Quadrilaterals

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