Chapter 8 of 14

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IGCSE 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 fractions, decimals and percentages, and say which fractions convert exactly
2Find a percentage of a quantity, and reverse the process to find the whole from a part
3Use the multiplier form — increase by r% means × (1 + r/100), decrease means × (1 − r/100)
4Reverse a percentage change by dividing by the multiplier, never by adding the percentage back
5Distinguish 'fallen by 85%' from 'fallen to 85%', and '250% of' from 'increased by 250%'
6Compute profit and loss percentages, stating whether the base is cost price or revenue
7Chain percentage changes by multiplying multipliers, and explain why they do not simply add
8Compare simple and compound interest, and explain why compounding depends on how often interest is added back

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Why this chapter matters

The chapter's title is its main idea: a percentage is just a fraction wearing a disguise, with the denominator fixed at 100. Fixing the denominator is what makes comparison possible — 9/34 against 13/45 tells you nothing at a glance, but 26.47% against 28.88% is instantly readable. Everything that follows (discounts, GST, inflation, interest) is that one idea applied to money and time.

Before you start — revise these

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

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Fractions — simplifying, and converting to decimals

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Ratio and proportion (Chapter 7, Proportional Reasoning — 1)

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Multiplying and dividing by decimals

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Forming and solving a simple linear equation

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.

Fraction to percentage

(a/b) × 100 %

Exact only when b divides 100 — that is, when b is built from 2s and 5s.

Percentage of a quantity

p% of q = (p/100) × q

Find 1% or 10% first, then scale — far faster than multiplying by a decimal.

Percentage change

(change ÷ ORIGINAL) × 100

The base is always the starting value, for both rises and falls.

Increase / decrease multiplier

× (1 + r/100)  or  × (1 − r/100)

18% GST is × 1.18; a 25% discount is × 0.75.

Reversing a change

divide by the multiplier

₹4,40,000 after a 15% discount means the original was 440000 ÷ 0.85.

Chained changes

multiply the multipliers

+3% then +4% gives 1.03 × 1.04 = 1.0712, an overall 7.12% — not 7%.

Simple interest

amount = p(1 + rt)

Only the product rt matters: 10% for 2 years equals 5% for 4 years.

Compound interest

amount = p(1 + r)ᵗ

Depends on how many times interest is added back, so 5% for 4 years beats 10% for 2.

Commutativity of percentages

x% of y = y% of x

Both equal xy/100. Use it to swap 4% of 75 for the easier 75% of 4.

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Common mistakes & fixes

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

WATCH OUT

✗ Confusing 'fallen by 85%' with 'fallen to 85%'

✓ Rewrite the sentence as 'what fraction is left?' before computing. The same trap appears in '250% of' versus 'increased by 250%'.

WATCH OUT

✗ Reversing a percentage change by adding the percentage back

✓ Divide by the multiplier: 440000 ÷ 0.85 = ₹5,17,647. Check the direction — after a discount the original must be LARGER.

WATCH OUT

✗ Adding percentage changes instead of multiplying them

✓ Convert each change to a multiplier and multiply them. The gap matters most for large percentages: two 50% rises give 125%, not 100%.

WATCH OUT

✗ Measuring a percentage change against the new value

✓ Percentage change = change ÷ ORIGINAL, always.

WATCH OUT

✗ Not stating whether a profit margin is on cost or on revenue

✓ Always name the base. The chapter uses margin on cost for shopkeepers and margin on revenue for companies.

WATCH OUT

✗ Averaging percentages taken on different bases

✓ Find each cost price separately, then compare total cost with total revenue. The answer here is a 3.08% loss.

WATCH OUT

✗ Treating a percentage of a percentage as an addition

✓ When one percentage is taken OF another, multiply. Drawing a 100-student diagram makes this immediate.

WATCH OUT

✗ Confusing percentage points with per cent

✓ Say 'percentage points' for the difference between two percentages, and 'per cent' for a relative change.

NCERT exercises (with solutions)

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

Figure it Out — Fractions as Percentages

Questions

Figure it Out — Percentage of Some Quantity

Questions

Figure it Out — Profit, Loss and Discount

Questions

Figure it Out — Percentage Change, Profit and Loss

Questions

Figure it Out — Interest and Compounding

Questions

Figure it Out — Applications and Challenges

Questions

Practice problems

Work through this chapter's problems as a readiness check — reveal each solution, mark yourself honestly, and get your gap report at the end.

Readiness check

Are you exam-ready for Fractions in Disguise?

8 problems from this chapter. Try each one, reveal the worked solution, mark yourself honestly — get your gap report at the end.

8 questions~6 min worth ~10 marks in IGCSE exams

5-minute revision

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

•A percentage is a fraction with denominator 100. 'Per cent' is from the Latin per centum, 'out of a hundred'.
•Convert exactly only when the denominator divides 100 (2, 4, 5, 10, 20, 25, 50, 100); 1/3 and 5/11 give recurring decimals.
•Find 1% or 10% first, then scale — faster and safer than multiplying by a decimal.
•Percentage change = (change ÷ original) × 100. The base is always the starting value.
•Increase by r% means × (1 + r/100); decrease means × (1 − r/100). Reverse by DIVIDING.
•Percentage changes multiply, they never add. +3% then +4% gives 7.12%.
•A p% markup followed by a p% discount always ends in a loss, since the discount acts on the larger figure.
•x% of y = y% of x, because both equal xy/100 — useful for swapping 4% of 75 into 75% of 4.
•Simple interest: amount = p(1 + rt), so only the product rt matters.
•Compound interest: amount = p(1 + r)ᵗ, so the number of compounding periods matters. 5% for 4 years beats 10% for 2.
•The idea of 'per hundred' appears as early as the 4th century BCE in Kautilya's Arthashastra, describing monthly interest rates.

IGCSE 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

Where this shows up in the real world

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

GST on every bill, income tax slabs, and TDS

GST on every bill, income tax slabs, and TDS — all specified as percentages

Bank interest on savings

Bank interest on savings, fixed deposits and loans, where compounding decides the real cost

Shop discounts and sale offers

Shop discounts and sale offers, including 'buy one get one' deals that are 50% not 100%

Inflation

Inflation — the percentage rise in prices that determines what your money buys next year

Exam marks

Exam marks, batteries, download progress and opinion polls, all reported per hundred

Nutrition labels and medicine concentrations

Nutrition labels and medicine concentrations, given as percentages by weight or volume

Exam strategy

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

Five question types recur. (1) Convert or compare — find 1% first, and use landmarks (a half, a quarter) to compare without computing. (2) Find the whole from a part — the reverse direction, so DIVIDE by the multiplier; check the answer moves in the right direction. (3) Profit, loss and discount — always state the base (cost or revenue), and take a cost price of ₹100 when only a percentage is asked for. (4) Chained changes — multiply the multipliers and never add the percentages; the marks are in showing the combined multiplier. (5) Interest — set out a year-by-year table for compounding, since it earns method marks even if the arithmetic slips. Read the wording with care: 'by' versus 'to', and 'of' versus 'more than', decide several questions in this chapter on their own.

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.

Because our number system is base 10, so 100 converts neatly to and from decimals — 31% = 31/100 = 0.31. It is also large enough to give useful detail but small enough to grasp mentally. The book notes you could say 'per mille' (per 1000) or 'per decem' (per 10), and statistics sometimes does — 'per lakh' for rare events.

A fall BY 85% removes 85%, leaving 15% of the original. A fall TO 85% leaves 85%. The same trap appears as '250% of' (multiply by 2.5) versus 'increased by 250%' (multiply by 3.5). Read the preposition before you calculate.

Because each change is taken on a different base. A 3% rise followed by a 4% rise gives 1.03 × 1.04 = 1.0712, an overall 7.12% — the extra 0.12% being the 4% charged on the 3% already added. For small percentages adding is a fair estimate, but two 50% rises give 125%, not 100%.

Divide by the multiplier. If a price is ₹4,40,000 after a 15% discount, the original is 440000 ÷ 0.85 = ₹5,17,647 — NOT 440000 × 1.15. Adding 15% of the smaller number cannot undo subtracting 15% of the larger one.

Simple interest depends only on the product rt, which is 20% in both cases. Compounding depends on how many times interest is added back: (1.05)⁴ = 1.2155 against (1.10)² = 1.21. More compounding periods means more interest earning interest.

No — it is 50%. You still pay for one of the two items you take home. Compare the free items with the TOTAL received: 1 free out of 2 is 50%, 1 out of 3 is 33.33%, 1 out of 4 is 25%.

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