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Fractions in Disguise

NCERT Class 8 · Mathematics Based on NCERT Class 8 Mathematics textbook · Free CBSE study kit

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Chapter Notes

Overview
This chapter introduces the concept of percentages as a special type of fraction where the denominator is always 100. It explains why percentages are useful for easy comparison, how to convert fractions to percentages and vice versa, and how to find a percentage of a given quantity. The big idea is that percentages are simply 'fractions in disguise' — they make comparing proportions and understanding real-world situations like discounts, profit-loss, and interest rates intuitive and quick.
Key Concepts (24)
Per cent
Per cent means 'by the hundred' or 'out of hundred'. The symbol '%' is read as per cent.
e.g. 25% means 25 out of every 100, like 25 marks out of 100 marks.
Percentage as a fraction
A percentage is a fraction where the denominator is 100. For example, 20% = 20/100 = 1/5.
e.g. 50% = 50/100 = 1/2 of a quantity.
Expressing a fraction as a percentage
To express a fraction as a percentage, find an equivalent fraction with denominator 100, or multiply the fraction by 100.
e.g. 3/4 = (3×25)/(4×25) = 75/100 = 75%. Also, 3/4 × 100 = 75%.
Percentage of a quantity
y% of a quantity is found by multiplying the quantity by y/100.
e.g. 25% of 120 = (25/100) × 120 = 30.
Free-hand computations
Using relationships between percentages to calculate mentally, e.g., 20% is double of 10%, and 20% + 5% = 25%.
e.g. To find 15% of a value, add 10% and 5% of that value.
FDP Trio (Fractions, Decimals, Percentages)
The three ways of representing the same value: fraction, decimal, and percentage are interchangeable.
e.g. 50% = 50/100 = 1/2 = 0.5.
Percentages greater than 100
A percentage can be more than 100, meaning the quantity is more than the whole or the reference value.
e.g. If target is ₹5000 and sales are ₹6000, then sales = (6000/5000) × 100 = 120% of target.
Percentage increase
Percentage increase = (amount of increase / original amount) × 100.
e.g. Price of tomatoes increased from ₹30 to ₹42. Increase = ₹12. Percentage increase = (12/30) × 100 = 40%.
Percentage decrease
Percentage decrease = (amount of decrease / original amount) × 100.
e.g. Footfall decreased from 160 to 100. Decrease = 60. Percentage decrease = (60/160) × 100 = 37.5%.
Cost Price (CP)
The price at which an item is bought by a shopkeeper or seller.
e.g. Kishanlal buys a sweater from a wholesaler at ₹300. That is his cost price.
Marked Price (MP)
The price quoted by the shopkeeper on the item, often before any discount.
e.g. Kishanlal marks the sweater at ₹480.
Selling Price (SP)
The price at which the item is actually sold to the customer.
e.g. After bargaining, the sweater is sold for ₹430.
Profit
Profit = Selling Price - Cost Price, when SP > CP.
e.g. ₹430 - ₹300 = ₹130 profit on the sweater.
Loss
Loss = Cost Price - Selling Price, when CP > SP.
e.g. Rice bought at ₹350, sold at ₹300. Loss = ₹50.
Profit percentage
Profit percentage = (Profit / Cost Price) × 100.
e.g. Profit of ₹130 on CP of ₹300 gives (130/300) × 100 = 43.3%.
Loss percentage
Loss percentage = (Loss / Cost Price) × 100.
e.g. Loss of ₹50 on CP of ₹350 gives (50/350) × 100 = 14.28%.
Discount
A reduction on the marked price. Discount percentage is calculated on the marked price.
e.g. 30% off means the price is reduced by 30% of the marked price.
Gross Profit
The difference between total sales and the cost of goods sold, before deducting other expenses.
e.g. Sales ₹80,000 - cost of goods ₹48,000 = gross profit ₹32,000.
Net Profit
Gross profit minus other expenses (like transport, salary, electricity).
e.g. Gross profit ₹32,000 - other expenses ₹8,000 = net profit ₹24,000.
Interest (Simple)
Extra money paid on a deposit or loan, calculated only on the original principal amount.
e.g. ₹6000 at 10% p.a. for 3 years gives interest = 6000 × 0.10 × 3 = ₹1800.
Principal (P)
The original amount of money deposited or borrowed.
e.g. In a fixed deposit of ₹6000, the principal is ₹6000.
Rate of Interest (r)
The percentage of the principal paid as interest per time period (usually per annum).
e.g. 10% p.a. means ₹10 interest per ₹100 per year.
Compounding
The process where interest earned is added back to the principal, so that subsequent interest is earned on the increased principal.
e.g. ₹6000 at 10% p.a. compounded annually: after 1 year ₹6600, after 2 years ₹7260, after 3 years ₹7986.
Depreciation
The reduction in the value of an asset over time due to use and age.
e.g. A TV bought for ₹21,000 depreciates by 5% in one year. Its value after one year = ₹21,000 - (5% of 21000) = ₹19,950.
Formulas & Equations (10)
Percentage = (Fraction) × 100
Reads as: Percentage equals fraction multiplied by 100
Variables: Fraction: a part of a whole (dimensionless); Percentage: the value out of 100
Use when: To convert any given fraction into a percentage.
⚠ Students forget to multiply by 100 and just write the fraction as the percentage.
y% of a quantity = (y/100) × quantity
Reads as: y percent of a quantity equals y divided by 100 multiplied by the quantity
Variables: y: the percentage number; quantity: the whole value
Use when: To find the actual value represented by a given percentage of a quantity.
⚠ Students write y% of quantity = y × quantity, forgetting to divide by 100.
Percentage increase = (Amount of increase / Original amount) × 100
Reads as: Percentage increase equals amount of increase divided by original amount multiplied by 100
Variables: Amount of increase: new value - original value; Original amount: the starting value
Use when: To find the percentage by which a quantity has increased.
⚠ Students divide by the new value instead of the original value.
Percentage decrease = (Amount of decrease / Original amount) × 100
Reads as: Percentage decrease equals amount of decrease divided by original amount multiplied by 100
Variables: Amount of decrease: original value - new value; Original amount: the starting value
Use when: To find the percentage by which a quantity has decreased.
⚠ Students divide by the new value instead of the original value.
Profit = Selling Price - Cost Price
Reads as: Profit equals selling price minus cost price
Variables: Selling Price (SP): the price at which item is sold; Cost Price (CP): the price at which item is bought
Use when: To calculate the profit made on a transaction when SP > CP.
⚠ Students subtract in the wrong order or use marked price instead of selling price.
Loss = Cost Price - Selling Price
Reads as: Loss equals cost price minus selling price
Variables: Cost Price (CP): the price at which item is bought; Selling Price (SP): the price at which item is sold
Use when: To calculate the loss incurred on a transaction when CP > SP.
⚠ Students subtract in the wrong order or use marked price instead of selling price.
Profit percentage = (Profit / Cost Price) × 100
Reads as: Profit percentage equals profit divided by cost price multiplied by 100
Variables: Profit: SP - CP; Cost Price: the buying price
Use when: To express profit as a percentage of the investment (cost price).
⚠ Students calculate profit percentage on selling price instead of cost price.
Loss percentage = (Loss / Cost Price) × 100
Reads as: Loss percentage equals loss divided by cost price multiplied by 100
Variables: Loss: CP - SP; Cost Price: the buying price
Use when: To express loss as a percentage of the investment (cost price).
⚠ Students calculate loss percentage on selling price instead of cost price.
Simple Interest: Amount = P (1 + rt)
Reads as: Amount equals P times 1 plus r times t
Variables: P: Principal (initial deposit); r: rate of interest per year (as a decimal); t: time in years
Use when: To calculate the total amount (principal + interest) when interest is not compounded (simple interest).
⚠ Students forget to add the principal back to the interest, or use r as a percentage instead of decimal.
Compound Interest: Amount = P (1 + r)^t
Reads as: Amount equals P times 1 plus r raised to the power t
Variables: P: Principal; r: rate of interest per compounding period (as a decimal); t: number of compounding periods
Use when: To calculate the total amount when interest is compounded annually (or per period).
⚠ Students multiply P by (1 + r) × t instead of raising (1 + r) to the power t.
How It Works
Converting a fraction to a percentage
  1. Step 1: Write the given fraction.
  2. Step 2: Multiply the fraction by 100.
  3. Step 3: Add the '%' symbol to the result.
💡 You can also find an equivalent fraction with denominator 100, then write the numerator followed by '%'.
Finding a percentage of a quantity
  1. Step 1: Write the percentage as a fraction with denominator 100.
  2. Step 2: Multiply this fraction by the given quantity.
  3. Step 3: Simplify if needed to get the final value.
💡 Always divide by 100 first, then multiply, to avoid large numbers.
Calculating profit or loss percentage
  1. Step 1: Identify the Cost Price (CP) and Selling Price (SP).
  2. Step 2: If SP > CP, Profit = SP - CP. If CP > SP, Loss = CP - SP.
  3. Step 3: Profit% = (Profit / CP) × 100. Loss% = (Loss / CP) × 100.
💡 Profit and loss percentages are always calculated on the Cost Price, not on the Selling Price.
Calculating simple interest (no compounding)
  1. Step 1: Identify Principal (P), rate of interest per year (r as decimal), and time in years (t).
  2. Step 2: Interest = P × r × t.
  3. Step 3: Total Amount = P + Interest = P (1 + rt).
💡 The principal remains the same every year; interest is paid out or not added back.
Calculating compound interest (compounded annually)
  1. Step 1: Identify Principal (P), rate of interest per year (r as decimal), and number of years (t).
  2. Step 2: Amount after 1 year = P × (1 + r).
  3. Step 3: For each subsequent year, multiply the previous amount by (1 + r).
  4. Step 4: After t years, Amount = P × (1 + r)^t.
💡 Interest is added to the principal each year, so the principal grows and interest increases each year.
Calculating depreciation
  1. Step 1: Identify the current value and the depreciation rate (as a decimal).
  2. Step 2: Value after one period = Current value × (1 - depreciation rate).
  3. Step 3: Repeat for each time period if needed.
💡 Depreciation reduces the value, so multiply by (1 - rate), not (1 + rate).
Exam Traps — Don't Make These Mistakes
⚠ Confusing the base for percentage increase/decrease
✗ Percentage increase = (increase / new value) × 100
→ The base (denominator) should always be the original amount, not the new amount.
✓ Percentage increase = (amount of increase / original amount) × 100
⚠ Calculating profit percentage on selling price
✗ Profit% = (Profit / Selling Price) × 100
→ Profit percentage is always calculated on the cost price (the investment), not on the selling price.
✓ Profit% = (Profit / Cost Price) × 100
⚠ Forgetting to add principal in simple interest
✗ Amount after t years = P × r × t
→ This gives only the interest earned, not the total amount. The principal must be added back.
✓ Amount = P + (P × r × t) = P (1 + rt)
⚠ Using multiplication instead of exponentiation in compound interest
✗ Amount after 3 years = P × (1 + r) × 3
→ Compounding multiplies the factor (1+r) repeatedly, not just once per year. It is P × (1+r)^t.
✓ Amount = P × (1 + r)^t
⚠ Treating percentage as a whole number instead of fraction
✗ 25% of 120 = 25 × 120 = 3000
→ 25% means 25/100, not 25. You must divide by 100.
✓ 25% of 120 = (25/100) × 120 = 30
⚠ Confusing 'percentage of' with 'percentage increase'
✗ If a number increases by 20% to become 90, then the original number is 90 × 0.20 = 18.
→ A 20% increase means the new number is 120% of the original. So original = 90 / 1.20, not 90 × 0.20.
✓ Original number = 90 / 1.20 = 75
Compare & Contrast
How interest is calculated over time
Simple Interest
Compound Interest
In simple interest, interest is calculated only on the original principal every year. In compound interest, interest is calculated on the principal plus previously earned interest, so the amount grows faster.
Basis of calculation
Profit Percentage
Loss Percentage
Both are calculated on the Cost Price. Profit% uses (SP - CP)/CP × 100, while Loss% uses (CP - SP)/CP × 100.
Direction of change
Percentage Increase
Percentage Decrease
Percentage increase measures growth from original to new (new > original), while percentage decrease measures reduction (new < original). Both use the original value as the denominator.
Exam Focus
Marks pattern: 2 MCQs (1 mark each), 1 SA (3 marks) on percentage of quantity or profit/loss, 1 LA (5 marks) on simple/compound interest or multi-step percentage problem
Most asked: Converting fractions to percentages and vice versa · Finding percentage of a quantity · Profit and loss percentage problems · Simple and compound interest calculations · Percentage increase and decrease in real-life contexts
Don't skip: Word problems involving profit/loss percentage with discount; Compound interest problems comparing with simple interest; Finding the original value given a percentage increase or decrease
One-Line Revision (read aloud before the exam)
  1. Per cent means 'out of hundred', so 25% = 25/100 = 1/4.
  2. To convert a fraction to a percentage, multiply the fraction by 100.
  3. To find y% of a quantity, multiply the quantity by y/100.
  4. Percentage increase = (increase / original) × 100.
  5. Percentage decrease = (decrease / original) × 100.
  6. Profit = Selling Price - Cost Price.
  7. Loss = Cost Price - Selling Price.
  8. Profit percentage = (Profit / Cost Price) × 100.
  9. Loss percentage = (Loss / Cost Price) × 100.
  10. Simple Interest: Amount = P (1 + rt).
  11. Compound Interest: Amount = P (1 + r)^t.
  12. Depreciation: Value after one year = Current value × (1 - depreciation rate).
  13. A percentage greater than 100 means the quantity is more than the whole or reference.
  14. Discount is calculated on the marked price, not on the cost price.
  15. 20% of a value is double of 10% of the same value.

MCQs — 10 Questions with Answers

Q1. What does 35% mean?

  • A. 35 out of every 100 ✓
  • B. 35 out of every 10
  • C. 35 out of every 1000
  • D. 35 out of every 50

Answer: A — Percent means 'per hundred', so 35% = 35/100, which means 35 out of every 100.

Q2. Which of the following is equivalent to 50%?

  • A. 1/3
  • B. 1/2 ✓
  • C. 2/5
  • D. 3/4

Answer: B — 50% = 50/100 = 1/2, so 50% is the same as the fraction one-half.

Q3. Express 3/5 as a percentage.

  • A. 30%
  • B. 50%
  • C. 60% ✓
  • D. 75%

Answer: C — 3/5 = (3 × 20)/(5 × 20) = 60/100 = 60%, or multiply: (3/5) × 100 = 60.

Q4. What is 25% of 80?

  • A. 16
  • B. 20 ✓
  • C. 25
  • D. 40

Answer: B — 25% of 80 = (25/100) × 80 = (1/4) × 80 = 20.

Q5. Nandini has 25 marbles, of which 15 are white. What percentage of her marbles are white?

  • A. 40%
  • B. 50%
  • C. 60% ✓
  • D. 75%

Answer: C — Percentage = (15/25) × 100 = (3/5) × 100 = 60%.

Q6. In a school, 15 of the 80 students come by walking. What percentage of students walk to school?

  • A. 15.75%
  • B. 18.75% ✓
  • C. 20%
  • D. 25%

Answer: B — Percentage = (15/80) × 100 = (3/16) × 100 = 18.75%.

Q7. Madhav ate 95 g of biscuits with 35% sugar. How much sugar did he consume?

  • A. 29.75 g
  • B. 33.25 g ✓
  • C. 35 g
  • D. 38.5 g

Answer: B — 35% of 95 g = (35/100) × 95 = 33.25 g.

Q8. A test has maximum marks 75. If a student needs 80% to pass, how many marks must they score?

  • A. 50
  • B. 55
  • C. 60 ✓
  • D. 65

Answer: C — 80% of 75 = (80/100) × 75 = (4/5) × 75 = 60 marks.

Q9. Which statement correctly compares 50% and 5%?

  • A. 50% = 5%
  • B. 50% < 5%
  • C. 50% > 5% ✓
  • D. Cannot be compared without knowing the total

Answer: C — 50% (50 out of 100) is greater than 5% (5 out of 100) as percentages are proportions independent of the base.

Q10. Madhu ate 120 g of biscuits with 25% sugar and Madhav ate 95 g with 35% sugar. Who consumed more sugar?

  • A. Madhu (30 g > 33.25 g)
  • B. Madhav (33.25 g > 30 g) ✓
  • C. Both consumed equal sugar
  • D. Cannot determine without more information

Answer: B — Madhu consumed 25% of 120 g = 30 g sugar; Madhav consumed 35% of 95 g = 33.25 g sugar; therefore Madhav consumed more.

Flashcards

What does the symbol '%' mean and where does the word come from?

The symbol '%' means 'per cent' which comes from Latin 'per centum' meaning 'out of hundred' or 'by the hundred'.

How do you convert the fraction 3/4 into a percentage?

Multiply 3/4 by 100 to get (3/4) × 100 = 75%, or find the equivalent fraction with denominator 100: (3 × 25)/(4 × 25) = 75/100 = 75%.

Write the general formula to find y% of some value z.

y% of z = (y/100) × z.

What is the relationship between 25%, 1/4, and 0.25?

They are all equivalent: 25% = 25/100 = 1/4 = 0.25.

Why is 100 chosen as the denominator for percentages instead of 10 or 1000?

100 is the sweet spot because our number system is base 10, it connects easily to decimals, it is large enough to give detail, yet simple enough to understand mentally.

If Madhu ate 120 g of biscuits with 25% sugar, how much sugar did she eat?

25% of 120 g = (25/100) × 120 = 30 g of sugar.

When comparing two percentages of different wholes, what must you do first?

You must find the actual quantities (not just the percentages) because percentage is a proportion and depends on the total amount.

Express 24% as a fraction in simplest form.

24% = 24/100 = 6/25.

If 80% of a test has maximum marks 75, what is the minimum score for grade A?

80% of 75 = (80/100) × 75 = 60 marks.

What relationship always holds between 20%, 5%, and 25% of any value y?

(20% of y) + (5% of y) = 25% of y, because (20/100 + 5/100) × y = (25/100) × y.

Important Board Questions

What does 45% mean? [1 mark]

Per cent means 'out of hundred'. Write as a fraction with denominator 100.

Convert 7/20 into a percentage. Show your method. [2 marks]

Find equivalent fraction with denominator 100 by multiplying numerator and denominator by 5, or multiply the fraction by 100.

In a school library, there are 240 books. If 30% of the books are fiction, how many fiction books are there? Show your working step by step. [3 marks]

Use formula: 30% of 240 = (30/100) × 240. Simplify the fraction first to make calculation easier, or convert to decimal 0.3 × 240.

A shopkeeper gives a 20% discount on a shirt that costs ₹500. How much does the customer pay? Explain the concept of percentage discount and show all your steps. Also state why percentages are more useful than fractions for comparing such quantities. [5 marks]

First find 20% of 500 using (20/100) × 500. Subtract from original price. Explain that percentage makes comparison standard (out of 100) regardless of original value, unlike fractions which vary with different denominators. Also mention that the number system base 10 makes 100 convenient for understanding.

Assertion-Reason Questions for Fractions in Disguise

Q1. Assertion (A):

Percentages are particularly useful for comparing fractions with different denominators because they convert all fractions to have 100 as the denominator.

Reason (R):

The number 100 was chosen as the standard denominator for percentages because it is a round number that works well with our base-10 decimal system and is easy to work with mentally.

Answer: A — Both A and R are true, and R is the correct explanation of A — Both the assertion and reason are true, and the reason correctly explains why percentages are useful for comparison. The denominator 100 makes all percentages directly comparable (50% vs 26.47% is immediately clear), which is exactly because of the properties mentioned in the reason.

Q2. Assertion (A):

To express the fraction 2/5 as a percentage, we can multiply 2/5 by 100 to get 40%.

Reason (R):

A percentage is defined as a fraction where the denominator is always 100, so any fraction can be converted by finding its equivalent form with 100 as denominator.

Answer: B — Both A and R are true, but R is NOT the correct explanation of A — Both statements are true. The assertion shows the correct calculation (2/5 × 100 = 40%). The reason correctly defines percentage as a fraction with denominator 100. However, the reason doesn't explain WHY multiplying by 100 works—it's because we're creating an equivalent fraction. The statements are related but the reason isn't the best explanation for the assertion.

Q3. Assertion (A):

If Madhu ate 120 g of biscuits with 25% sugar and Madhav ate 95 g of biscuits with 35% sugar, then Madhav definitely consumed more sugar than Madhu.

Reason (R):

35% is greater than 25%, therefore any quantity with 35% will contain more of that component than a quantity with only 25%.

Answer: D — A is false but R is true — The assertion is false because we must calculate actual amounts: Madhu consumed 25% × 120 = 30 g sugar, while Madhav consumed 35% × 95 = 33.25 g sugar. Madhav did consume more, but the assertion uses 'definitely' based only on percentage comparison. The reason is also false because it ignores that the original quantities (120 g vs 95 g) matter. You cannot compare percentages directly without considering the base quantities.

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Case-Based Questions (CBQ) for Fractions in Disguise

Case Study:

Rajesh manages a small grocery store in Delhi. On a particular day, he has different fruits in stock. Out of 80 apples, 60 are fresh. Out of 50 oranges, 40 are fresh. Out of 120 bananas, 96 are fresh. Rajesh wants to display the freshness percentage of each fruit type to attract customers and ensure transparency about fruit quality.

Q1. What percentage of apples are fresh?

  • A. 60%
  • B. 70%
  • C. 75% ✓
  • D. 80%

Answer: C

Q2. What percentage of oranges are fresh?

  • A. 70%
  • B. 80% ✓
  • C. 90%
  • D. 85%

Answer: B

Q3. What percentage of bananas are fresh?

  • A. 75%
  • B. 80% ✓
  • C. 85%
  • D. 90%

Answer: B

Q4. Rajesh claims that 'All fruits have more than 70% freshness, so they are all good for sale.' Which statement best validates or challenges this claim?

  • A. The claim is false because apples have only 60 fresh out of 80
  • B. The claim is partially true but depends on customer preference
  • C. The claim cannot be verified without knowing the total quantity
  • D. The claim is true because all three fruits exceed 70% freshness (75%, 80%, 80%) ✓

Answer: D

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