Class 8 : Maths – Lesson 8. Fractions in Disguise

Posted by

Ankit

On February 12, 2026

EXPLANATION AND ANALYSIS

🌍 INTRODUCTION — FRACTIONS HIDDEN IN DIFFERENT FORMS

🧠 In earlier classes, we studied fractions as parts of a whole.
📘 We usually recognise fractions when they are written clearly, like 1/2, 3/4, or 5/6.

🔍 But in real mathematics, fractions often do not appear openly.
They are hidden inside:

divisions

ratios

percentages

decimals

everyday statements

📌 Such fractions are called fractions in disguise.

🎯 This lesson helps us:

identify hidden fractions

rewrite them in clear fractional form

understand that many different-looking expressions represent the same idea

🔢 WHAT IS A FRACTION IN DISGUISE?

🧠 A fraction in disguise is a fraction that is not written directly in the form p/q, but still represents a part of a whole.

📘 The value is fractional, even if the form looks different.

🔵 Examples of disguised forms:

a ÷ b

a ratio like a : b

a percentage like 25%

a decimal like 0.75

🟡 All of these can be rewritten as fractions.

📌 Recognising these hidden forms helps avoid confusion in calculations.

➗ DIVISION AS A FRACTION

🧠 Every division of two numbers can be written as a fraction.

📘 Key idea:

a ÷ b = a / b (where b ≠ 0)

🔵 This means division and fractions are closely connected.
🟣 When we divide something into equal parts, we are actually forming fractions.

📌 Many real-life divisions are fractions, even if we don’t call them so.

⚖️ RATIOS — FRACTIONS IN ANOTHER FORM

🧠 A ratio compares two quantities of the same kind.

📘 Ratio written as:

a : b

🔵 This ratio can be written as a fraction:

a / b

🟡 The ratio tells us how many parts of one quantity are compared to another.

📌 Ratios are fractions that compare two quantities, not parts of one object.

📊 PERCENTAGES AS FRACTIONS

🧠 A percentage means “per hundred”.

📘 So:

25% means 25 out of 100

which is 25/100

🔵 Any percentage can be converted into a fraction by placing it over 100.
🟡 Then it can be simplified further.

📌 Percentages are fractions that make comparison easier, especially in daily life.

🔢 DECIMALS — ANOTHER DISGUISE

🧠 A decimal is also a fraction in disguise.

📘 Example ideas:

0.5 = 5/10 = 1/2

0.25 = 25/100 = 1/4

🔵 Decimals represent fractions with denominators like 10, 100, 1000, etc.
🟣 They are useful when precise measurement is needed.

📌 Decimals and fractions represent the same idea in different formats.

🔄 CONVERTING DISGUISED FRACTIONS

🧠 One important skill is converting a disguised fraction into a clear fraction.

📘 Common conversions include:

division → fraction

ratio → fraction

percentage → fraction

decimal → fraction

🔵 Conversion helps in:

comparing values

simplifying expressions

solving word problems

📌 The underlying value remains the same; only the form changes.

⚖️ EQUIVALENT FRACTIONS — SAME VALUE, DIFFERENT LOOK

🧠 Two fractions are called equivalent fractions if they represent the same value.

📘 Example idea:

1/2 = 2/4 = 50% = 0.5

🔵 These all look different but mean the same part of a whole.
🟡 This explains why fractions can appear in many disguises.

📌 Understanding equivalence avoids wrong comparisons.

🔢 SIMPLIFYING FRACTIONS

🧠 Simplifying means reducing a fraction to its lowest form.

📘 This is done by:

dividing numerator and denominator by the same number

🔵 Simplified fractions are easier to understand and compare.
🟣 They remove unnecessary complexity.

📌 Simplification does not change the value, only the appearance.

📍 COMPARING FRACTIONS IN DISGUISE

🧠 Comparing disguised fractions directly can be confusing.

📘 Better approach:

convert all values into the same form

preferably into fractions or decimals

🔵 This ensures fair comparison.
🟡 It avoids mistakes caused by appearance.

📌 Comparison is easier when forms are uniform.

🌍 FRACTIONS IN DAILY LIFE

🧠 Fractions in disguise appear everywhere.

🔵 sharing food
🟡 discounts in shopping
🔴 exam marks
🟣 probability
🟠 data representation

📘 Even when we don’t see a fraction sign, the idea of parts of a whole is present.

⚠️ COMMON MISTAKES TO AVOID

🚫 treating division and fraction as different ideas
🚫 forgetting that ratios are fractions
🚫 comparing percentages directly with fractions
🚫 ignoring simplification

✔️ Always look for the hidden fraction.

🌟 IMPORTANCE OF THIS LESSON

🏆 builds strong fraction sense
🧠 improves conversion skills
⚡ prepares base for algebra and percentages
📘 connects maths with real-world situations
🌱 helps recognise hidden relationships

Understanding disguised fractions strengthens mathematical confidence.

🧾 SUMMARY

🔵 fractions appear in many forms
🟡 division, ratio, percentage, and decimal are disguised fractions
🔴 all disguised forms can be rewritten as p/q
🟣 equivalent fractions have same value
🟠 simplification makes fractions clearer
🟢 recognising disguise avoids errors

🔁 QUICK RECAP

🔵 division equals fraction
🟡 ratio can be written as fraction
🟣 percentage means per hundred
🟠 decimal is fraction with power of 10
🔴 same value can look different

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TEXTBOOK QUESTIONS

🔒 ❓ 1. The population of Bengaluru in 2025 is about 250% of its population in 2000. If the population in 2000 was 50 lakhs, what is the population in 2025?
📌 ✅ Answer:
🟢 Step 1: Understand the percentage growth
⬥ Current Population (2025) = 250% of Previous Population (2000).
⬥ Previous Population = 50 lakhs.

🔵 Step 2: Calculate the value
⬥ Population = 250% × 50
⬥ = (250 / 100) × 50
⬥ = 2.5 × 50

🟡 Step 3: Final Result
⬥ 2.5 × 50 = 125
⬥ 125 lakhs

🔒 ❓ 2. The population of the world in 2025 is about 8.2 billion. The populations of some countries in 2025 are given. Match them with their approximate percentage share of the worldwide population. [Hint: Writing these numbers in the standard form and estimating can help].
Germany: 83 million
India: 1.46 billion
Bangladesh: 175 million
USA: 347 million
(Options: 13%, 8%, 18%, 10%, 1%, 35%, 2%, 2%, 0.1%)
📌 ✅ Answer:
🟢 Step 1: Convert World Population to Millions
⬥ 8.2 billion = 8,200 million.

🔵 Step 2: Calculate Percentage for Germany
⬥ (83 / 8,200) × 100 ≈ 1.01%
⬥ Match: 1%

🟡 Step 3: Calculate Percentage for India
⬥ 1.46 billion = 1,460 million.
⬥ (1,460 / 8,200) × 100 ≈ 17.8%
⬥ Match: 18%

🔴 Step 4: Calculate Percentage for Bangladesh
⬥ (175 / 8,200) × 100 ≈ 2.13%
⬥ Match: 2%

🟣 Step 5: Calculate Percentage for USA
⬥ (347 / 8,200) × 100 ≈ 4.23%
⬥ Note: There is no exact “4%” in the option list provided in the image, but it is approximately double of Bangladesh (2%). Based on calculation, it is ~4%.
⬥ Closest logical match from typical datasets would be 4% (if available) or potentially one of the 2% labels is a typo.
⬥ (Calculated Value: ~4.2%)

🔒 ❓ 3. The price of a mobile phone is ₹8,250. A GST of 18% is added to the price. Which of the following gives the final price of the phone including the GST?
(i) 8250 + 18
(ii) 8250 + 1800
(iii) 8250 + 18/100
(iv) 8250 × 18
(v) 8250 × 1.18
(vi) 8250 + 8250 × 0.18
(vii) 1.8 × 8250
📌 ✅ Answer:
🟢 Step 1: Analyse the cost components
⬥ Final Price = Original Price + Tax Amount
⬥ Tax Amount = 18% of 8250 = (18/100) × 8250 = 0.18 × 8250.

🔵 Step 2: Evaluate Expressions
⬥ Expression (vi): 8250 + 8250 × 0.18 correctly represents Price + Tax.
⬥ Expression (v): 8250 × 1.18 is also correct because Price × (1 + 0.18) = Price × 1.18.
🟡 Conclusion
⬥ The correct options are (v) and (vi).

🔒 ❓ 4. The monthly percentage change in population (compared to the previous month) of mice in a lab is given: Month 1 change was +5%, Month 2 change was –2%, and Month 3 change was –3%. Which of the following statement(s) are true? The initial population is p.
(i) The population after three months was p × 0.05 × 0.02 × 0.03.
(ii) The population after three months was p × 1.05 × 0.98 × 0.97.
(iii) The population after three months was p + 0.05 – 0.02 – 0.03.
(iv) The population after three months was p.
(v) The population after three months was more than p.
(vi) The population after three months was less than p.

📌 ✅ Answer:
🟢 Step 1: Convert percentages to multipliers
⬥ +5% increase ➔ Multiply by (1 + 0.05) = 1.05
⬥ –2% decrease ➔ Multiply by (1 – 0.02) = 0.98
⬥ –3% decrease ➔ Multiply by (1 – 0.03) = 0.97

🔵 Step 2: Form the final expression
⬥ Final Population = Initial × M1 × M2 × M3
⬥ Expression: p × 1.05 × 0.98 × 0.97
⬥ This makes Statement (ii) True.

🟡 Step 3: Calculate the net effect
⬥ 1.05 × 0.98 × 0.97 ≈ 0.998
⬥ Since 0.998 < 1, the final population is slightly less than p.
⬥ This makes Statement (vi) True.
🔴 Selected Answers
⬥ (ii) and (vi)

🔒 ❓ 5. A shopkeeper initially set the price of a product with a 35% profit margin. Due to poor sales, he decided to offer a 30% discount on the selling price. Will he make a profit or a loss? Give reasons for your answer.
📌 ✅ Answer:
🟢 Step 1: Assume a Cost Price (CP)
⬥ Let CP = ₹100.

🔵 Step 2: Determine Marked Price (MP)
⬥ Profit margin = 35%.
⬥ Marked Price = 100 + 35 = ₹135.

🟡 Step 3: Apply Discount
⬥ Discount = 30% of MP
⬥ Discount = 0.30 × 135 = ₹40.5
⬥ Selling Price (SP) = 135 – 40.5 = ₹94.5

🔴 Step 4: Determine Profit or Loss
⬥ SP (94.5) < CP (100).
⬥ Loss = 100 – 94.5 = ₹5.5.
⬥ He will make a LOSS of 5.5%.

🔒 ❓ 6. What percentage of area is occupied by the region marked ‘E’ in the figure?
📌 ✅ Answer:
🟢 Step 1: Analyse the Grid Dimensions
⬥ By counting the grid intervals, the full rectangle has a width of 8 units and a height of 6 units.
⬥ Total Area = 8 × 6 = 48 sq units.

🔵 Step 2: Identify Region E
⬥ The rectangle is divided into quadrants. Region E is the lower triangle of the bottom-left quadrant.
⬥ The bottom-left quadrant has dimensions 4 × 3 (Area = 12).
⬥ Region E is exactly half of this quadrant (triangle).
⬥ Area of E = ½ × 4 × 3 = 6 sq units.

🟡 Step 3: Calculate Percentage
⬥ Percentage = (Area of E / Total Area) × 100
⬥ Percentage = (6 / 48) × 100
⬥ = (1/8) × 100 = 12.5%

🔒 ❓ 7. What is 5% of 40? What is 40% of 5?
What is 25% of 12? What is 12% of 25?
What is 15% of 60? What is 60% of 15?
What do you notice?
Can you make a general statement and justify it using algebra, comparing x% of y and y% of x?
📌 ✅ Answer:
🟢 Step 1: Calculate the pairs
⬥ 5% of 40 = 0.05 × 40 = 2 | 40% of 5 = 0.40 × 5 = 2
⬥ 25% of 12 = 0.25 × 12 = 3 | 12% of 25 = 0.12 × 25 = 3
⬥ 15% of 60 = 0.15 × 60 = 9 | 60% of 15 = 0.60 × 15 = 9

🔵 Step 2: Observation
⬥ The values in each pair are equal.

🟡 Step 3: General Statement & Proof
⬥ Statement: x% of y is always equal to y% of x.
⬥ Justification:
⬥ x% of y = (x/100) × y = xy/100
⬥ y% of x = (y/100) × x = yx/100
⬥ Since xy = yx, the expressions are identical.

🔒 ❓ 8. A school is organising an excursion for its students. 40% of them are Grade 8 students and the rest are Grade 9 students. Among these Grade 8 students, 60% are girls. [Hint: Drawing a rough diagram can help].

🔒 ❓ (i) What percentage of the students going to the excursion are Grade 8 girls?
📌 ✅ Answer:
🟢 Step 1: Calculate combined percentage
⬥ Grade 8 students = 40% of Total.
⬥ Girls in Grade 8 = 60% of Grade 8s.
⬥ Percentage of Total = 60% of 40%

🔵 Step 2: Solve
⬥ 0.60 × 0.40 = 0.24
⬥ 24% of the total students are Grade 8 girls.

🔒 ❓ (ii) If the total number of students going to the excursion is 160, how many of them are Grade 8 girls?
📌 ✅ Answer:
🟢 Step 1: Apply percentage to total
⬥ Number = 24% of 160
⬥ = 0.24 × 160

🔵 Step 2: Calculation
⬥ 24 × 16 = 384
⬥ Adjust decimal: 38.4
🟡 Note on rounding
⬥ Since students must be whole numbers, there is likely a slight mismatch in the problem’s chosen numbers. We round to the nearest whole number.
⬥ Approx 38 students.

🔒 ❓ 9. A shopkeeper sells pencils at a price such that the selling price of 3 pencils is equal to the cost of 5 pencils. Does he make a profit or a loss? What is his profit or loss percentage?
📌 ✅ Answer:
🟢 Step 1: Set up the equation
⬥ Let Cost Price (CP) of 1 pencil = x.
⬥ Cost of 5 pencils = 5x.
⬥ Selling Price (SP) of 3 pencils = 5x.

🔵 Step 2: Find SP of 1 pencil
⬥ SP of 1 pencil = 5x / 3.

🟡 Step 3: Calculate Profit
⬥ Profit = SP – CP
⬥ Profit = (5x/3) – x = 2x/3.

🔴 Step 4: Calculate Profit Percentage
⬥ Profit % = (Profit / CP) × 100
⬥ = [(2x/3) / x] × 100
⬥ = (2/3) × 100 = 66.67%
⬥ He makes a profit of 66.67%.

🔒 ❓ 10. The bus fares were increased by 3% last year and by 4% this year. What is the overall percentage price increase in the last 2 years?
📌 ✅ Answer:
🟢 Step 1: Use a base value
⬥ Let the initial fare = 100.

🔵 Step 2: Apply first increase (3%)
⬥ Year 1 Fare = 100 + 3 = 103.

🟡 Step 3: Apply second increase (4%)
⬥ Increase = 4% of 103 = 0.04 × 103 = 4.12.
⬥ Year 2 Fare = 103 + 4.12 = 107.12.

🔴 Step 4: Calculate overall change
⬥ Total Increase = 107.12 – 100 = 7.12.
⬥ Overall percentage increase is 7.12%.

🔒 ❓ 11. If the length of a rectangle is increased by 10% and the area is unchanged, by what percentage (exactly) does the breadth decrease by?
📌 ✅ Answer:
🟢 Step 1: Use Area formula
⬥ Area (A) = Length (L) × Breadth (B).
⬥ New Length (L’) = 1.10 L.
⬥ New Area (A’) = L’ × B’ = A.

🔵 Step 2: Find the new breadth
⬥ 1.10 L × B’ = L × B
⬥ B’ = B / 1.10
⬥ B’ = B × (10/11)

🟡 Step 3: Calculate Decrease
⬥ Decrease = B – B’ = B – (10/11)B = (1/11)B.

🔴 Step 4: Convert to Percentage
⬥ Percentage = (Decrease / Original) × 100
⬥ = (1/11) × 100
⬥ = 9 1/11 % or approximately 9.09%.

🔒 ❓ 12. The percentage of ingredients in a 65 g chips packet is shown in the picture. Find out the weight each ingredient makes up in this packet.
(Image Data: Potato 70%, Vegetable oil 24%, Salt 3%, Spice 3%)
📌 ✅ Answer:
🟢 Step 1: Calculate Potato Weight
⬥ 70% of 65g = 0.70 × 65 = 45.5 g

🔵 Step 2: Calculate Vegetable Oil Weight
⬥ 24% of 65g = 0.24 × 65 = 15.6 g

🟡 Step 3: Calculate Salt Weight
⬥ 3% of 65g = 0.03 × 65 = 1.95 g

🔴 Step 4: Calculate Spice Weight
⬥ 3% of 65g = 0.03 × 65 = 1.95 g

🔒 ❓ 13. Three shops sell the same items at the same price. The shops offer deals as follows:
Shop A: “Buy 1 and get 1 free”
Shop B: “Buy 2 and get 1 free”
Shop C: “Buy 3 and get 1 free”
Answer the following:

🔒 ❓ (i) If the price of one item is ₹100, what is the effective price per item in each shop? Arrange the shops from cheapest to costliest.
📌 ✅ Answer:
🟢 Shop A
⬥ You get 2 items for the price of 1 (₹100).
⬥ Effective Price = 100 ÷ 2 = ₹50.
🔵 Shop B
⬥ You get 3 items for the price of 2 (₹200).
⬥ Effective Price = 200 ÷ 3 ≈ ₹66.67.
🟡 Shop C
⬥ You get 4 items for the price of 3 (₹300).
⬥ Effective Price = 300 ÷ 4 = ₹75.
🔴 Order:
⬥ Shop A < Shop B < Shop C (Cheapest to Costliest).

🔒 ❓ (ii) For each shop, calculate the percentage discount on the items.
📌 ✅ Answer:
🟢 Shop A: (50 off 100) ➔ 50%
🔵 Shop B: (33.33 off 100) ➔ 33.33%
🟡 Shop C: (25 off 100) ➔ 25%

🔒 ❓ (iii) Suppose you need 4 items. Which shop would you choose? Why?
📌 ✅ Answer:
🟢 Calculate Cost for 4 items:
⬥ Shop A: Buy 2, Get 2 Free. Cost = 2 × 100 = ₹200.
⬥ Shop B: Buy 2 Get 1 Free (3 items for ₹200). Need 1 more (Pay ₹100). Total = ₹300.
⬥ Shop C: Buy 3 Get 1 Free (4 items). Cost = 3 × 100 = ₹300.
🔵 Conclusion
⬥ You would choose Shop A because it is the cheapest (₹200 vs ₹300).

🔒 ❓ 14. In a room of 100 people, 99% are left-handed. How many left-handed people have to leave the room to bring that percentage down to 98%?
📌 ✅ Answer:
🟢 Step 1: Analyze Initial Counts
⬥ Total = 100.
⬥ Left-Handed (LH) = 99.
⬥ Right-Handed (RH) = 1.

🔵 Step 2: Understand the Goal
⬥ We want the same 1 Right-Handed person to represent 2% of the new total (since 100% – 98% LH = 2% RH).

🟡 Step 3: Calculate New Total
⬥ If 1 person = 2% of Total, then:
⬥ Total = 1 ÷ 0.02 = 50 people.

🔴 Step 4: Calculate number to leave
⬥ Original Total = 100. New Total = 50.
⬥ People to leave = 100 – 50 = 50 people.
⬥ (Since the RH person stays, all 50 leavers are left-handed).

🔒 ❓ 15. Look at the following graph.
(Graph shows “Ability to use computer by age and gender”)
Based on the graph, which of the following statement(s) are valid?
(i) People in their twenties are the most computer-literate among all age groups.
(ii) Women lag behind in the ability to use computers across age groups.
(iii) There are more people in their twenties than teenagers.
(iv) More than a quarter of people in their thirties can use computers.
(v) Less than 1 in 10 aged 60 and above can use computers.
(vi) Half of the people in their twenties can use computers.

📌 ✅ Answer:
🟢 Analysis of Statements:
⬥ (i) Valid: The bars for “Twenties” (both male and female) are higher than any other age group.
⬥ (ii) Valid: In every age category, the blue bar (Female) is shorter than the orange bar (Male).
⬥ (iii) Invalid: The graph shows percentages of ability, not total population numbers.
⬥ (iv) Invalid: Thirties data shows Male ~25% and Female ~14%. The average is below 25%.
⬥ (v) Valid: Seniors (60+) show 2% and 4%. Both are well below 10% (1 in 10).
⬥ (vi) Invalid: Twenties data shows ~26% and ~37%. Neither reaches 50% (half).
🔵 Conclusion
⬥ The valid statements are (i), (ii), and (v).
✔️ All questions and answers belong to this lesson only.
✔️ All answers are rechecked twice and found correct.

Important Notice : The lessons and study materials up to July–August 2025 have been sent. Soon, after receiving feedback from teachers and students, we will deliver the latest and more innovative study materials for the upcoming months.

Important Notice : The lessons and study materials up to July–August 2025 have been sent. Soon, after receiving feedback from teachers and students, we will deliver the latest and more innovative study materials for the upcoming months.

Class 8 : Maths – Lesson 8. Fractions in Disguise

EXPLANATION AND ANALYSIS

TEXTBOOK QUESTIONS

OTHER IMPORTANT QUESTIONS

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