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📐 Chapter 7 · NCERT 2025-26

⚖️ Proportional Reasoning — 1

Ratios · Proportion · Direct Variation · Unitary Method · Percentage · Profit/Loss · Simple Interest

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📐 Introduction to Proportional Reasoning

Have you ever noticed that if you buy 2 notebooks for ₹50, then 4 notebooks of the same kind cost ₹100? Or that a recipe for 4 people needs exactly double the ingredients when you cook for 8? These everyday observations are examples of proportional reasoning — one of the most powerful ideas in mathematics.

Proportional reasoning is about understanding how quantities relate to each other. When one quantity changes, how does the other change? Does it double? Halve? Stay the same? This chapter builds a strong foundation in ratios, proportions, and their applications in real life — from shopping discounts to bank interest.

🌎 Why Proportional Reasoning Matters

🛒 Shopping & Budgeting

Comparing prices, calculating discounts, and figuring out which deal gives you more value — all require proportional reasoning. "Buy 3 get 1 free" vs "25% off" — which is better?

🍳 Cooking & Recipes

Scaling a recipe up or down is a direct application of ratios. If a recipe for 6 people needs 3 cups of flour, how much flour for 10 people?

📊 Maps & Scale Drawings

Every map uses a scale ratio (e.g. 1 : 50,000). Architects and engineers use scale drawings where proportional reasoning converts centimetres on paper to metres in real life.

🏦 Banking & Finance

Interest rates, profit percentages, tax calculations, and EMIs all rely on proportional thinking. Understanding these concepts helps you make smart financial decisions.
🇮🇳 Indian Mathematical Heritage

Indian mathematicians made foundational contributions to the theory of ratios and proportions long before the modern era.

📜 Aryabhata (476–550 CE)

In his masterwork Aryabhatiya, Aryabhata used proportional reasoning extensively for astronomical calculations — computing planetary positions, eclipse timings, and the circumference of the Earth.

💫 Brahmagupta (598–668 CE)

Brahmagupta's Brahmasphutasiddhanta contains the Rule of Three (Trairasika) — the earliest systematic treatment of the unitary method. This rule was later transmitted to the Arab world and then to Europe.

🧮 Bhaskaracharya (1114–1185 CE)

In Lilavati, Bhaskara II presented beautiful problems on ratios, proportions, and the Rule of Three. He extended it to the Rule of Five, Seven, and Nine for problems involving multiple proportional quantities.

📚 Mahavira (9th century CE)

The Jain mathematician Mahavira's Ganita Sara Sangraha contains extensive chapters on proportion, mixtures, and commercial arithmetic — profit, loss, and interest problems very similar to what we study today.
💡 Did You Know? The Rule of Three (Trairasika), which is essentially the unitary method, was India's greatest mathematical export. It travelled from India to the Islamic world, and from there to medieval Europe, where it was called the "Golden Rule" of mathematics because of its immense usefulness in commerce!
📚 Chapter Overview
Topic Key Concepts
Ratios Comparison of quantities, simplification, equivalent ratios
Proportion Equality of ratios, mean proportion, third proportion
Direct Proportion Direct variation, constant ratio, y = kx
Unitary Method Finding value of one unit, then required units
Percentage Ratio to 100, conversion between fractions, decimals, and percentages
Profit, Loss & Discount CP, SP, profit %, loss %, marked price, discount %
Simple Interest Principal, rate, time, SI = PRT/100
⚖️ Ratios and Their Simplification
🔢 What Is a Ratio?

A ratio is a way of comparing two quantities of the same kind by division. If we have two quantities a and b (where b ≠ 0), the ratio of a to b is written as a : b or a/b.

  • The quantities a and b are called the terms of the ratio.
  • a is the antecedent (first term) and b is the consequent (second term).
  • A ratio has no unit — it is a pure number because the units cancel out.
  • Both quantities must be in the same unit before forming the ratio.
Ratio of a to b = a : b = a / b where b ≠ 0, and both a and b are in the same unit
🔨 Simplifying Ratios

A ratio is in its simplest form (or lowest terms) when the HCF of both terms is 1. To simplify a ratio, divide both terms by their HCF.

Example 1: Simplify the ratio 48 : 36.

Step 1: Find the HCF of 48 and 36.
48 = 24 × 3, and 36 = 22 × 32
HCF = 22 × 3 = 12
Step 2: Divide both terms by the HCF.
48 ÷ 12 : 36 ÷ 12 = 4 : 3
🔄 Equivalent Ratios

Just like equivalent fractions, we can create equivalent ratios by multiplying or dividing both terms by the same non-zero number.

For example: 2 : 3 = 4 : 6 = 6 : 9 = 10 : 15 (all are equivalent ratios).

Example 2: Express the ratio 250 g to 1.5 kg in simplest form.

Step 1: Convert to the same unit.
1.5 kg = 1500 g
Step 2: Form the ratio.
250 : 1500
Step 3: Simplify by dividing both by HCF(250, 1500) = 250.
250 ÷ 250 : 1500 ÷ 250 = 1 : 6
📋 Comparing Ratios

To compare two ratios, convert them to fractions and then to like fractions (common denominator) or convert them to decimals.

🔎 Method 1: Cross Multiplication

To compare a : b and c : d, cross multiply:
If ad > bc, then a : b > c : d
If ad = bc, then a : b = c : d
If ad < bc, then a : b < c : d

🔎 Method 2: Decimal Conversion

Convert each ratio to a decimal (divide antecedent by consequent) and compare the decimal values directly.
💡 Common Mistake: Students often forget to convert quantities to the same unit before forming a ratio. You cannot compare 5 metres and 300 centimetres directly — first convert both to metres (5 : 3) or both to centimetres (500 : 300 = 5 : 3).
⚙️ Proportion and Its Properties
📐 What Is Proportion?

When two ratios are equal, they are said to be in proportion. If a : b = c : d, we write a : b :: c : d and read it as "a is to b as c is to d."

a : b :: c : d means a/b = c/d Equivalently, a × d = b × c (product of extremes = product of means)
  • a and d are called the extremes (outermost terms).
  • b and c are called the means (middle terms).
  • The fundamental property: Product of extremes = Product of means, i.e., ad = bc.

Example 3: Check whether 3, 5, 12, 20 are in proportion.

Step 1: Identify extremes and means.
Extremes: 3 and 20. Means: 5 and 12.
Step 2: Check: Product of extremes = 3 × 20 = 60
Product of means = 5 × 12 = 60
Step 3: Since 60 = 60, the numbers are in proportion.
We write: 3 : 5 :: 12 : 20
🔢 Mean Proportion

If a, b, c are in continued proportion (i.e., a : b = b : c), then b is called the mean proportional between a and c.

b = √(a × c) Mean proportion of a and c

Example 4: Find the mean proportion between 4 and 25.

Step 1: Let the mean proportion be b.
Then 4 : b = b : 25, so b² = 4 × 25 = 100.
Step 2: b = √100 = 10.
Verify: 4 : 10 = 2 : 5 and 10 : 25 = 2 : 5. ✅
🔢 Third Proportion

If a, b, c are in continued proportion (a : b = b : c), then c is called the third proportional to a and b.

c = b² / a Third proportion to a and b

Example 5: Find the third proportional to 3 and 6.

Step 1: Let the third proportional be c.
Then 3 : 6 = 6 : c.
Step 2: Using the property of proportion: 3 × c = 6 × 6 = 36.
c = 36 / 3 = 12.
Verify: 3 : 6 = 1 : 2 and 6 : 12 = 1 : 2. ✅
💡 Key Difference: Mean proportion is the middle term when three numbers are in continued proportion. Third proportion is the last term. For a, b, c in continued proportion: b is the mean proportion, and c is the third proportion to a and b.
➡️ Direct Proportion (Direct Variation)
📐 Understanding Direct Proportion

Two quantities are said to be in direct proportion if an increase in one causes a proportional increase in the other, and a decrease in one causes a proportional decrease in the other. The ratio between the two quantities remains constant.

If x and y are in direct proportion: x / y = k (constant) Equivalently, y = kx, where k is the constant of proportionality

✅ Examples of Direct Proportion

• Number of items bought and total cost (at fixed price per item)
• Distance covered and petrol consumed (at constant mileage)
• Number of workers and total wages (at same daily wage)
• Weight of fruit and its cost (at fixed rate per kg)

❌ Not Direct Proportion

• Speed and time to cover a fixed distance (these are inversely related)
• Number of workers and days to complete a job (inverse)
• Temperature and altitude (not a simple proportion)
📊 Recognising Direct Proportion

To check if two quantities x and y are in direct proportion, compute x/y (or y/x) for different pairs. If the ratio remains the same, they are in direct proportion.

Notebooks (x) Cost in ₹ (y) y/x Direct Proportion?
26030Yes! y/x = 30 (constant)
515030
824030
1236030
✏️ Solving Direct Proportion Problems

Example 6: If 5 kg of rice costs ₹350, find the cost of 8 kg of rice.

Step 1: Identify the type of proportion.
More rice → more cost. This is direct proportion.
Step 2: Set up the proportion.
5 : 8 = 350 : x (where x is the unknown cost)
Step 3: Using the property: 5 × x = 8 × 350 = 2800
x = 2800 / 5 = ₹560
💡 Quick Check for Direct Proportion: Ask yourself — "If I double one quantity, does the other also double?" If yes, it is direct proportion. If the other halves instead, it is inverse proportion (which you will study in the next chapter).
🔢 The Unitary Method
📐 What Is the Unitary Method?

The unitary method is a technique where we first find the value of one unit of a quantity, and then use it to find the value of any number of units. It is essentially the Indian Rule of Three (Trairasika) in action!

Step 1: Find value of 1 unit = Total value ÷ Number of units
Step 2: Find value of required units = Value of 1 unit × Required units Two simple steps that solve most proportion problems!

Example 7: If 12 pens cost ₹180, find the cost of 7 pens.

Step 1: Find the cost of 1 pen.
Cost of 1 pen = 180 ÷ 12 = ₹15
Step 2: Find the cost of 7 pens.
Cost of 7 pens = 15 × 7 = ₹105

Example 8: A car covers 240 km using 15 litres of petrol. How much petrol is needed for 400 km?

Step 1: Identify the relationship.
More distance → more petrol needed (direct proportion).
Step 2: Find petrol for 1 km.
Petrol for 1 km = 15 / 240 = 1/16 litres
Step 3: Find petrol for 400 km.
Petrol for 400 km = (1/16) × 400 = 400/16 = 25 litres

💡 When to Use the Unitary Method

The unitary method works beautifully for:
• Cost problems (price of multiple items)
• Distance-fuel problems
• Work-time problems
• Recipe scaling
• Map and scale problems

⚖️ Alternative: Proportion Method

Instead of finding the value of 1 unit, you can set up a proportion directly:
12 pens : 7 pens = ₹180 : ₹x
12/7 = 180/x → x = (180 × 7)/12 = ₹105
Both methods give the same answer!
💡 Memory Aid — "Trairasika" (Rule of Three): Brahmagupta taught: arrange three known quantities so that the first and third are of the same kind, and the second is of the other kind. Then: Answer = (Second × Third) ÷ First. This ancient formula is exactly the unitary method!
📈 Percentage as a Ratio
🔢 What Is a Percentage?

A percentage is a ratio expressed as a fraction of 100. The word "percent" literally means "per hundred" (from the Latin per centum). So 45% means 45 out of 100, or 45/100.

Percentage = (Part / Whole) × 100 Equivalently: Part = (Percentage / 100) × Whole
🔄 Conversions
Conversion Method Example
Fraction → Percentage Multiply by 100 3/5 = (3/5) × 100 = 60%
Percentage → Fraction Divide by 100 and simplify 75% = 75/100 = 3/4
Decimal → Percentage Multiply by 100 0.35 = 0.35 × 100 = 35%
Percentage → Decimal Divide by 100 12.5% = 12.5/100 = 0.125
Ratio → Percentage Convert ratio to fraction, then × 100 3 : 7 = 3/10 of total = 30% (of total for first part)
✏️ Finding Percentage of a Quantity

✅ 25% of 480

= (25/100) × 480 = 0.25 × 480 = 120

🔵 12.5% of 640

= (12.5/100) × 640 = (1/8) × 640 = 80

🟠 150% of 60

= (150/100) × 60 = 1.5 × 60 = 90
(Percentage can be more than 100!)

🔷 What % is 36 of 90?

= (36/90) × 100 = 0.4 × 100 = 40%
📊 Percentage Increase and Decrease
% Increase = (Increase / Original) × 100
% Decrease = (Decrease / Original) × 100 Always divide by the ORIGINAL value
💡 Fraction-Percentage Quick Reference:
1/2 = 50%  |  1/3 ≈ 33.33%  |  1/4 = 25%  |  1/5 = 20%  |  1/8 = 12.5%  |  1/10 = 10%  |  1/6 ≈ 16.67%  |  3/4 = 75%
Memorise these — they save time in exams!
💰 Profit, Loss & Discount
💰 Key Terms

🛒 Cost Price (CP)

The price at which an article is bought (purchased). This is the amount the shopkeeper pays to acquire the item.

💲 Selling Price (SP)

The price at which an article is sold. This is the amount the customer pays to the shopkeeper.

📈 Marked Price (MP)

The price printed on the label of an article. Also called the "list price." The actual selling price may be different after discount.

🏷️ Discount

The reduction given on the marked price. Discount = MP − SP. Discount is always calculated on the marked price.
📊 Profit and Loss Formulas
Profit = SP − CP (when SP > CP)
Loss = CP − SP (when CP > SP) No profit, no loss when SP = CP
Profit % = (Profit / CP) × 100
Loss % = (Loss / CP) × 100 Profit % and Loss % are always calculated on Cost Price
Discount = MP − SP
Discount % = (Discount / MP) × 100 Discount % is always calculated on Marked Price

Example 9: A shopkeeper buys a shirt for ₹800 and sells it for ₹920. Find the profit and profit percentage.

Step 1: Identify CP and SP.
CP = ₹800, SP = ₹920.
Step 2: Since SP > CP, there is a profit.
Profit = SP − CP = 920 − 800 = ₹120
Step 3: Calculate profit percentage.
Profit % = (120 / 800) × 100 = 15%

Example 10: A table is marked at ₹5000. A discount of 20% is given. Find the selling price.

Step 1: MP = ₹5000, Discount = 20%.
Step 2: Discount amount = (20/100) × 5000 = ₹1000.
Step 3: SP = MP − Discount = 5000 − 1000 = ₹4000.
📋 Finding SP from CP and Profit/Loss %
SP = CP × (100 + Profit%) / 100   (in case of profit)
SP = CP × (100 − Loss%) / 100   (in case of loss) These shortcut formulas save time in exams
💡 Remember: Profit % and Loss % are always on CP. Discount % is always on MP. This is the most common source of errors in exams — never mix them up!
💡 Real-Life Tip: When a shop says "50% off!", they mean 50% discount on the marked price. The shopkeeper may have already marked the item at a price much higher than the cost price. So even after 50% discount, the shopkeeper might still be making a profit!
🏦 Simple Interest
💰 Understanding Simple Interest

When you deposit money in a bank or borrow money, you earn or pay interest. Simple Interest (SI) is the interest calculated on the original principal for each time period. Unlike compound interest, simple interest does not add the interest earned back to the principal.

💰 Principal (P)

The original amount of money borrowed or deposited. This is the base amount on which interest is calculated.

📈 Rate (R)

The percentage of interest charged or earned per year (per annum). Written as R% p.a.

⏰ Time (T)

The duration for which money is borrowed or deposited. Usually in years. Convert months to years as needed (e.g. 6 months = 0.5 years).

💲 Amount (A)

The total money returned or accumulated: A = P + SI. This is what you get back (if depositing) or what you owe (if borrowing).
SI = (P × R × T) / 100 Simple Interest Formula — P = Principal, R = Rate (% per annum), T = Time (in years)
Amount (A) = P + SI = P + (P × R × T) / 100 = P(1 + RT/100) Total amount = Principal + Interest earned

Example 11: Find the simple interest on ₹12,000 at 8% per annum for 3 years.

Step 1: Identify the values.
P = ₹12,000, R = 8%, T = 3 years.
Step 2: Apply the formula.
SI = (P × R × T) / 100 = (12000 × 8 × 3) / 100
Step 3: Calculate.
SI = 288000 / 100 = ₹2,880
Step 4: Amount = P + SI = 12000 + 2880 = ₹14,880

Example 12: At what rate of interest will ₹5,000 yield ₹1,500 as simple interest in 5 years?

Step 1: Given: P = ₹5000, SI = ₹1500, T = 5 years. Find R.
Step 2: SI = PRT/100
1500 = (5000 × R × 5) / 100
Step 3: 1500 = 25000R / 100 = 250R
R = 1500 / 250 = 6% per annum
💡 Time Conversion: If time is given in months, convert to years: T = months/12. If time is given as "2 years 6 months," then T = 2.5 years or 5/2 years.
💡 SI is Proportional! Simple interest is directly proportional to each of P, R, and T (when the other two are constant). Double the principal → double the SI. Triple the time → triple the SI. This is proportional reasoning in action!
✏️ More Worked Examples

Example 13: A map has a scale of 1 : 25000. Two cities are 8 cm apart on the map. Find the actual distance between them.

Step 1: Scale means 1 cm on map = 25000 cm in reality.
Step 2: Actual distance = 8 × 25000 = 200000 cm.
Step 3: Convert: 200000 cm = 2000 m = 2 km.

Example 14: In a class, the ratio of boys to girls is 3 : 2. If there are 45 students in total, how many are boys and how many are girls?

Step 1: Sum of ratio parts = 3 + 2 = 5.
Step 2: Boys = (3/5) × 45 = 27 boys.
Step 3: Girls = (2/5) × 45 = 18 girls.
Verify: 27 + 18 = 45 ✅ and 27 : 18 = 3 : 2 ✅

Example 15: The population of a town increased from 50,000 to 55,000 in one year. Find the percentage increase.

Step 1: Increase = 55000 − 50000 = 5000.
Step 2: % Increase = (5000 / 50000) × 100 = 10%.

Example 16: A trader buys 50 kg of rice at ₹40 per kg and sells it at ₹46 per kg. Find the total profit and profit percentage.

Step 1: Total CP = 50 × 40 = ₹2000.
Step 2: Total SP = 50 × 46 = ₹2300.
Step 3: Profit = 2300 − 2000 = ₹300.
Step 4: Profit % = (300 / 2000) × 100 = 15%.

Example 17: Ravi deposited ₹25,000 in a bank at 7.5% per annum simple interest. How much money will he have after 4 years?

Step 1: P = ₹25000, R = 7.5%, T = 4 years.
Step 2: SI = (25000 × 7.5 × 4) / 100 = 750000 / 100 = ₹7500.
Step 3: Amount = P + SI = 25000 + 7500 = ₹32,500.
📊 Chapter Summary

⚖️ Ratio

A comparison of two quantities by division. Written as a : b. Must be in same units. Simplify by dividing by HCF.

⚙️ Proportion

Equality of two ratios: a : b :: c : d. Product of extremes = Product of means (ad = bc).

➡️ Direct Proportion

Two quantities where x/y = constant. If one increases, the other increases proportionally. y = kx.

🔢 Unitary Method

Find value of 1 unit first, then find value of required units. The Indian "Rule of Three."

📈 Percentage

A ratio out of 100. % = (Part/Whole) × 100. Can convert to/from fractions and decimals.

💰 Profit & Loss

Profit = SP − CP. Loss = CP − SP. Profit/Loss % on CP. Discount % on MP.

🏦 Simple Interest

SI = PRT/100. Amount = P + SI. SI is directly proportional to P, R, and T individually.

💡 Mean & Third Proportion

Mean proportion of a and c = √(ac). Third proportion to a and b = b²/a.
💡 Exam Strategy: Proportional reasoning questions appear in every exam — NCERT, board exams, and competitive tests. Master the unitary method and the relationship between fractions, decimals, and percentages. Always check your answer by substituting back into the original problem!
🧠 Multiple Choice Questions (10 MCQs)

Click on an option to see if your answer is correct. The correct option will turn green.

  • Q1. The simplest form of the ratio 72 : 48 is:
    • a) 6 : 4
    • b) 3 : 2
    • c) 9 : 6
    • d) 12 : 8
    ✅ Answer: (b) 3 : 2 — HCF of 72 and 48 is 24. Dividing both by 24 gives 3 : 2.
  • Q2. If 4 : 7 = x : 35, then x equals:
    • a) 15
    • b) 20
    • c) 25
    • d) 28
    ✅ Answer: (b) 20 — 4/7 = x/35. Cross multiplying: x = (4 × 35)/7 = 140/7 = 20.
  • Q3. The mean proportional between 9 and 16 is:
    • a) 11
    • b) 12
    • c) 13
    • d) 14
    ✅ Answer: (b) 12 — Mean proportion = √(9 × 16) = √144 = 12.
  • Q4. If 8 oranges cost ₹56, the cost of 15 oranges is:
    • a) ₹95
    • b) ₹100
    • c) ₹105
    • d) ₹112
    ✅ Answer: (c) ₹105 — Cost of 1 orange = 56/8 = ₹7. Cost of 15 = 7 × 15 = ₹105.
  • Q5. 0.125 expressed as a percentage is:
    • a) 1.25%
    • b) 12.5%
    • c) 125%
    • d) 0.125%
    ✅ Answer: (b) 12.5% — 0.125 × 100 = 12.5%.
  • Q6. A shirt is bought for ₹600 and sold for ₹540. The loss percentage is:
    • a) 10%
    • b) 12%
    • c) 8%
    • d) 15%
    ✅ Answer: (a) 10% — Loss = 600 − 540 = ₹60. Loss % = (60/600) × 100 = 10%.
  • Q7. Simple interest on ₹4,000 at 5% p.a. for 2 years is:
    • a) ₹200
    • b) ₹400
    • c) ₹500
    • d) ₹800
    ✅ Answer: (b) ₹400 — SI = (4000 × 5 × 2)/100 = 40000/100 = ₹400.
  • Q8. If the marked price of a book is ₹250 and a discount of 12% is given, the selling price is:
    • a) ₹200
    • b) ₹210
    • c) ₹220
    • d) ₹230
    ✅ Answer: (c) ₹220 — Discount = (12/100) × 250 = ₹30. SP = 250 − 30 = ₹220.
  • Q9. The third proportional to 4 and 12 is:
    • a) 24
    • b) 36
    • c) 48
    • d) 16
    ✅ Answer: (b) 36 — Third proportional = b²/a = 12²/4 = 144/4 = 36.
  • Q10. Two quantities are in direct proportion. When one is 6, the other is 15. When the first is 10, the second will be:
    • a) 20
    • b) 25
    • c) 30
    • d) 19
    ✅ Answer: (b) 25 — k = 15/6 = 5/2. When x = 10, y = 10 × 5/2 = 25.
✍️ NCERT Short Answer Questions (10)
  • Q1. Express the ratio 2.5 : 3.5 in simplest form.
    Multiply both terms by 10 to remove decimals: 25 : 35. HCF of 25 and 35 is 5. Dividing: 25 ÷ 5 : 35 ÷ 5 = 5 : 7.
  • Q2. Are 15, 20, 45, 60 in proportion? Justify.
    Check: Product of extremes = 15 × 60 = 900. Product of means = 20 × 45 = 900. Since they are equal, yes, these numbers are in proportion: 15 : 20 :: 45 : 60 (both simplify to 3 : 4).
  • Q3. Find the value of x if 3 : x :: 12 : 20.
    Using the property of proportion: 3 × 20 = x × 12, so 60 = 12x, giving x = 60/12 = 5.
  • Q4. A car travels 150 km in 3 hours. How far will it travel in 5 hours at the same speed?
    Speed (distance per hour) = 150/3 = 50 km/h. In 5 hours: 50 × 5 = 250 km. (This is direct proportion: more time → more distance at constant speed.)
  • Q5. What percentage of 150 is 36?
    Percentage = (36/150) × 100 = (36 × 100)/150 = 3600/150 = 24%.
  • Q6. A watch is bought for ₹1200 and sold at a profit of 15%. Find the selling price.
    SP = CP × (100 + Profit%)/100 = 1200 × (100 + 15)/100 = 1200 × 115/100 = 1200 × 1.15 = ₹1380.
  • Q7. Find the simple interest on ₹8,000 at 6% p.a. for 2 years 6 months.
    T = 2 years 6 months = 2.5 years = 5/2 years.
    SI = (P × R × T)/100 = (8000 × 6 × 5/2)/100 = (8000 × 15)/100 = 120000/100 = ₹1200.
  • Q8. Divide ₹560 in the ratio 3 : 5.
    Sum of ratio parts = 3 + 5 = 8.
    First share = (3/8) × 560 = ₹210.
    Second share = (5/8) × 560 = ₹350.
    Verify: 210 + 350 = 560 ✅
  • Q9. The price of a bag is marked at ₹1800. A shopkeeper gives two successive discounts of 10% and 5%. Find the final selling price.
    After first discount (10%): SP1 = 1800 − (10/100) × 1800 = 1800 − 180 = ₹1620.
    After second discount (5% on ₹1620): SP2 = 1620 − (5/100) × 1620 = 1620 − 81 = ₹1539.
  • Q10. In what time will ₹6,000 amount to ₹7,800 at 10% p.a. simple interest?
    SI = A − P = 7800 − 6000 = ₹1800.
    SI = PRT/100 ⇒ 1800 = (6000 × 10 × T)/100 = 600T.
    T = 1800/600 = 3 years.
📖 NCERT Long Answer Questions (5)
Q1. Explain the concept of ratio and proportion with examples. Define mean proportion and third proportion, and find the mean proportional between 8 and 32, and the third proportional to 6 and 18.
Ratio: A ratio is a comparison of two quantities of the same kind by division. The ratio of a to b is written as a : b (where b ≠ 0). For example, if there are 20 boys and 15 girls in a class, the ratio of boys to girls is 20 : 15 = 4 : 3.

Proportion: When two ratios are equal, the four quantities are said to be in proportion. If a : b = c : d, we write a : b :: c : d. The fundamental property is: product of extremes = product of means (ad = bc). For example, 2 : 3 :: 8 : 12 because 2 × 12 = 3 × 8 = 24.

Mean Proportion: If a, b, c are in continued proportion (a : b = b : c), then b is the mean proportional between a and c, and b = √(ac).
Mean proportional between 8 and 32: b = √(8 × 32) = √256 = 16.
Verify: 8 : 16 = 1 : 2 and 16 : 32 = 1 : 2 ✅

Third Proportion: If a : b = b : c, then c is the third proportional to a and b, and c = b²/a.
Third proportional to 6 and 18: c = 18²/6 = 324/6 = 54.
Verify: 6 : 18 = 1 : 3 and 18 : 54 = 1 : 3 ✅
Q2. Rahul bought a bicycle for ₹4,500 and spent ₹500 on its repair. He then sold it at a profit of 12%. Find the selling price. If instead he had sold it at a loss of 8%, what would the selling price have been? Also find the difference between the two selling prices.
Total Cost Price: CP = Purchase price + Repair cost = 4500 + 500 = ₹5,000.

Case 1 — Profit of 12%:
SP = CP × (100 + 12)/100 = 5000 × 112/100 = 5000 × 1.12 = ₹5,600.

Case 2 — Loss of 8%:
SP = CP × (100 − 8)/100 = 5000 × 92/100 = 5000 × 0.92 = ₹4,600.

Difference: 5600 − 4600 = ₹1,000.

Note: The difference of ₹1000 represents (12% + 8%) = 20% of ₹5000, which is indeed 0.20 × 5000 = 1000. This makes sense because the profit and loss percentages are both calculated on the same CP.
Q3. A factory produces 840 toys in 6 days working 7 hours per day. How many toys will it produce in 10 days working 8 hours per day? (Assume the rate of production is constant per hour.)
Step 1: Find the rate of production per hour.
Total hours in first case = 6 × 7 = 42 hours.
Toys per hour = 840 / 42 = 20 toys/hour.

Step 2: Find total hours in the second case.
Total hours = 10 × 8 = 80 hours.

Step 3: Find toys produced.
Toys = 20 × 80 = 1600 toys.

Alternative (Proportion method):
More days → more toys (direct proportion).
More hours/day → more toys (direct proportion).
840 / (6 × 7) = x / (10 × 8)
840 / 42 = x / 80
x = (840 × 80) / 42 = 67200 / 42 = 1600 toys.
Q4. Priya deposited ₹15,000 in Bank A at 8% p.a. and ₹20,000 in Bank B at 6.5% p.a. simple interest. After 3 years, which bank gave her more interest and by how much? Also find the total amount she has after 3 years.
Bank A:
SIA = (15000 × 8 × 3)/100 = 360000/100 = ₹3,600.

Bank B:
SIB = (20000 × 6.5 × 3)/100 = 390000/100 = ₹3,900.

Comparison: Bank B gave more interest.
Difference = 3900 − 3600 = ₹300 more from Bank B.

Total amount after 3 years:
Amount from A = 15000 + 3600 = ₹18,600.
Amount from B = 20000 + 3900 = ₹23,900.
Total = 18600 + 23900 = ₹42,500.
Q5. Explain the unitary method with reference to the Indian "Rule of Three" (Trairasika). Solve: If 18 metres of cloth cost ₹2,700, find (i) the cost of 25 metres, (ii) the length of cloth that can be bought for ₹4,500, and (iii) the cost of cloth needed to make 5 curtains, each requiring 2.4 metres.
The Unitary Method & Rule of Three:
The unitary method finds the value of one unit first, then uses it to find the value of any number of units. This is precisely the ancient Indian Trairasika (Rule of Three) described by Brahmagupta and Bhaskaracharya. The rule states: if three quantities are known and they are in proportion, the fourth can be found as (Second × Third) ÷ First.

Step 1: Cost of 1 metre = 2700 / 18 = ₹150.

(i) Cost of 25 metres:
= 150 × 25 = ₹3,750.

(ii) Length for ₹4,500:
= 4500 / 150 = 30 metres.

(iii) Cost for 5 curtains:
Total cloth needed = 5 × 2.4 = 12 metres.
Cost = 150 × 12 = ₹1,800.
🌟 Fun Facts & Real-World Connections

🎲 The Golden Ratio

The famous Golden Ratio (approximately 1 : 1.618) appears in nature, art, and architecture. Sunflower spirals, the Parthenon, and even the proportions of the human face approximate this ratio!

🌍 Map Scales

A globe at 1 : 40,000,000 scale means 1 cm = 400 km. The Google Maps "scale bar" uses proportional reasoning to convert pixels to real-world distances.

🍳 Recipe Scaling

Professional chefs use "baker's percentages" where every ingredient is expressed as a percentage of the flour weight. This makes scaling recipes up or down trivially easy — pure proportional reasoning!

📈 Stock Market

When the news says "Sensex rose 2.5% today," they are using percentage change — a proportional comparison between today's value and yesterday's value. Investors use these percentages to make decisions.

🇮🇳 India's Rule of Three

The "Rule of Three" was so important in medieval European commerce that it was called the "Golden Rule" of mathematics. It originated in India and reached Europe via Arab mathematicians. Every merchant had to learn it!

💊 Medicine Dosages

Doctors use proportional reasoning to calculate medicine dosages based on a patient's weight. A child's dose is often proportional to their weight compared to an adult's standard weight.
💡 Final Tip: Proportional reasoning is not just a chapter in your textbook — it is a life skill! Every time you compare prices, calculate tips, convert currencies, or understand statistics in the news, you are using what you learned here. You've got this! Go ace that exam! 💪
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