# Comparing Quantities

## Overview

Comparing Quantities is a fundamental chapter that bridges arithmetic with real-world applications in commerce, finance, and daily life. This chapter explores various methods of comparing quantities through ratios, percentages, discounts, taxes, and interest calculations. Students will develop skills essential for understanding business mathematics, financial literacy, and practical problem-solving scenarios encountered in everyday life.

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## Key Topics Covered

### 1. Recalling Ratios and Percentages

#### 1.1 Understanding Ratios

##### Definition and Basic Concepts
- **Ratio**: A comparison of two quantities of the same kind
- **Expression**: For quantities a and b, ratio is written as a : b or $\frac{a}{b}$
- **Reading**: "a is to b"

##### Practical Applications
**Example**: In a basket with 20 apples and 5 oranges:
- Ratio of oranges to apples = 5 : 20 = 1 : 4
- This means oranges are $\frac{1}{4}$ of the apples
- Alternatively, apples are 4 times the oranges

#### 1.2 Converting Ratios to Percentages

##### Method 1: Direct Calculation
For ratio a : b, percentage of a = $\frac{a}{a+b} \times 100\%$

**Example**: With 5 oranges and 20 apples (25 fruits total):
- Percentage of oranges = $\frac{5}{25} \times 100\% = 20\%$
- Percentage of apples = $\frac{20}{25} \times 100\% = 80\%$

##### Method 2: Unitary Method
- Out of 25 fruits, 5 are oranges
- Out of 100 fruits, oranges would be $\frac{5 \times 100}{25} = 20$
- Therefore, 20% are oranges

#### 1.3 Problem-Solving with Percentages

##### Finding Total from Given Percentage
**Example**: If 60% of students are girls and there are 18 girls, find total students.

**Solution**:
- Let total students = x
- 60% of x = 18
- $\frac{60}{100} \times x = 18$
- $x = \frac{18 \times 100}{60} = 30$ students

### 2. Finding Discounts

#### 2.1 Basic Discount Concepts

##### Key Terms
- **Marked Price (MP)**: Original price displayed on an item
- **Sale Price (SP)**: Actual price paid by customer
- **Discount**: Reduction given on marked price

##### Formula
$$\text{Discount} = \text{Marked Price} - \text{Sale Price}$$

#### 2.2 Calculating Discount Percentage

##### Formula
$$\text{Discount Percentage} = \frac{\text{Discount}}{\text{Marked Price}} \times 100\%$$

**Example**: Item marked at ₹840, sold for ₹714
- Discount = ₹840 - ₹714 = ₹126
- Discount% = $\frac{126}{840} \times 100\% = 15\%$

#### 2.3 Finding Sale Price from Discount Percentage

##### Method 1: Calculate Discount First
If MP = ₹220 and discount = 20%:
- Discount amount = $\frac{20}{100} \times 220 = ₹44$
- Sale Price = ₹220 - ₹44 = ₹176

##### Method 2: Direct Calculation
- 20% discount means customer pays 80% of MP
- Sale Price = $\frac{80}{100} \times 220 = ₹176$

#### 2.4 Estimation in Percentages

##### Quick Estimation Technique
For bill amount ₹577.80 with 15% discount:
1. Round to nearest ten: ₹580
2. Find 10%: ₹58
3. Find 5% (half of 10%): ₹29
4. Add: ₹58 + ₹29 = ₹87 (approximate discount)

### 3. Sales Tax/Value Added Tax/Goods and Services Tax

#### 3.1 Sales Tax (ST)

##### Characteristics
- Charged by government on sale of items
- Collected by shopkeeper from customer
- Added to the bill amount
- Calculated on selling price

##### Formula
$$\text{Sales Tax} = \frac{\text{Tax Rate}}{100} \times \text{Bill Amount}$$
$$\text{Total Amount} = \text{Bill Amount} + \text{Sales Tax}$$

**Example**: Roller skates cost ₹450, sales tax 5%
- Sales Tax = $\frac{5}{100} \times 450 = ₹22.50$
- Total Bill = ₹450 + ₹22.50 = ₹472.50

#### 3.2 Value Added Tax (VAT)

##### Key Difference from Sales Tax
- VAT is **included** in the displayed price
- Sales Tax is **added** to the displayed price

##### Finding Original Price from VAT-Inclusive Price
If price including 10% VAT is ₹3300:
- Let original price = ₹100, then VAT-inclusive price = ₹110
- When VAT-inclusive price is ₹3300, original price = $\frac{100 \times 3300}{110} = ₹3000$

#### 3.3 Goods and Services Tax (GST)

##### Introduction
- Introduced in India from July 1, 2017
- Levied on supply of goods or services or both
- Replaces multiple taxes

##### Calculation Method
Similar to VAT calculation:
If GST-inclusive price is ₹784 with 12% GST:
- Original price = $\frac{100 \times 784}{112} = ₹700$

### 4. Compound Interest

#### 4.1 Simple vs Compound Interest

##### Simple Interest Characteristics
- Interest calculated only on principal
- Interest amount remains constant each year
- Formula: SI = $\frac{P \times R \times T}{100}$

##### Compound Interest Characteristics
- Interest calculated on previous year's amount (Principal + Interest)
- Interest amount increases each year
- More realistic for real-world scenarios

#### 4.2 Calculating Compound Interest Step-by-Step

**Example**: ₹20,000 borrowed for 2 years at 8% compounded annually

##### Year 1
- Principal (P₁) = ₹20,000
- Interest (SI₁) = $\frac{20000 \times 8}{100} = ₹1,600$
- Amount at end of Year 1 = ₹20,000 + ₹1,600 = ₹21,600

##### Year 2
- Principal (P₂) = ₹21,600 (previous year's amount)
- Interest (SI₂) = $\frac{21600 \times 8}{100} = ₹1,728$
- Amount at end of Year 2 = ₹21,600 + ₹1,728 = ₹23,328

##### Total Compound Interest
CI = ₹1,600 + ₹1,728 = ₹3,328

##### Comparison with Simple Interest
SI for 2 years = $\frac{20000 \times 8 \times 2}{100} = ₹3,200$
Difference = ₹3,328 - ₹3,200 = ₹128

#### 4.3 Compound Interest Formula

##### Derivation
For principal P, rate R% per annum, time n years:
$$A = P\left(1 + \frac{R}{100}\right)^n$$
$$CI = A - P = P\left(1 + \frac{R}{100}\right)^n - P$$

##### Application
**Example**: CI on ₹12,600 for 2 years at 10% per annum
$$A = 12600 \times \left(1 + \frac{10}{100}\right)^2 = 12600 \times \left(\frac{11}{10}\right)^2 = ₹15,246$$
$$CI = ₹15,246 - ₹12,600 = ₹2,646$$

### 5. Applications of Compound Interest Formula

#### 5.1 Population Growth

##### Application of CI Formula
Population growth follows compound interest pattern:
$$\text{Final Population} = \text{Initial Population} \times \left(1 + \frac{\text{Growth Rate}}{100}\right)^{\text{Years}}$$

**Example**: City population 20,000 in 1997, growing at 5% annually
Population in 2000 = $20000 \times \left(1 + \frac{5}{100}\right)^3 = 20000 \times \left(\frac{21}{20}\right)^3 = 23,153$

#### 5.2 Depreciation

##### Understanding Depreciation
- Reduction in value due to use and age
- Follows compound interest formula with negative rate

##### Formula for Depreciation
$$\text{Final Value} = \text{Initial Value} \times \left(1 - \frac{\text{Depreciation Rate}}{100}\right)^{\text{Years}}$$

**Example**: TV bought for ₹21,000, depreciates 5% annually
Value after 1 year = $21000 \times \left(1 - \frac{5}{100}\right) = 21000 \times \frac{19}{20} = ₹19,950$

#### 5.3 Other Applications

##### Bacterial Growth
- Rate of increase known
- Use compound interest formula with appropriate time units

##### Investment Growth
- Money invested with compound returns
- Application in financial planning

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## New Terms and Simple Definitions

| Term | Simple Definition |
|------|------------------|
| Ratio | Comparison of two quantities expressed as a:b |
| Percentage | Parts per hundred, expressed with % symbol |
| Marked Price | Original price tag on an item |
| Sale Price | Actual price paid by customer after discount |
| Discount | Reduction given on marked price |
| Sales Tax | Tax added to bill amount by government |
| VAT | Value Added Tax included in the price |
| GST | Goods and Services Tax levied on supply |
| Simple Interest | Interest calculated only on principal amount |
| Compound Interest | Interest calculated on previous year's amount |
| Principal | Original amount of money |
| Depreciation | Decrease in value over time |
| Appreciation | Increase in value over time |
| Overhead Expenses | Additional costs after purchasing |
| Unitary Method | Finding value of one unit to solve problems |

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## Discussion Questions

### Conceptual Understanding
1. Why is compound interest more realistic than simple interest in real-world scenarios?
2. How do discounts and sales taxes affect the final price paid by consumers?
3. What is the difference between sales tax and VAT in terms of calculation?
4. Why do banks use compound interest for both loans and deposits?

### Application-based Questions
1. If a shopkeeper gives 20% discount but adds 18% GST, what is the effective change in price?
2. How would you calculate the actual interest rate when processing fees are added to a loan?
3. In which scenarios would you prefer simple interest over compound interest?
4. How do compound interest principles apply to population growth studies?

### Critical Thinking
1. Is it better for a consumer to have a high discount with high tax or low discount with low tax?
2. How does the frequency of compounding (annual, quarterly, monthly) affect the final amount?
3. What economic factors influence discount rates offered by retailers?

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## Practice Problems

### Basic Level
1. Find the ratio of 15 minutes to 1 hour
2. Convert 3:4 to percentage
3. Find 15% of ₹240
4. A dress marked ₹500 is sold for ₹400. Find discount%

### Intermediate Level
1. If 72% of 25 students like mathematics, how many don't like it?
2. Find sale price if MP = ₹1200 and discount = 25%
3. Calculate bill amount if article costs ₹800 and sales tax is 12%
4. Find CI on ₹5000 for 2 years at 6% per annum

### Advanced Level
1. A article costing ₹1500 including 20% VAT. Find price before VAT
2. Population of a town is 50,000. If it grows at 8% annually, find population after 3 years
3. Calculate effective price change if 15% discount is given but 18% GST is added
4. Compare CI and SI on ₹8000 for 3 years at 10% per annum

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## Learning Outcomes

### Knowledge and Understanding
- Understand relationships between ratios, percentages, and fractions
- Know different types of taxes and their calculation methods
- Comprehend compound interest concepts and applications
- Recognize real-world applications of mathematical concepts

### Skills and Application
- Calculate discounts, taxes, and final prices accurately
- Apply compound interest formula to various scenarios
- Use unitary method for problem-solving
- Estimate percentages for quick calculations

### Mathematical Reasoning
- Compare different calculation methods for efficiency
- Analyze the impact of compounding on growth/decay
- Make connections between mathematical concepts and real-life situations
- Justify choice of calculation method based on context

### Communication
- Express mathematical relationships using proper terminology
- Explain calculation steps clearly and logically
- Present solutions with appropriate units and formatting
- Communicate mathematical reasoning effectively

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## Real-world Connections

### Business and Commerce
1. **Retail Industry**: Understanding markup, discount strategies, and pricing
2. **Banking**: Interest calculations for loans and deposits
3. **Insurance**: Premium calculations and policy valuations
4. **Real Estate**: Property appreciation and depreciation

### Personal Finance
1. **Shopping**: Comparing prices with different discount and tax structures
2. **Investments**: Understanding compound returns on investments
3. **Loans**: Calculating EMIs and total interest payments
4. **Savings**: Planning for long-term financial goals

### Economics and Social Sciences
1. **Population Studies**: Demographic projections and planning
2. **Economic Growth**: GDP growth rates and economic indicators
3. **Inflation**: Understanding price changes over time
4. **Resource Management**: Consumption and conservation planning

### Cross-curricular Links
1. **Science**: Exponential growth in biology, radioactive decay in physics
2. **Geography**: Population density studies and urbanization
3. **History**: Economic changes and their mathematical representation
4. **Social Studies**: Understanding economic policies and their impacts

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## Assessment and Evaluation

### Formative Assessment
- Quick calculation exercises with real-world scenarios
- Peer explanation of discount and tax calculations
- Error analysis in compound interest problems
- Mental math estimation activities

### Summative Assessment
- Problem-solving tests with mixed question types
- Project work on financial planning or business mathematics
- Case study analysis of real business scenarios
- Mathematical modeling of growth/decay situations

### Self-reflection Questions
1. How confident am I in calculating percentages mentally?
2. Can I explain the difference between various types of taxes?
3. Do I understand when to use simple vs compound interest?
4. How well can I apply these concepts to real-life situations?

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## Extensions and Enrichment

### Advanced Topics
- Compound interest with different compounding frequencies
- Effective interest rates and annual percentage rates
- Introduction to present value and future value concepts
- Mathematical modeling of economic phenomena

### Mathematical Investigations
1. Investigating the "Rule of 72" for doubling time
2. Exploring the mathematics behind credit card interest
3. Analyzing inflation rates and purchasing power
4. Comparing different investment strategies mathematically

### Project Ideas
1. Create a business plan with pricing, discounts, and tax calculations
2. Research and present different countries' tax systems
3. Design a savings plan for a specific financial goal
4. Investigate historical population growth patterns

### Technology Integration
1. Using spreadsheets for complex compound interest calculations
2. Creating graphs to visualize exponential growth/decay
3. Programming simple interest and discount calculators
4. Using financial calculators and apps for real-world problems

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## Problem-Solving Strategies

### Step-by-Step Approach
1. **Identify** the type of problem (ratio, percentage, discount, interest)
2. **Extract** given information and required answer
3. **Choose** appropriate formula or method
4. **Calculate** step by step with proper units
5. **Verify** answer for reasonableness

### Common Pitfalls and Solutions
1. **Confusing percentage of vs percentage more/less**: Practice with clear examples
2. **Mixing up marked price and sale price**: Always identify what price is given
3. **Forgetting to add tax or subtract discount**: Follow systematic calculation steps
4. **Compound interest calculation errors**: Use formula rather than repeated calculation

### Mental Math Techniques
- Use benchmark percentages (10%, 25%, 50%) for quick estimation
- Round numbers for estimation before exact calculation
- Use the relationship between fractions and percentages
- Break complex percentages into simpler parts

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## Historical and Cultural Context

### Historical Development
- Evolution of percentage concept from ancient fraction systems
- Development of compound interest in medieval banking
- Introduction of modern tax systems
- Role of mathematics in commercial revolution

### Cultural Applications
- Different cultural approaches to bargaining and discounts
- Traditional methods of calculating interest in various cultures
- Role of mathematics in trade and commerce across civilizations
- Modern digital payment systems and mathematical accuracy

### Mathematical Connections
- Connection to exponential functions and logarithms
- Relationship with geometric progressions
- Links to probability and statistics in risk assessment
- Foundation for higher-level financial mathematics