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Introduction

Comparing quantities is a crucial part of mathematics that helps us make sense of everyday situations. Whether we are analyzing marks in exams, measuring heights, comparing prices, or calculating profits, understanding how to compare quantities is essential. This chapter explores various methods ratios, percentages, increases and decreases, profit and loss, interest, and more offering practical examples and exercises for mastery.

1. Ratio and Proportion

Understanding Ratio

A ratio is a way to compare two quantities of the same kind by division. If there are a apples and b oranges, their ratio is written as a : b (read as "a to b").

Properties of Ratios

  • Both quantities must be in the same units.
  • Ratios can be simplified by dividing both numbers by their greatest common divisor.
  • Order matters: 2:3 is different from 3:2.

Example

If there are 12 boys and 8 girls in a class, the ratio of boys to girls is 12:8, which simplifies to 3:2.

Proportion

When two ratios are equal, we say they are in proportion. For example, if a : b = c : d, then a, b, c, d are in proportion.

Example

2:5 = 4:10 (both simplify to 0.4)

2. Percentages

Definition

Percentage means "per hundred." It is a way to express a number as a fraction of 100. For example, 45% means 45 out of 100.

Converting Fractions and Decimals to Percentages

  • Fraction to Percent: Multiply by 100 and add the % sign.
    Example: 3/5 × 100 = 60%
  • Decimal to Percent: Move the decimal two places right and add the % sign.
    Example: 0.35 × 100 = 35%

Converting Percentages to Fractions or Decimals

  • Percent to Fraction: Divide by 100 and simplify.
    Example: 75% = 75/100 = 3/4
  • Percent to Decimal: Divide by 100.
    Example: 64% = 0.64

Percentage Increase and Decrease

To find how much a quantity increases or decreases as a percentage:

Percentage Change = (Change / Original Value) × 100%

  • Increase Example: If a salary goes from ₹10,000 to ₹12,000, the increase is 2,000.
    Percentage Increase = (2,000 / 10,000) × 100% = 20%
  • Decrease Example: If a value drops from 80 to 64, the decrease is 16.
    Percentage Decrease = (16 / 80) × 100% = 20%

3. Profit, Loss, and Discount

Profit and Loss

  • Cost Price (CP): The original price of an item.
  • Selling Price (SP): The price at which the item is sold.
  • Profit: If SP > CP, Profit = SP - CP
    Loss: If SP < CP, Loss = CP - SP
  • Profit % = (Profit / CP) × 100%
    Loss % = (Loss / CP) × 100%

Example

A shopkeeper buys a shirt for ₹500 and sells it for ₹650.
Profit = ₹650 - ₹500 = ₹150
Profit % = (150 / 500) × 100% = 30%

Discount

  • Marked Price (MP): The price before discount.
  • Discount: The amount by which the marked price is reduced.
  • Selling Price = Marked Price - Discount
  • Discount % = (Discount / Marked Price) × 100%

Example

An item is marked at ₹800 and sold at ₹640.
Discount = ₹800 - ₹640 = ₹160
Discount % = (160 / 800) × 100% = 20%

4. Simple Interest

Definition

When money is borrowed or lent for a certain period, a sum is paid or earned as interest. Simple interest is calculated only on the principal amount.

Simple Interest (SI) = (Principal × Rate × Time) / 100

  • Principal (P): The amount of money borrowed or invested.
  • Rate (R): Interest rate per year (as a percentage).
  • Time (T): Number of years (or portion of a year).

Example

If ₹2,000 is invested at 6% per annum for 3 years:
SI = (2000 × 6 × 3) / 100 = ₹360

5. Compound Interest

Definition

In compound interest, interest is calculated on the principal plus any interest already earned. It is commonly used in banking and investments.

Compound Interest (CI) = Amount - Principal
Amount = Principal × (1 + Rate/100)Time

Example

If ₹1,000 is deposited at 10% per annum for 2 years (compounded yearly):
Amount = 1000 × (1 + 10/100)2 = 1000 × 1.21 = ₹1,210
CI = ₹1,210 - ₹1,000 = ₹210

Difference Between Simple and Compound Interest

  • Simple interest is always calculated on the original principal.
  • Compound interest is calculated on the principal plus any interest that has already been added.
  • For one year, SI and CI are the same. For more than one year, CI is always higher (if the rate and time are the same).

6. Real-Life Applications

  • Shopping: Calculating discounts and sale prices.
  • Banking: Understanding interest on savings and loans.
  • Business: Figuring out profit, loss, and pricing strategies.
  • Statistics: Comparing marks, populations, and percentages.
  • Personal Finance: Managing budgets, expenses, and investments.

7. Important Formulas & Shortcuts

  • Percentage Change: ((New Value - Old Value) / Old Value) × 100%
  • Profit %: (Profit / Cost Price) × 100%
  • Loss %: (Loss / Cost Price) × 100%
  • Discount %: (Discount / Marked Price) × 100%
  • Simple Interest: (Principal × Rate × Time) / 100
  • Compound Interest: Amount - Principal, where Amount = Principal × (1 + Rate/100)Time
  • To find a percentage of a quantity: (Percentage × Quantity) / 100

8. Common Mistakes to Avoid

  • Not converting units to the same base before comparing (e.g., grams to kilograms).
  • Mixing up cost price and selling price when calculating profit or loss.
  • Calculating interest without converting time to years (if the rate is per annum).
  • Applying percentage directly to the wrong value (e.g., discount on cost price instead of marked price).
  • Forgetting to compound interest each year in CI problems.

9. Practice Problems

Basic

  1. Express 3/4 as a percentage.
  2. If the ratio of apples to oranges is 5:7, how many apples are there if there are 35 oranges?
  3. Calculate the profit percentage: CP = ₹400, SP = ₹500.
  4. Find the loss and loss % if a bicycle bought for ₹1,200 is sold for ₹900.

Intermediate

  1. An article is marked at ₹2,500. It is sold at a discount of 20%. Find the selling price.
  2. Find the simple interest on ₹5,000 at 8% per annum for 3 years.
  3. Find the amount and compound interest on ₹2,000 at 5% per annum for 2 years, compounded yearly.
  4. If a quantity increases from 240 to 300, what is the percentage increase?

Advanced

  1. A sum of ₹10,000 is invested at 12% per annum compound interest, compounded yearly. What will be the amount after 3 years?
  2. If the selling price of a pen is ₹54 after a discount of 10%, what was its marked price?
  3. A population increases by 8% in the first year and 10% in the second year. What is the overall percentage increase over two years?
  4. Calculate the loss percentage if an article is sold at a 15% loss and its cost price is ₹1,600.

10. Solutions

  1. 3/4 × 100 = 75%
  2. Let apples = x. 5/7 = x/35 ⇒ x = (5/7) × 35 = 25 apples
  3. Profit = ₹500 - ₹400 = ₹100; Profit % = (100/400) × 100 = 25%
  4. Loss = ₹1,200 - ₹900 = ₹300; Loss % = (300/1,200) × 100 = 25%
  5. Discount = 20% of ₹2,500 = ₹500; Selling Price = ₹2,500 - ₹500 = ₹2,000
  6. SI = (5,000 × 8 × 3)/100 = ₹1,200
  7. Amount = 2,000 × (1 + 5/100)2 = 2,000 × 1.1025 = ₹2,205
    CI = ₹2,205 - ₹2,000 = ₹205
  8. Increase = 300 - 240 = 60; Percentage increase = (60/240) × 100 = 25%
  9. Amount = 10,000 × (1 + 12/100)3 = 10,000 × 1.404928 = ₹14,049.28
  10. Let marked price = x; x - 10%x = 54 ⇒ 0.9x = 54 ⇒ x = ₹60
  11. Let original population = 100. After first year: 100 + 8 = 108.
    Second year: 108 + 10% of 108 = 108 + 10.8 = 118.8.
    Overall increase = 118.8 - 100 = 18.8%; Answer = 18.8%
  12. Loss = 15% of 1,600 = ₹240; Loss % = 15%

Conclusion

The ability to compare quantities is essential not only in mathematics but also in daily life. By mastering the concepts of ratios, percentages, profit and loss, and interest, students can make informed decisions, analyze real-world scenarios, and manage personal finances effectively. Keep practicing these skills, and apply them to everyday problems to build your confidence and numeracy!

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