Comparing Quantities- Notes and Solutions

Introduction- Comparing Quantities

Comparing quantities is an essential skill in mathematics and real life. This chapter teaches us how to compare two or more quantities using ratios, percentages, and other methods. It also introduces practical applications like calculating profit, loss, and interest. The main topics covered are:

1. Equivalent Ratios
2. Percentages
3. Converting Fractions, Decimals, and Ratios to Percentages
4. Use of Percentages (Profit and Loss)
5. Simple Interest

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1. Equivalent Ratios

• Definition: Two ratios are equivalent if they represent the same relationship between quantities. For example, 2:3 is equivalent to 4:6 because both show the same proportion.
• How to Find: Multiply or divide both terms of a ratio by the same number.
  • Example: 2:3 → Multiply by 2 → 4:6
• Checking Equivalence: Cross-multiply the terms. If 2:3 and 4:6 are equivalent, then 2 × 6 = 3 × 4 (12 = 12), which is true.
• Uses: Ratios help compare quantities like the number of boys to girls in a class or ingredients in a recipe.

Example:

• If a recipe uses 2 cups of flour and 3 cups of water, the ratio is 2:3. If you double the recipe, it becomes 4:6, still equivalent to 2:3.

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2. Percentages

• Definition: A percentage is a way of expressing a number as a fraction of 100. The symbol is “%”.
  • Example: 25% means 25 out of 100.
• Formula:
  • Percentage = (Part / Whole) × 100
• Importance: Percentages make it easy to compare quantities of different sizes.

Example:Comparing Quantities

• In a class of 40 students, 10 are girls. What percentage are girls?
  • Percentage = (10 / 40) × 100 = 25%
  • So, 25% of the students are girls.

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3. Converting Fractions, Decimals, and Ratios to Percentages

• Fraction to Percentage:
  • Multiply the fraction by 100 and add the % sign.
  • Example: 1/4 = (1/4) × 100 = 25%
• Decimal to Percentage:
  • Multiply the decimal by 100 and add the % sign.
  • Example: 0.75 = 0.75 × 100 = 75%
• Ratio to Percentage:
  • Convert the ratio to a fraction, then to a percentage.
  • Example: 3:5 = 3/5 → (3/5) × 100 = 60%

Example:Comparing Quantities

• Convert 0.2 to a percentage: 0.2 × 100 = 20%
• Convert 2:3 to a percentage: 2/3 = (2/3) × 100 = 66.67% (approx.)

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4. Use of Percentages: Profit and Loss

Percentages are widely used in buying and selling to calculate profit and loss.

Key Terms:Comparing Quantities

• Cost Price (CP): The price at which an item is bought.
• Selling Price (SP): The price at which an item is sold.
• Profit: When SP > CP, Profit = SP – CP
• Loss: When SP < CP, Loss = CP – SP
• Profit Percentage:
  • Profit % = (Profit / CP) × 100
• Loss Percentage:
  • Loss % = (Loss / CP) × 100

Examples:Comparing Quantities

1. Profit Calculation:
   • CP = ₹200, SP = ₹250
   • Profit = 250 – 200 = ₹50
   • Profit % = (50 / 200) × 100 = 25%
2. Loss Calculation:
   • CP = ₹300, SP = ₹270
   • Loss = 300 – 270 = ₹30
   • Loss % = (30 / 300) × 100 = 10%

Finding SP or CP:

• If profit or loss percentage is given:
  • SP = CP × (100 + Profit %) / 100 (for profit)
  • SP = CP × (100 – Loss %) / 100 (for loss)

Example:Comparing Quantities

• CP = ₹500, Profit % = 20%
• SP = 500 × (100 + 20) / 100 = 500 × 120 / 100 = ₹600

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5. Simple Interest

• Definition: Simple Interest (SI) is the extra money paid or earned when borrowing or lending money, calculated only on the initial amount (principal).
• Formula:
  • SI = (P × R × T) / 100
  • Where:
    • P = Principal (initial amount)
    • R = Rate of interest per year (in %)
    • T = Time (in years)
• Amount: The total money paid back or received.
  • Amount = Principal + SI

Example:Comparing Quantities

• P = ₹1000, R = 5%, T = 2 years
• SI = (1000 × 5 × 2) / 100 = ₹100
• Amount = 1000 + 100 = ₹1100

Finding Other Values:

• If SI, R, and T are given, find P:
  • P = (SI × 100) / (R × T)
• If SI, P, and T are given, find R:
  • R = (SI × 100) / (P × T)

Example:

• SI = ₹150, P = ₹500, T = 3 years
• R = (150 × 100) / (500 × 3) = 15000 / 1500 = 10%

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Practical Applications

1. Shopping: Discounts are given in percentages. For example, a 20% discount on a ₹500 item means:
   • Discount = (20 / 100) × 500 = ₹100
   • SP = 500 – 100 = ₹400
2. Exams: If a student scores 45 out of 50, their percentage is:
   • (45 / 50) × 100 = 90%
3. Savings: Banks pay interest on savings, calculated as simple interest.

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Summary of Key Formulas

1. Percentage: (Part / Whole) × 100
2. Profit: SP – CP
3. Loss: CP – SP
4. Profit %: (Profit / CP) × 100
5. Loss %: (Loss / CP) × 100
6. Simple Interest: (P × R × T) / 100
7. Amount: P + SI

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Solved Examples

1. Ratio Problem:
   • A bag has 4 red balls and 6 blue balls. What is the ratio of red to blue balls?
   • Ratio = 4:6 = 2:3
2. Percentage Problem:
   • 30 out of 50 students passed a test. What percentage passed?
   • Percentage = (30 / 50) × 100 = 60%
3. Profit Problem:
   • CP = ₹400, SP = ₹480. Find profit %.
   • Profit = 480 – 400 = ₹80
   • Profit % = (80 / 400) × 100 = 20%
4. Simple Interest Problem:
   • P = ₹2000, R = 4%, T = 3 years. Find SI and Amount.
   • SI = (2000 × 4 × 3) / 100 = ₹240
   • Amount = 2000 + 240 = ₹2240

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Tips for Solving Problems

• Read the question carefully to identify what is given and what is asked.
• Use the correct formula based on the type of problem (profit, loss, or interest).
• Double-check calculations to avoid mistakes.
• Practice converting between fractions, decimals, and percentages for fluency.

Download pdf notes and Sulutions of the Chapter:

Chapter 9: Rational Numbers

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