Class 8 Mathematics Notes Chapter 8 (Comparing Quantities) – Mathematics Book

Alright class, let's focus on Chapter 8, "Comparing Quantities." This is a very important chapter, not just for your Class 8 exams, but its concepts form the backbone of quantitative aptitude sections in many government exams. Understanding ratios, percentages, profit & loss, discounts, taxes, and interest calculations is crucial. Pay close attention, and let's break it down.

Chapter 8: Comparing Quantities - Detailed Notes for Government Exam Preparation

Core Idea: This chapter deals with different methods used to compare quantities, primarily focusing on ratios and percentages, and their applications in real-life financial scenarios.

1. Ratios and Percentages (Recap and Extension)

• Ratio: A method to compare two quantities of the same unit by division. Represented as a : b or a/b.
  • Quantities must be in the same unit (e.g., cm to cm, kg to kg). If not, convert them first.
  • Example: Ratio of 5 kg to 500 g = Ratio of 5000 g to 500 g = 5000/500 = 10/1 = 10 : 1.
• Percentage: A special ratio where the second term (denominator) is always 100. The symbol is %. It means "per hundred".
  • x % = x/100.
  • Converting Ratio to Percentage: Convert the ratio to a fraction, then multiply by 100. Example: 3 : 4 = 3/4 = (3/4) * 100 % = 75 %.
  • Converting Percentage to Ratio/Fraction: Divide by 100 and simplify. Example: 40 % = 40/100 = 4/10 = 2/5 (Ratio 2 : 5).
  • Finding Percentage of a Quantity: x % of Y = (x/100) * Y. Example: 20 % of 150 = (20/100) * 150 = 30.
  • Expressing one quantity as a percentage of another: (Quantity 1 / Quantity 2) * 100 %. (Ensure units are the same). Example: What percent of 50 is 10? (10 / 50) * 100 % = (1/5) * 100 % = 20 %.

2. Percentage Change (Increase or Decrease)

• Often used to express changes in price, population, production etc.
• Percentage Increase: (Increase in Quantity / Original Quantity) * 100 %
  • Increase = New Value - Original Value
  • New Value = Original Value * (1 + Increase % / 100)
• Percentage Decrease: (Decrease in Quantity / Original Quantity) * 100 %
  • Decrease = Original Value - New Value
  • New Value = Original Value * (1 - Decrease % / 100)
• Key: The comparison is always with the Original Quantity.

3. Buying and Selling: Profit and Loss

• Cost Price (CP): The price at which an item is purchased. Includes overhead expenses like repairs, transportation, etc., if any.
  • Effective CP = Purchase Price + Overhead Expenses
• Selling Price (SP): The price at which an item is sold.
• Profit (Gain): If SP > CP, then Profit = SP - CP.
• Loss: If CP > SP, then Loss = CP - SP.
• Profit Percentage (Profit %): (Profit / CP) * 100 %
• Loss Percentage (Loss %): (Loss / CP) * 100 %
• Crucial Point: Profit and Loss percentages are always calculated on the Cost Price (CP), unless specified otherwise.
• Finding SP when CP and Profit%/Loss% are known:
  • SP = CP * (1 + Profit % / 100)
  • SP = CP * (1 - Loss % / 100)
• Finding CP when SP and Profit%/Loss% are known:
  • CP = SP / (1 + Profit % / 100) or CP = SP * 100 / (100 + Profit %)
  • CP = SP / (1 - Loss % / 100) or CP = SP * 100 / (100 - Loss %)

4. Discounts

• Marked Price (MP) or List Price (LP): The price printed or tagged on an item.
• Discount: A reduction given on the Marked Price.
• Sale Price (SP): The price after deducting the discount from the Marked Price. SP = MP - Discount.
• Discount Percentage (Discount %): (Discount / MP) * 100 %
• Crucial Point: Discount is always calculated on the Marked Price (MP).
• Calculating Discount: Discount = Discount % of MP = (Discount % / 100) * MP.
• Calculating SP: SP = MP - Discount = MP * (1 - Discount % / 100).
• Calculating MP: MP = SP / (1 - Discount % / 100) or MP = SP * 100 / (100 - Discount %)

5. Taxes (Sales Tax, VAT, GST)

• Taxes like Sales Tax, Value Added Tax (VAT), or Goods and Services Tax (GST) are charged by the government on the sale of items.
• These are generally charged over and above the selling price (or list price before tax).
• Calculation: Tax Amount = Tax % of Selling Price (or List Price)
• Bill Amount (Final Price): Selling Price + Tax Amount
  • Bill Amount = Selling Price * (1 + Tax % / 100)

6. Simple Interest (SI)

• Interest calculated uniformly on the original principal throughout the loan period.
• Principal (P): The initial sum of money borrowed or lent.
• Rate of Interest (R): The interest paid per ₹100 per year (usually expressed as % per annum).
• Time (T): The duration for which the money is borrowed/lent (usually in years).
• Simple Interest (SI): SI = (P * R * T) / 100
• Amount (A): The total money paid back at the end of the period. A = Principal + Simple Interest = P + SI.

7. Compound Interest (CI)

• Interest calculated on the Principal plus the accumulated interest from previous periods. Interest earns interest.
• Calculation Year-by-Year: Calculate SI for the first year. Add it to the Principal to get the Principal for the second year. Calculate SI for the second year on this new Principal, and so on.
• Formula for Amount (A) when compounded annually:
  • A = P * (1 + R/100)^n
  • Where:
    • A = Amount after n years
    • P = Principal
    • R = Rate of interest per annum (%)
    • n = Number of years
• Compound Interest (CI): CI = Amount - Principal = A - P
  • CI = P * [(1 + R/100)^n - 1]
• Compounding Frequency: Interest can be compounded more frequently than annually (e.g., half-yearly, quarterly).
  • Half-yearly: Rate becomes R/2 per half-year. Time becomes 2n half-years.
    • A = P * (1 + (R/2)/100)^(2n)
  • Quarterly: Rate becomes R/4 per quarter. Time becomes 4n quarters.
    • A = P * (1 + (R/4)/100)^(4n)
• Key Difference: CI is always greater than or equal to SI for the same P, R, T (for T > 1 year). CI = SI for the first year/compounding period.

8. Applications of Compound Interest Formula

• The CI formula is used in various situations involving growth or decay over time.
• Population Growth: If population grows at a constant rate R % per year.
  • Population after n years = Initial Population * (1 + R/100)^n
• Depreciation: If the value of an item (like a car, machine) decreases at a constant rate R % per year.
  • Value after n years = Initial Value * (1 - R/100)^n (Note the minus sign for decrease)
• Bacterial Growth: Similar to population growth.

Key Takeaways for Exams:

• Be very clear about the base value for calculation: CP for Profit/Loss %, MP for Discount %, Original Value for % Increase/Decrease.
• Read questions carefully to identify P, R, T, n and whether SI or CI is required.
• Pay attention to compounding frequency (annually, half-yearly, etc.) in CI problems.
• Practice conversions between fractions, ratios, decimals, and percentages quickly.
• Understand the formulas for finding SP/CP/MP when other values are given.

Master these concepts with practice, and you'll be well-prepared to tackle questions from this chapter in your government exams. Now, let's test your understanding with some multiple-choice questions.

Multiple Choice Questions (MCQs)

1. The ratio of 2 meters to 50 centimeters is:
   a) 2 : 50
   b) 1 : 25
   c) 4 : 1
   d) 1 : 4

2. If 30% of x is 72, then the value of x is:
   a) 216
   b) 240
   c) 21.6
   d) 24

3. A shopkeeper buys an article for ₹400 and sells it for ₹460. His gain percentage is:
   a) 15%
   b) 13.04%
   c) 60%
   d) 10%

4. A fan marked at ₹1500 is sold for ₹1350. The percentage discount offered is:
   a) 15%
   b) 10%
   c) 12%
   d) 9%

5. The price of a scooter was ₹34,000 last year. It has increased by 20% this year. What is the price now?
   a) ₹40,800
   b) ₹38,800
   c) ₹6,800
   d) ₹40,000

6. A shopkeeper sells an item for ₹550 at a loss of ₹50. What is the cost price of the item?
   a) ₹500
   b) ₹600
   c) ₹550
   d) ₹450

7. Calculate the Simple Interest on ₹5000 for 2 years at 8% per annum.
   a) ₹400
   b) ₹800
   c) ₹5800
   d) ₹5400

8. What amount is to be repaid on a loan of ₹12,000 for 1.5 years at 10% per annum compounded half-yearly?
   a) ₹13,891.50
   b) ₹13,800
   c) ₹13,500
   d) ₹14,000

9. The cost of an article including 8% GST is ₹2160. What is the price of the article before GST was added?
   a) ₹2000
   b) ₹1987.20
   c) ₹2332.80
   d) ₹2100

10. The population of a town was 20,000 in 2020. If it increases at a rate of 5% per annum, what will be the population in 2022?
    a) 22,000
    b) 21,000
    c) 22,050
    d) 22,100

Answer Key:

1. c) 4 : 1 (2m = 200cm; 200:50 = 4:1)
2. b) 240 (0.30 * x = 72 => x = 72 / 0.30 = 720 / 3 = 240)
3. a) 15% (CP=400, SP=460, Profit=60; Profit% = (60/400)*100 = 15%)
4. b) 10% (MP=1500, SP=1350, Discount=150; Discount% = (150/1500)*100 = 10%)
5. a) ₹40,800 (Increase = 20% of 34000 = 6800; New Price = 34000 + 6800 = 40800)
6. b) ₹600 (Loss = CP - SP => 50 = CP - 550 => CP = 550 + 50 = 600)
7. b) ₹800 (SI = (5000 * 8 * 2) / 100 = 800)
8. a) ₹13,891.50 (P=12000, R=10% pa = 5% per half-year, n=1.5 years = 3 half-years. A = 12000 * (1 + 5/100)^3 = 12000 * (1.05)^3 = 12000 * 1.157625 = 13891.50)
9. a) ₹2000 (Let price before GST be x. x * (1 + 8/100) = 2160 => x * (108/100) = 2160 => x = 2160 * 100 / 108 = 20 * 100 = 2000)
10. c) 22,050 (P=20000, R=5%, n=2 years. A = 20000 * (1 + 5/100)^2 = 20000 * (1.05)^2 = 20000 * 1.1025 = 22050)

Go through these notes and practice the MCQs. Remember, consistent practice is key to mastering these concepts for your exams. Let me know if any part needs further clarification.