Class 7 Notes

Class 7 Mathematics Notes Chapter 8 (Comparing Quantities) – Mathematics Book

Philoid

Philoid

— 6 min read

Mathematics
Alright class, let's get straight into Chapter 8, 'Comparing Quantities'. This is a very important chapter, not just for your exams but also for everyday life and especially foundational for many government exam quantitative aptitude sections. We compare things constantly, and this chapter teaches us the mathematical ways to do it precisely.

Chapter 8: Comparing Quantities - Detailed Notes

1. Introduction
Comparing quantities means looking at two quantities and determining how they relate to each other. We might want to know how many times one quantity is of another, or what part one quantity is of another. The main tools we use for this are Ratios and Percentages.

2. Ratio

  • Definition: A ratio is a comparison of two quantities by division. It shows how much of one quantity there is compared to another.
  • Notation: The ratio of a quantity 'a' to a quantity 'b' is written as a : b or a/b. (Read as 'a is to b').
  • Important Condition: The two quantities being compared must be in the same units. If they are not, you must convert them to the same unit before finding the ratio.
    • Example: Ratio of 50 cm to 2 meters. First convert 2 meters to 200 cm. The ratio is 50 cm : 200 cm = 50:200.
  • Simplest Form: Ratios should always be expressed in their simplest form, like fractions. Divide both parts of the ratio by their Highest Common Factor (HCF).
    • Example: 50:200. HCF is 50. So, 50/50 : 200/50 = 1:4.
  • Equivalent Ratios: Ratios that represent the same comparison, even if the numbers are different. You get equivalent ratios by multiplying or dividing both parts of the ratio by the same non-zero number.
    • Example: 1:4 is equivalent to 2:8 (multiplied by 2) or 3:12 (multiplied by 3).
  • Comparing Ratios: To compare two ratios (like a:b and c:d), convert them into like fractions (fractions with the same denominator) or decimals and then compare.

3. Proportion

  • Definition: A proportion is an equality of two ratios. If the ratio a:b is equal to the ratio c:d, then a, b, c, and d are said to be in proportion.
  • Notation: a : b :: c : d (Read as 'a is to b as c is to d') or a/b = c/d.
  • Terms:
    • 'a' and 'd' are called the Extremes (outer terms).
    • 'b' and 'c' are called the Means (middle terms).
  • Key Property: In a proportion, the product of the extremes is equal to the product of the means.
    • If a : b :: c : d, then a × d = b × c.
  • Checking for Proportion: To check if four quantities are in proportion, check if the product of extremes equals the product of means.
    • Example: Are 2, 4, 6, 12 in proportion?
      • Extremes: 2, 12. Product = 2 × 12 = 24.
      • Means: 4, 6. Product = 4 × 6 = 24.
      • Since the products are equal, 2, 4, 6, 12 are in proportion (2:4 :: 6:12).

4. Percentage

  • Definition: 'Percent' means 'per hundred' or 'out of one hundred'. It's a special type of ratio where the second term (denominator) is always 100.
  • Symbol: %
  • Meaning: x% means x/100.
  • Uses: Percentages are widely used for comparison because they provide a common base (100).

4.1 Conversions involving Percentages:

  • Fraction to Percentage: Multiply the fraction by 100 and put the '%' sign.
    • Example: 3/4 = (3/4) × 100 % = 75%.
  • Decimal to Percentage: Multiply the decimal by 100 (or shift the decimal point two places to the right) and put the '%' sign.
    • Example: 0.65 = 0.65 × 100 % = 65%.
  • Percentage to Fraction: Remove the '%' sign and divide by 100. Simplify the fraction.
    • Example: 40% = 40/100 = 2/5.
  • Percentage to Decimal: Remove the '%' sign and divide by 100 (or shift the decimal point two places to the left).
    • Example: 85% = 85/100 = 0.85.

4.2 Using Percentages:

  • Finding Percentage of a Quantity: Convert the percentage to a fraction or decimal and multiply by the quantity.
    • Example: Find 20% of 150.
      • Method 1 (Fraction): (20/100) × 150 = (1/5) × 150 = 30.
      • Method 2 (Decimal): 0.20 × 150 = 30.
  • Expressing One Quantity as a Percentage of Another: (Quantity 1 / Quantity 2) × 100 %. (Ensure both quantities are in the same unit).
    • Example: What percentage of 200 is 50?
      • (50 / 200) × 100 % = (1/4) × 100 % = 25%.
  • Percentage Increase/Decrease:
    • Increase Amount = New Value - Original Value
    • Decrease Amount = Original Value - New Value
    • Percentage Increase = (Increase Amount / Original Value) × 100 %
    • Percentage Decrease = (Decrease Amount / Original Value) × 100 %
    • Example: Price increased from ₹80 to ₹100.
      • Increase Amount = 100 - 80 = ₹20.
      • Percentage Increase = (20 / 80) × 100 % = (1/4) × 100 % = 25%.

5. Applications of Percentage

5.1 Profit and Loss

  • Cost Price (CP): The price at which an item is purchased.
  • Selling Price (SP): The price at which an item is sold.
  • Overhead Expenses: Additional costs like repairs, transportation, etc., are added to the CP to get the effective CP.
  • Profit (P): If SP > CP, there is a profit. P = SP - CP.
  • Loss (L): If CP > SP, there is a loss. L = CP - SP.
  • Profit Percentage (P%): (Profit / CP) × 100 %
  • Loss Percentage (L%): (Loss / CP) × 100 %
    • Important: Profit and Loss percentages are always calculated on the Cost Price (CP).
  • Finding SP:
    • If Profit P% is made: SP = CP × (1 + P/100) = CP × (100 + P%)/100
    • If Loss L% is incurred: SP = CP × (1 - L/100) = CP × (100 - L%)/100
  • Finding CP:
    • If Profit P% is made: CP = SP / (1 + P/100) = SP × 100 / (100 + P%)
    • If Loss L% is incurred: CP = SP / (1 - L/100) = SP × 100 / (100 - L%)

5.2 Simple Interest (SI)

  • Principal (P): The initial amount of money borrowed or lent.
  • Rate of Interest (R): The interest paid on ₹100 for one year (usually expressed as a percentage per annum, p.a.).
  • Time (T): The duration for which the money is borrowed or lent (must be in years). If given in months or days, convert to years.
  • Simple Interest (SI): The additional money paid for using the principal. It is calculated uniformly on the original principal throughout the loan period.
    • Formula: SI = (P × R × T) / 100
  • Amount (A): The total money paid back at the end of the loan period.
    • Formula: Amount (A) = Principal (P) + Simple Interest (SI)
    • A = P + (P × R × T) / 100 = P × (1 + (R × T) / 100)

Key Formulas Summary:

  • Ratio: a:b = a/b (Same units)
  • Proportion: a:b :: c:d => a × d = b × c
  • Percentage: x% = x/100
  • % Increase = (Increase / Original) × 100%
  • % Decrease = (Decrease / Original) × 100%
  • Profit% = (Profit / CP) × 100%
  • Loss% = (Loss / CP) × 100%
  • SP (Profit) = CP × (100 + P%)/100
  • SP (Loss) = CP × (100 - L%)/100
  • CP (Profit) = SP × 100 / (100 + P%)
  • CP (Loss) = SP × 100 / (100 - L%)
  • SI = (P × R × T) / 100
  • Amount = P + SI

Tips for Exam Preparation:

  • Understand the core concepts of Ratio, Proportion, and Percentage deeply.
  • Memorize the formulas, especially for Profit/Loss and Simple Interest. Understand how they are derived.
  • Pay close attention to units when dealing with ratios.
  • Always calculate Profit% or Loss% on the Cost Price.
  • Ensure Time (T) is in years for Simple Interest calculations.
  • Practice a wide variety of problems, including word problems, to build speed and accuracy.

Now, let's test your understanding with some multiple-choice questions.

Multiple Choice Questions (MCQs)

  1. The ratio of 25 minutes to 1.5 hours is:
    A) 5:18
    B) 1:3
    C) 25:150
    D) 5:3

  2. If 4, x, 12, 18 are in proportion, what is the value of x?
    A) 6
    B) 8
    C) 9
    D) 5

  3. What is 0.04 expressed as a percentage?
    A) 0.4%
    B) 4%
    C) 40%
    D) 0.04%

  4. Out of 40 students in a class, 32 passed. What percentage of students failed?
    A) 80%
    B) 20%
    C) 8%
    D) 10%

  5. A shopkeeper buys a toy for ₹150 and sells it for ₹180. What is his profit percentage?
    A) 15%
    B) 20%
    C) 25%
    D) 30%

  6. If the price of a book is decreased from ₹50 to ₹45, what is the percentage decrease?
    A) 5%
    B) 10%
    C) 9%
    D) 11.11%

  7. What is the Simple Interest on ₹5000 for 2 years at the rate of 8% per annum?
    A) ₹400
    B) ₹800
    C) ₹1000
    D) ₹5800

  8. A man sells an article for ₹550, incurring a loss of ₹50. What is his loss percentage?
    A) 10%
    B) 9.09%
    C) 8.33%
    D) 11.11%

  9. What amount will be received on a principal of ₹10,000 after 3 years at a simple interest rate of 5% per annum?
    A) ₹1500
    B) ₹10500
    C) ₹11000
    D) ₹11500

  10. If 30% of x is 72, what is the value of x?
    A) 21.6
    B) 216
    C) 240
    D) 300

Answer Key for MCQs:

  1. A (Convert 1.5 hours to 1.5 * 60 = 90 minutes. Ratio = 25:90 = 5:18)
  2. A (Product of extremes = Product of means => 4 * 18 = x * 12 => 72 = 12x => x = 6)
  3. B (0.04 * 100% = 4%)
  4. B (Passed = 32, Failed = 40 - 32 = 8. % Failed = (8/40) * 100% = (1/5) * 100% = 20%)
  5. B (CP = 150, SP = 180. Profit = 180 - 150 = 30. Profit % = (30/150) * 100% = (1/5) * 100% = 20%)
  6. B (Decrease = 50 - 45 = 5. % Decrease = (5/50) * 100% = (1/10) * 100% = 10%)
  7. B (SI = (P * R * T) / 100 = (5000 * 8 * 2) / 100 = 50 * 16 = 800)
  8. C (SP = 550, Loss = 50. CP = SP + Loss = 550 + 50 = 600. Loss % = (Loss / CP) * 100% = (50 / 600) * 100% = (1/12) * 100% = 8.33%)
  9. D (SI = (10000 * 5 * 3) / 100 = 100 * 15 = 1500. Amount = P + SI = 10000 + 1500 = 11500)
  10. C (30% of x = 72 => (30/100) * x = 72 => (3/10) * x = 72 => x = (72 * 10) / 3 => x = 24 * 10 = 240)

Make sure you understand the reasoning behind each answer. Practice these concepts thoroughly!

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