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Mathematics: Class 8 (Part 2) — Chapter 1: Fractions in Disguise

1.1 Fractions as Percentages

Concept of Percentage

  • Per Cent: Derived from the Latin phrase per centum, meaning "by the hundred" or "out of hundred". Represented by the symbol %.
  • Percentage: Simply a fraction where the denominator is always 100. For example, 25% means 25 out of 100.
  • Conversion Rule: To express any fraction as a percentage, multiply the fraction by 100. Formula: z% = z / 100.

Methods of Conversion

  • Method 1 (Equivalent Fraction): Multiply the numerator and denominator by a number that makes the denominator 100. Example: 3/4 = (3 * 25) / (4 * 25) = 75/100 = 75%.
  • Method 2 (Algebraic): Set the fraction equal to x/100 and solve for x. Example: 3/4 = x/100 implies x = (3/4) * 100 = 75.
1.2 Percentage of Some Quantity

Calculating Percentage of a Value

  • General Formula: y% of a value V is given by (y / 100) * V.
  • Additive Property: Percentages of the same quantity can be added directly: (a% of y) + (b% of y) = (a + b)% of y.
  • FDP Trio: Fractions, Decimals, and Percentages are different representations of the same value (e.g., 1/2 = 0.5 = 50%).
1.3 Using Percentages

Comparing Proportions

  • Percentages standardize different wholes to a common base of 100, making comparison intuitive and direct.

Percentage Increase or Decrease

  • Used to describe the rate of change of quantities over time.
  • Formula: Percentage Change = (Amount of Change / Original Amount) * 100.

Profit and Loss

  • Cost Price (CP): The price at which an item is bought by the seller.
  • Marked Price (MP): The price quoted or displayed on the item.
  • Selling Price (SP): The actual price paid by the customer after any discounts.
  • Profit: Occurs when SP is greater than CP. Profit = SP - CP.
  • Loss: Occurs when CP is greater than SP. Loss = CP - SP.
  • Profit Percentage: (Profit / CP) * 100.
  • Loss Percentage: (Loss / CP) * 100.
  • Discount: Reduction offered on the Marked Price. Discount = MP - SP.

Growth and Compounding

  • Principal (P): The initial amount of money deposited or borrowed.
  • Interest (I): The extra money paid or earned over time.
  • Rate of Interest (r): The percentage of the principal paid as interest per annum (p.a.).
  • Simple Interest (No Compounding): Interest is calculated only on the initial principal. Total Amount: A = P * (1 + r * t).
  • Compound Interest (Compounding): Interest earned is added back to the principal for the next period. Total Amount: A = P * (1 + r)^t.

Depreciation

  • Depreciation: The reduction in the financial value of an asset over time due to wear, tear, or age. Formula for value after t periods: A = P * (1 - r)^t.