Ratio and Proportion

• Ratio

  • Definition
    • Comparison of two quantities of the same kind
    • Uses division
  • Notation
    • Denoted by symbol $:$ (colon)
    • Example: Ratio of $a$ to $b$ is written as $a : b$
  • Important Properties
    • Order of terms is very important ($a : b \neq b : a$)
    • Must be in the same units (convert if different, e.g., meters to cm)
    • Has no units of its own (dimensionless number)
  • Simplification
    • Can be reduced to lowest terms (like fractions)
    • Divide both terms by their HCF

• Equivalent Ratios

  • Concept
    • Ratios that represent the same value
    • Obtained by multiplying or dividing both terms by the same non-zero number
  • Example
    • $2 : 3$ is equivalent to $4 : 6$ or $6 : 9$
  • Checking Equivalence
    • Using fraction form or cross multiplication

• Proportion

  • Definition
    • Equality of two ratios
    • Denoted by symbol $::$ or $=$
  • Terminology
    • For four quantities $a, b, c, d$ in proportion: $a : b :: c : d$
    • $a$ and $d$ are called extremes
    • $b$ and $c$ are called means
  • Rule of Proportion
    • Product of Extremes = Product of Means
    • $ad = bc$

• Unitary Method

  • Definition
    • Method to find the value of a single unit first, and then finding the value of the required number of units
  • Steps
    • Step 1: Find the value of one unit using division
    • Step 2: Find the value of multiple units using multiplication
  • Application
    • Solving everyday word problems (e.g., cost of items, distance traveled)