Ratio, Rate, and Proportion

Ratio

  • Definition: A ratio is a comparison of two quantities of the same kind.
    • Symbolic forms: fraction \frac{a}{b} or colon a:b.
  • Reading convention: “a:b” is read as “a is to b.”
  • Reduced (simplest) form: divide both terms by their greatest common divisor (GCD).
  • Order matters.
    • The first number in the ratio corresponds to the first item mentioned.
    • The second number in the ratio corresponds to the second item mentioned.
  • Example (Coin Purse):
    • Contents: 30 coins in total – 10 five-peso coins, 20 one-peso coins.
    • Requested ratio: five-peso coins to one-peso coins.
    • Raw ratio: 10:20 or \frac{10}{20}.
    • Simplified by GCD 10 → 1:2 or \frac12.
    • Read as “1 is to 2.”
  • Conceptual reminders:
    • Treat ratios as numbers; you can multiply or divide both terms by the same non-zero factor without changing the value of the ratio.
    • Ratios are dimensionless because they compare like quantities.

Rate

  • Definition: A rate is a ratio of two measurements that have different units.
    • General form: \text{quantity}1 : \text{quantity}2 with different units (e.g.
      distance per time, cost per item).
  • The value of a rate often answers a “per …” question.
  • Example (Falling Object):
    • Scenario: An object falls the full height of a 60-storey building in 5 min.
    • Rate expression: \frac{60\;\text{storeys}}{5\;\text{min}}.
    • Simplified rate: \frac{60}{5}=12, so 12\;\text{storeys/min}.
    • Read as “12 storeys per minute.”
  • Practical notes:
    • Units must be carried and simplified where possible.
    • Converting one unit (e.g.
      storeys to metres) changes the numerical value but not the physical meaning.

Proportion

  • Definition: A proportion states that two ratios (or two rates) are equal.
    • Notation: \frac{a}{b}=\frac{c}{d} or a:b=c:d.
  • Terms:
    • Means: the two “middle” numbers b and c.
    • Extremes: the two “outer” numbers a and d.
  • Example: 39:21 = 13:7
    • Means → 21 and 13.
    • Extremes → 39 and 7.
  • Fundamental property (Cross-Multiplication):
    • If \frac{a}{b}=\frac{c}{d} then ad = bc.
    • Conversely, if ad = bc and all quantities are non-zero, the two ratios are proportional.

Types of Proportion

  1. Direct Proportion
    • Condition: a:b = c:d implies ad=bc (product of extremes equals product of means).
    • All four quantities scale in the same direction; if one term of a ratio increases while maintaining the proportion, its corresponding term in the other ratio also increases.
    • Real-world cues: “more → more,” “less → less.”
  2. (Inverse proportion is not covered on this page but is often introduced next, where one quantity increases while the other decreases such that their product remains constant.)

Additional Study Tips / Connections

  • Ratios simplify to fractions; everything you know about fractions (simplifying, equivalent fractions, cross-multiplying) applies.
  • Rates are just ratios with units; unit analysis (dimensional analysis) is a powerful tool to avoid mistakes.
  • Proportions underpin many practical tasks: map reading (scale), converting currencies, mixing solutions, and solving physics problems involving uniform motion.
  • Always write units explicitly when dealing with rates or mixed-unit proportions to keep track of what is being compared.
  • Ethical / practical note: Real-world decision-making (e.g.
    medication dosage, financial interest rates) often depends on accurate ratio and rate calculations, so precision and proper simplification are crucial.