Ratio and Proportion (Including Properties and Uses)

Introduction

• Revisits concepts of Ratio and Proportion initially introduced in Classes 8–9; Class 10 deepens understanding and extends applications.
• Central idea: Comparison of quantities of the same kind using division (ratio) and establishing equality of two ratios (proportion).
• Vital for topics such as similar triangles, trigonometry, commercial math (profit–loss, discount, simple/compound interest), mixtures, speed–time–distance, and various competitive-exam problems.

Ratio

• Definition: The ratio of two quantities of the same kind in the same units is the quotient obtained on dividing one quantity by the other.
  • If the quantities are $a$ and $b$, the ratio is written $a:b$ or $\frac{a}{b}$ (read as “a is to b”).
• Order matters: $a:b \neq b:a$ unless $a = b$.
• Terms:
  • $a$ → Antecedent (first term).
  • $b$ → Consequent (second term).
• Unit-free: Because identical units cancel during division, the ratio itself has no unit.
• Simplest (Lowest) Form:
  • Divide both terms by their $\text{HCF}$ (Highest Common Factor).
  • A ratio is said to be in lowest terms when $\text{HCF}(a,b)=1$.
  • Example: $48:60 \rightarrow \frac{48\div12}{60\div12}=4:5$.
• Types of Ratios:
  • Equivalent ratios: $a:b = ka:kb$ for any $k \neq 0$.
  • Compound ratio: Ratio obtained by multiplying two or more ratios term-wise, e.g.
  $(a:b) \times (c:d)=ac:bd$.
  • Inverse (Reciprocal) ratio: $b:a$.
  • Duplicate, Triplicate, Sub-duplicate, etc.:
  – Duplicate of $a:b$ is $a^2:b^2$.
  – Triplicate is $a^3:b^3$, etc.
• Comparison of Ratios: Convert to identical consequent (or antecedent) or to decimals/fractions; the larger quotient corresponds to the greater ratio.

Proportion

• Definition: Two ratios $a:b$ and $c:d$ are in proportion if $a:b = c:d$, written $a:b::c:d$ or $a:b = c:d$.
  • Means: $b,c$.
  • Extremes: $a,d$.
  • Fundamental property: $a\times d = b\times c$.
• Types of Proportion:
  • Direct (Simple) proportion: Increase in one quantity causes proportional increase in another.
  • Inverse (Indirect) proportion: Increase in one causes proportional decrease in another; product remains constant.
  • Continued proportion: $a:b = b:c$ ⇒ $b^2 = ac$; $b$ is the mean proportional (geometric mean) of $a$ and $c$.
  • Compound, Partitive, and Mixed proportions appear in mixture or partnership contexts.

Algebraic Properties of Proportion (Valid when $a:b::c:d$)

• Invertendo: $b:a = d:c$.
• Alternendo: $a:c = b:d$.
• Componendo: $\frac{a+b}{b} = \frac{c+d}{d}$.
• Dividendo: $\frac{a-b}{b} = \frac{c-d}{d}$.
• Componendo-Dividendo: $\frac{a+b}{a-b} = \frac{c+d}{c-d}$.
• Substitendo (or Convertendo): $\frac{a-b}{a} = \frac{c-d}{c}$.
• Equality of Products (Cross-multiplication): $ad = bc$, fundamental for solving proportion equations.

Solving Problems Involving Proportion

• Direct proportion setup: $\frac{x<em>1}{y</em>1} = \frac{x<em>2}{y</em>2}$ ⇒ unknown found by cross-multiplication.
• Inverse proportion setup: $x<em>1 y</em>1 = x<em>2 y</em>2$.
• Unitary method: Reduce to “quantity per one unit”, then scale up.
• Partnership problems: Share $P$ in ratio of time-weighted capitals: $P \times \frac{C<em>1T</em>1}{\sum C<em>iT</em>i}$.
• Mixture/alligation: Use rule of alligation; mean concentration sits between two given concentrations.

Common Examples & Illustrations

• Example 1 (simplifying ratios):
  • Simplify $154:198$.
  • $\text{HCF}(154,198)=22\Rightarrow 154\div22=7,\;198\div22=9$ ⇒ $7:9$.
• Example 2 (direct proportion): If 5 workers finish a task in 8 days, how long for 20 workers?
  • Direct ratio of workers to inverse ratio of days: $5:20 = d:8$ ⇒ $5\times8 = 20\times d$ ⇒ $d=2$ days.
• Example 3 (mean proportional): Find mean proportional between 72 and 8.
  • $b^2 = ac$ ⇒ $b = \sqrt{72\times8}=\sqrt{576}=24$.
• Example 4 (componendo-dividendo): If $a:b = 3:4$, find $\frac{a+b}{a-b}$.
  • $\frac{3+4}{3-4}=\frac{7}{-1}=-7$.

Real-World Connections

• Finance: Interest rates, currency conversions, taxation percentages rely on ratio logic.
• Science & Engineering: Gear ratios, dilution of solutions, stoichiometric coefficients in chemistry.
• Maps & Models: Scale ratios (e.g., $1:50\,000$ in cartography).
• Photography & Screens: Aspect ratios (16:9, 4:3) define framing.
• Ethics & Fairness: Proportional allocation underpins equitable division of resources, revenue sharing, and societal tax structures.

Common Mistakes & Misconceptions

• Mixing different units within a ratio (e.g., $\text{cm}$ with $\text{m}$) → convert first.
• Assuming order doesn’t matter (writing $b:a$ when $a:b$ is required).
• Applying direct proportion where relation is actually inverse (speed vs. time).
• Forgetting to simplify ratios to lowest terms before comparison.
• Misusing cross-multiplication when terms are not in proportion.

Key Formulae (LaTeX Syntax)

• Ratio: $a:b = \frac{a}{b}$, provided $b \neq 0$.
• Proportion condition: $a:b::c:d \;\Longleftrightarrow\; ad = bc$.
• Mean proportional $m$ of $a$ and $b$: $m = \sqrt{ab}$.
• Fourth proportional $x$ to $a,b,c$ (i.e.
  $a:b::c:x$): $x = \frac{bc}{a}$.
• Third proportional $x$ to $a,b$ (i.e.
  $a:b::b:x$): $x = \frac{b^2}{a}$.
• Partnership share $S<em>i$: $S</em>i = P \times \frac{C<em>i T</em>i}{\sum C<em>j T</em>j}$.

Quick Reference Cheatsheet

• Simplify the ratio first.
• Cross-multiply to verify or solve a proportion.
• Direct proportion ⇒ quantities change in same direction.
• Inverse proportion ⇒ product remains constant.
• Use properties (invertendo, alternendo, etc.) to manipulate/solve complex ratio statements efficiently.