Chapter in a nutshell: A ratio compares two quantities ($a:b$); a proportion states two ratios are equal ($a:b::c:d$). Key tools are the mean proportional, product of extremes = product of means, and the manipulation laws — invertendo, alternendo, componendo, dividendo and componendo-dividendo.
1. Key Ideas & Terms
- Ratio $a:b=\dfrac{a}{b}$ (b ≠ 0); has no units; unchanged if both terms are multiplied/divided by the same number.
- Proportion: $a:b::c:d \Leftrightarrow \dfrac{a}{b}=\dfrac{c}{d}$. Here a, d = extremes; b, c = means.
- Continued proportion: $a:b::b:c$ → b is the mean proportional and c is the third proportional.
2. Formulas (quick reference)
- Proportion: $\dfrac{a}{b}=\dfrac{c}{d}\Rightarrow ad=bc$ (product of extremes = product of means).
- Mean proportional of a and c: $b=\sqrt{ac}$ (so $b^2=ac$).
- Compound ratio of $a:b$ and $c:d$ = $ac:bd$.
- Duplicate ratio of $a:b=a^2:b^2$; triplicate $=a^3:b^3$; sub-duplicate $=\sqrt a:\sqrt b$; sub-triplicate $=\sqrt[3]a:\sqrt[3]b$.
- Manipulation laws (if $\frac{a}{b}=\frac{c}{d}$):
3. Method Note
Componendo–dividendo is the workhorse for ICSE problems: when $\dfrac{x}{y}=\dfrac{a}{b}$, apply it to simplify expressions like $\dfrac{x+y}{x-y}$.4. Worked Example (one, for the method)
Q. Find the mean proportional between 9 and 16. Solution: $b=\sqrt{9\times16}=\sqrt{144}=\mathbf{12}$.5. Prerequisite Ideas
- Fractions / simplification; square and cube roots (for duplicate/sub-duplicate ratios).
6. Common Mistakes to Avoid
- Treating a ratio as having units (it is unitless; quantities must be in the same unit first).
- Mixing means (b, c) and extremes (a, d).
- Using $b=ac$ instead of $b=\sqrt{ac}$ for the mean proportional.
- Applying componendo-dividendo with the wrong signs.
7. Likely Exam Questions (with crisp answers)
- State the property of a proportion. → Product of extremes = product of means ($ad=bc$).
- Define mean proportional of a and c. → $\sqrt{ac}$.
- What is the duplicate ratio of $a:b$? → $a^2:b^2$.
- State componendo–dividendo. → $\frac{a+b}{a-b}=\frac{c+d}{c-d}$.
- Find the mean proportional of 4 and 25. → 10.
- What is invertendo? → $\frac{a}{b}=\frac{c}{d}\Rightarrow\frac{b}{a}=\frac{d}{c}$.
- Compound ratio of 2:3 and 4:5? → 8:15.
- Third proportional to a and b? → $\frac{b^2}{a}$ (since $a:b::b:c$).
- Is a ratio changed by multiplying both terms by 3? → No.
- Sub-duplicate ratio of 9:16? → 3:4.
Formulas — Current & Prerequisite
Current
- Proportion: $a:b::c:d \Rightarrow ad=bc$.
- Continued proportion: $a:b=b:c \Rightarrow b^2=ac$ (b = mean proportional $=\sqrt{ac}$); c = third proportional.
- Componendo & dividendo: if $\dfrac{a}{b}=\dfrac{c}{d}$ then $\dfrac{a+b}{a-b}=\dfrac{c+d}{c-d}$.
- Alternendo $\dfrac{a}{c}=\dfrac{b}{d}$; Invertendo $\dfrac{b}{a}=\dfrac{d}{c}$; Componendo $\dfrac{a+b}{b}=\dfrac{c+d}{d}$; Dividendo $\dfrac{a-b}{b}=\dfrac{c-d}{d}$.
Prerequisite
- Simplifying ratios; cross-multiplication; algebraic fractions.