Ratio & Proportion – Study Notes (Tamil Nadu PSC – Quantitative Aptitude)
Target Exam: Tamil Nadu Public Service Commission (PSC)
Domain: Quantitative Aptitude
Difficulty Level: 5–7 out of 10 (IIT‑JEE style depth)
1. Concept & Definitions
| Term | Definition |
|---|---|
| Ratio | A comparison of two numbers or quantities expressed as (a:b) or (\frac{a}{b}). |
| Proportion | An equation stating that two ratios are equal: (\frac{a}{b} = \frac{c}{d}). |
| Direct Proportion | (y) increases as (x) increases: (y = kx). |
| Inverse Proportion | (y) decreases as (x) increases: (y = \frac{k}{x}). |
| Compound Ratio | A ratio of ratios, e.g., (\frac{a:b}{c:d}). |
| Ratio of Whole to Parts | If (a:b:c=1:2:3), then (a = \frac{1}{6} \times (a+b+c)). |
2. Basic Properties
| Property | Formula | Example |
|---|---|---|
| Transitivity | If (a:b = c:d) and (b:e = d:f) then (a:e = c:f). | If (2:3 = 4:6) and (3:5 = 6:10), then (2:5 = 4:10). |
| Substitution | If (a:b = c:d) then (a \times d = b \times c). | (3:4 = 6:8 \Rightarrow 3\times 8 = 4\times 6). |
| Componendo | (a:b = c:d \implies a+b:a = c+d:c). | (\frac{2}{3} = \frac{4}{6} \Rightarrow \frac{2+3}{2} = \frac{4+6}{4}). |
| Dividendo | (a:b = c:d \implies a:b(a+b) = c:d(c+d)). | (\frac{1}{2} = \frac{3}{6} \Rightarrow \frac{1}{2(1+2)} = \frac{3}{6(3+6)}). |
| Alternendo | (a:b = c:d \implies a:c = b:d). | (\frac{3}{4} = \frac{6}{8} \implies \frac{3}{6} = \frac{4}{8}). |
3. Derivations & Proofs
3.1 Transitivity Proof
If (\frac{a}{b} = \frac{c}{d}) and (\frac{b}{e} = \frac{d}{f})
[ \frac{a}{b} \times \frac{b}{e} = \frac{c}{d} \times \frac{d}{f} ] [ \Rightarrow \frac{a}{e} = \frac{c}{f} ]
3.2 Componendo Derivation
Starting from (\frac{a}{b} = \frac{c}{d}):
[ \frac{a}{b} = \frac{c}{d} \Rightarrow ad = bc ]
Add (b) to the numerator and denominator of the left ratio, (d) to the numerator and denominator of the right:
[ \frac{a+b}{b} = \frac{c+d}{d} ]
Cross‑multiply:
[ (a+b)d = (c+d)b \Rightarrow ad + bd = cb + db ]
Since (ad = bc), the equation holds. Thus ( \frac{a+b}{a} = \frac{c+d}{c}).
(Proof uses the equality (ad = bc) and simple algebra.)
4. General Solution Patterns
| Problem Type | Approach | Key Formula |
|---|---|---|
| Finding a missing term | Use substitution: (x = \frac{a \times d}{b}) if (a:b = x:d). | (x = \frac{a \cdot d}{b}). |
| Checking if a set is in proportional | Verify (a:b = c:d) via cross‑multiplication. | (ad = bc). |
| Finding ratio of whole to parts | If ratios given as (p:q:r = 1:2:3), then whole (= p+q+r). Ratio of part (p) to whole (= p/(p+q+r)). | (p/(p+q+r)). |
| Compound ratio | Compute numerator and denominator first, then simplify. | (\frac{a:b}{c:d} = \frac{a \cdot d}{b \cdot c}). |
| Inverse proportion | If (x) and (y) are inversely proportional, (x \times y = k). | (y = \frac{k}{x}). |
| Direct proportion | If (x) and (y) are directly proportional, (y = kx). | (k = \frac{y}{x}). |
5. Tricky Points & Common Mistakes
| Mistake | Why It Happens | Remedy |
|---|---|---|
| Confusing ratio vs proportion | Thinking any division is a proportion. | Remember: proportion is an equality of ratios. |
| Incorrect cross‑multiplication | Mixing up numerators/denominators. | Always write (\frac{a}{b} = \frac{c}{d} \Rightarrow ad = bc). |
| Ignoring units | Ratios of different units (e.g., kg to m) are invalid. | Ensure same units or convert before forming ratios. |
| Assuming commutativity | Swapping terms in a ratio changes the value. | Keep order: (a:b \neq b:a). |
| Using wrong formula for compound ratio | Forgetting to multiply numerator with numerator and denominator |