Ratio and Proportion — SSC CGL Quantitative Aptitude

Ratio questions are rarely hard; they are frequent — directly, and hidden inside partnership, mixtures, time–work, and DI. The skill that separates fast candidates is refusing to solve for actual values: stay in ratio units ($3x$, $5x$) until the last line, and let one given rupee/year/mark figure convert units to reality.

1. What SSC actually asks

Tier 1: 1–2 Q · Tier 2: 2–3 Q directly, and ratio arithmetic silently powers another 6–8 questions across the paper. Direct types: chaining ($a:b$, $b:c$ → $a:c$), dividing an amount, proportionals (mean/third/fourth), coin bags, ages, income–expenditure–savings, and ratio-change equations ("if 9 is added to each…").

2. Chaining and combining ratios

To combine $a:b=3:4$ and $b:c=5:6$, make $b$ equal in both — LCM(4,5) = 20:

$a:b=15:20,b:c=20:24\Rightarrow a:b:c=15:20:24$

Shortcut: $a:b:c=(3\times 5):(4\times 5):(4\times 6)$ — multiply through the bridge term.

Dividing an amount in a fraction ratio ($\frac{1}{2}:\frac{2}{3}:\frac{3}{4}$): multiply by the LCM of denominators (12) first → $6:8:9$, then divide normally.

3. The proportionals

For $a:b::c:d$ ("$a$ is to $b$ as $c$ is to $d$"): product of extremes = product of means, $ad=bc$.

4. Componendo–dividendo — the one-line identity

$\frac{a}{b}=\frac{c}{d}\Rightarrow \frac{a+b}{a-b}=\frac{c+d}{c-d}$

Whenever a question hands you $\frac{a}{b}$ and asks about $\frac{a+b}{a-b}$ (or gives the latter and wants the former — apply it in reverse), this identity replaces cross-multiplication entirely. It is the single biggest time-saver in this chapter and reappears in trigonometry ($\tan$ half-angle forms) and algebra.

5. The three word-problem templates

Coins: a bag holds ₹1, 50p, 25p coins in ratio $5:6:8$. Convert counts to value: value ratio $=5x:3x:2x$ (50p halves, 25p quarters). Total value ₹210 → $10x=210$ → $x=21$ → each count follows.

Ages: present ratio $4:5$, after 6 years $5:6$: $\frac{4x+6}{5x+6}=\frac{5}{6}$ → cross-multiply → $x=6$. One equation, always.

Income–expenditure: incomes $9:7$, expenditures $4:3$, both save ₹2000: $9x-4y=2000,7x-3y=2000\Rightarrow x=2000\Rightarrow \text{incomes }18000,14000.$ Savings = income − expenditure gives two linear equations; eliminate and done.

6. Solved PYQ-style examples

Q1. $a:b=3:4$, $b:c=5:6$. Find $a:c$. Solution. $a:b:c=15:20:24$ → $a:c=15:24=$ 5 : 8.

Q2. Mean proportional between 4 and 25? Solution. $\sqrt{4\times 25}=\sqrt{100}=$ 10.

Q3. Two numbers are in ratio 3:5. Adding 9 to each makes the ratio 3:4. Find the numbers. Solution. $\frac{3x+9}{5x+9}=\frac{3}{4}\Rightarrow 12x+36=15x+27\Rightarrow x=3$ → 9 and 15. (Check: 18/24 = 3/4 ✓)

Q4. If $\frac{a}{b}=\frac{5}{3}$, find $\frac{a+b}{a-b}$. Solution. Componendo–dividendo: $\frac{5+3}{5-3}=$ 4.

Q5. 20% of x = 30% of y. Find $x:y$. Solution. $\frac{x}{y}=\frac{30}{20}=\frac{3}{2}$ → 3 : 2. (The percentages swap sides — a five-second question once you expect the flip.)

7. Exam protocol

1. Never compute actual values until forced — carry $3x,5x$ and let the one rupee/year figure fix $x$ at the end.
2. Chain ratios through the bridge term's LCM; for three-term chains multiply through.
3. See $\frac{a+b}{a-b}$ anywhere? componendo–dividendo, one line.
4. Coin problems: convert counts to value ratio before using the total.
5. Sanity-check ratio-change answers by substituting back — it takes five seconds and catches sign slips.