A ratio is a comparison between two quantities of the same kind. When you say the ratio of sugar to flour in a recipe is 2 : 3, you are saying for every 2 parts sugar, there are 3 parts flour — proportions being irrelevant to actual amounts. The actual quantity could be 200g to 300g, or 2kg to 3kg, but the relationship stays fixed at 2 : 3.
A proportion is a statement that two ratios are equal. If a : b = c : d, then a, b, c, d are said to be in proportion. The outer terms (a and d) are called extremes, and the inner terms (b and c) are called means. The key rule here: product of extremes = product of means, i.e., ad = bc. This single identity solves probably 60% of proportion problems in SSC MTS.
Look — the analogy that sticks best: think of a ratio as a gear ratio in a bicycle. If your front gear has 40 teeth and rear has 20 teeth, the ratio is 2 : 1. Double the front gear size, you must double the rear to keep the same ratio. That is what proportion does — it locks two ratios together.
Why this matters for SSC MTS specifically: Ratio and proportion is one of those topics that bleeds into almost everything else — mixtures, partnerships, ages, income comparisons. If your ratio fundamentals are clean, roughly 4-5 question types across the paper become easier to recognise and solve. At this level, questions rarely go beyond two-step reasoning, so a systematic approach almost guarantees full marks on these.
One more term to know upfront: continued proportion (also called continuous proportion). Three numbers a, b, c are in continued proportion when a : b = b : c. The middle term b is called the mean proportional, and b² = ac. This is tested directly and frequently.
Rule 1 — Make the common term equal when combining two ratios.
If A : B = 3 : 4 and B : C = 5 : 6, you cannot just write A : B : C = 3 : 4 : 5 : 6. The B values (4 and 5) don't match. You need to find LCM(4, 5) = 20, then scale:
A : B = 3 : 4 = 15 : 20B : C = 5 : 6 = 20 : 24A : B : C = 15 : 20 : 24This exact technique shows up in PYQ 69ecc48e94785de145938d68.
Rule 2 — Assign a variable to the ratio unit.
If a : b = 2 : 3, let a = 2k and b = 3k. Every ratio question in SSC MTS becomes a simple algebra problem the moment you assign k. Don't skip this step trying to solve mentally — you will make sign errors.
Rule 3 — Cross-multiplication is your default proportion tool.
If (2x + 3y) : (x + 2y) = 9 : 5, write it as a fraction:
Cross-multiply: 5(2x + 3y) = 9(x + 2y) → 10x + 15y = 9x + 18y → x = 3y → x : y = 3 : 1.
Never try to solve these by inspection or substitution when variables appear in both numerator and denominator — cross-multiply every single time.
Type 1 — Finding combined A : B : C from two pairwise ratios.
Already covered above (LCM method). Takes about 25 seconds once you have the rhythm.
Type 2 — Age/income ratio problems.
These combine ratio with a time shift (ages 5 years from now, income 25% more, etc.). The pattern is always: set up the ratio using k, build the equation from the condition, solve for k. See the income problem in PYQ 69ecc48e94785de145938d6c and the age problem in PYQ 69ecc48e94785de145938d6e.
Type 3 — Expressions involving squared ratios.
When the question asks (a² + b²) : (b² + c²), first find a : b : c using the pairwise LCM method, assign values, then compute squares. Never try to simplify the ratio algebra symbolically for these — numerical substitution is faster and error-free. See PYQ 69ecc48e94785de145938d6f.
Three numbers a, b, c in continued proportion:
So b = \sqrt{ac} (take positive root in exam unless told otherwise).
This is really the geometric mean formula. If a = 4 and c = 36, then b = \sqrt{144} = 12.
If a/b = c/d, then by componendo-dividendo:
This is useful when a question gives you the ratio of (sum : difference) and asks you to find individual numbers. See PYQ 69ecc48e94785de145938d74 — you can either cross-multiply directly or apply componendo-dividendo. Both work; direct cross-multiplication is usually faster at this level.
SSC MTS loves disguising ratio problems as percentage problems (like PYQ 69ecc48e94785de145938d6c). The translation is instant:
5/44/5x/100Always set the reference quantity (usually C or B, whoever appears in both statements) as 100 or a round number, then compute forward.
When combining A:B and B:C, find LCM of the two B-values and scale both ratios. Pattern: identify the shared variable → LCM → multiply each ratio so the shared variable matches → read off A:B:C in one line.
Micro-example: A:B = 3:4, B:C = 5:6. LCM(4,5) = 20. Scale first ratio ×5: 15:20. Scale second ratio ×4: 20:24. Result: 15:20:24.
Standard method (working it out from scratch each time): ~45 seconds. LCM bridge after 5 practice runs: ~15 seconds.
Whenever a problem chains two percentage changes to find a final ratio, set the "anchor" person's value as 100. You avoid fractions entirely.
Micro-example (from PYQ 69ecc48e94785de145938d6c): C = 100 → B = 80 (20% less) → A = 80 × 1.25 = 100. A:C = 1:1. Done in 3 steps, no algebra needed.
Standard method (algebraic with variables for A, B, C): 4-5 steps, ~50 seconds. Reference = 100 method: 3 steps, ~20 seconds.
For continued proportion (a, b, c), the middle term b = √(ac). No need to set up a : b = b : c and solve the equation each time.
Micro-example: a = 4, c = 36 → b = √(4×36) = √144 = 12. One multiplication and one square root, nothing else.
Equation method (a/b = b/c, cross-multiply b² = ac, solve): 3 written steps, ~30 seconds. Direct formula: 1 mental step, ~8 seconds.
When a question gives you (expression1) : (expression2) = p : q with two unknowns, always cross-multiply immediately — never try to split the ratio into components first.
Micro-example: (2x+3y):(x+2y) = 9:5. Cross-multiply: 10x+15y = 9x+18y → x = 3y → x:y = 3:1. Solving in under 20 seconds once automatic.
Trying to substitute guesses first: typically 3-4 failed attempts before landing on the right values, ~60-90 seconds. Direct cross-multiplication: 3 algebraic lines, ~20 seconds.
Whenever current ages are in ratio p:q, let them be pk and qk. The future/past condition gives you one equation in k. Solve for k, then answer any question about the ages directly.
Micro-example (PYQ 69ecc48e94785de145938d6e): A:B = 5:6, so ages = 5k and 6k. After 8 years: (5k+8)/(6k+8) = 7/8 → 40k+64 = 42k+56 → k = 4. A's age = 5×4 = 20.
Mental guessing method: ~90 seconds, error-prone. k-substitution: 4 algebraic steps, ~25 seconds.
Read the question. Identify which type it is:
Is it asking for A:B:C from two pairwise ratios? → Use LCM bridge. 15 seconds.
Is it a "continued proportion / mean proportional" question? → Apply b = √(ac) directly. 10 seconds.
Does it involve percentage changes leading to a ratio? → Set anchor = 100, compute forward. 20 seconds.
Is it an expression ratio (2x+3y):(x+2y) = p:q? → Cross-multiply immediately, collect x and y terms, find x:y. 20 seconds.
Is it an age/quantity shift problem (ratio changes after adding/subtracting)? → Let current values = pk and qk. Build one equation from the new condition. Solve for k. 25 seconds.
Is it a squared-expression ratio? → Find numerical values of a, b, c using LCM bridge, then compute squares. Never try to simplify symbolically. 30 seconds.
One universal rule: as soon as you see a ratio, assign the variable k. Do not attempt ratio problems without k — it costs you 30+ seconds of circular thinking every time.
Why this question (PYQ 1): Tests whether you can combine pairwise ratios and then apply squared computations. Most errors happen because students forget to make b equal before combining.
Solving path: Identify the shared variable B. LCM(4,5) = 20. Scale A:B = 3:4 → 15:20. Scale B:C = 5:6 → 20:24. Read off A:B:C = 15:20:24. That is your answer — no further steps needed.
Why this question (PYQ 2): A pure continued proportion test. The formula b² = ac should fire automatically.
Solving path: a = 4, c = 36. Apply b² = ac = 144. Take square root: b = 12. The question has no traps — if you know the formula, it is a 10-second problem.
Why this question (PYQ 3): A disguised ratio question wrapped in percentage language. Students who do not spot the percentage-to-ratio translation waste 60+ seconds setting up equations.
Solving path: Set C = 100. B = 80 (20% less than C). A = 80 × 1.25 = 100 (25% more than B). A:C = 100:100 = 1:1. The "trick" is noticing that 20% less followed by 25% more brings you back to the original.
Why this question (PYQ 4): Tests k-substitution for age shift problems, the most common ratio template in SSC MTS.
Solving path: Let A = 5k, B = 6k. After 8 years: (5k+8)/(6k+8) = 7/8. Cross-multiply: 40k + 64 = 42k + 56. So 2k = 8, k = 4. A's present age = 5 × 4 = 20 years.
Why this question (PYQ 5): Tests cross-multiplication on an expression ratio — the fastest and most reliable approach.
Solving path: Write as fraction: (2x+3y)/(x+2y) = 9/5. Cross-multiply: 10x + 15y = 9x + 18y. Simplify: x = 3y. Therefore x:y = 3:1.
Not making the shared variable equal before combining ratios. If A:B = 3:4 and B:C = 5:6, writing A:B:C = 3:4:6 is a common slip. The B-values (4 and 5) are different — always LCM-bridge first.
Taking the wrong root in continued proportion. b² = ac has two roots: positive and negative. In all SSC MTS problems involving quantities (ages, lengths, money), take the positive root. You will not be asked to justify this.
Forgetting to multiply the entire ratio when scaling. When scaling A:B = 3:4 by 5, both terms become 15:20. Students sometimes scale only one term, getting 15:4 or 3:20.
Using the wrong anchor in chained percentage problems. In "A is 25% more than B, and B is 20% less than C", the anchor must be C (it appears only once, at the start of the chain). Setting A = 100 and working backward creates fractions and errors.
Cross-multiplying before fully writing out the equation. In expression ratio problems, write a/b = c/d as a clean fraction first, then cross-multiply. Students who skip the fraction form and go straight to mental multiplication almost always make sign errors.
Confusing "ratio of sum to difference" with "ratio of the numbers". If (a+b):(a-b) = 5:1, this does NOT mean a:b = 5:1. You must cross-multiply the sum-difference equation and use the additional constraint (like knowing a = 18) to find b.