A ratio is just a comparison of two quantities of the same kind. When you say a:b, you're answering the question: "for every a parts of the first thing, how many parts of the second?" That's it. No mystery.
A proportion is when two ratios are equal — a:b = c:d. This is the statement that makes the whole machinery work. Once you accept that two ratios are equal, you can find any unknown in the chain.
Here's a classroom analogy that sticks: think of a ratio like a recipe. If your dal recipe uses 2 cups of lentils for 5 cups of water, the ratio is 2:5. Now whether you cook for 4 people or 40, the ratio stays the same — that's proportion. You scale the ingredients, but the relationship between them stays locked.
The proportional relationship in notation:
If a:b = c:d, then:
a × d = b × c (cross-multiplication — the most-used tool in Agniveer questions)d is called the fourth proportional to a, b, cThe ratio unit trick is the fastest conceptual tool you'll use on this exam. If A:B = 5:2, it means A has 5 parts and B has 2 parts of the same unit. If you're told B = 18, you immediately know: 2 parts = 18, so 1 part = 9. From there, A = 5 × 9 = 45. Every age-ratio and partnership question on this paper runs on this single idea.
Look — the Agniveer CEE doesn't throw heavy algebra at you. The questions are clean, the numbers are manageable, and the fastest path is almost always the ratio-unit method. The candidate who can find "1 part = ?" in under 5 seconds and multiply up will outpace someone grinding through equations every time.
A ratio a:b is equivalent to the fraction a/b. Two ratios a:b and c:d are equal when their cross-products match: ad = bc.
Simplifying ratios: always divide both terms by their HCF. The ratio 56:40 simplifies to 7:5 (dividing both by 8). This matters because simplified ratios expose the "unit value" immediately.
Equivalent ratios: multiply or divide both terms by the same number. 3:4 = 6:8 = 9:12. In exam questions, when you're asked "find x if x:7 = 56:40", this is just a proportion equation solved by cross-multiplication.
Direct Proportion: as one quantity increases, the other increases proportionally. If 3 workers earn ₹900, then 5 workers earn ₹1500 (same rate). The formula: a₁/b₁ = a₂/b₂.
Inverse Proportion: as one increases, the other decreases. 4 workers finish a job in 12 days; 6 workers finish the same job in 8 days. Here a₁ × b₁ = a₂ × b₂.
Third Proportional: given a:b = b:x, then x = b²/a. Example: third proportional to 4 and 6 is 6²/4 = 9.
Mean Proportional: given a:x = x:b, then x = √(ab). Example: mean proportional between 4 and 9 is √36 = 6.
This is the question type that trips most candidates — the A:B, B:C → A:B:C chain. There's a systematic method, not guesswork.
Method — Make B Common:
Given A:B = 11:7 and B:C = 4:19, B appears in both ratios but as different numbers (7 and 4). To combine them:
A:B = 11:7 → 44:28 (multiply both by 4)B:C = 4:19 → 28:133 (multiply both by 7)A:B:C = 44:28:133Shortcut for a:c when given a:b and b:c:
a/c = (a/b) × (b/c) — just multiply the fractions. Cancel if possible.
Given a:b = 7:9 and b:c = 5:7:
a/c = (7/9) × (5/7) = 5/9
So a:c = 5:9. Note how the 7s cancelled. This happens frequently — scan for cancellation before multiplying.
These are pure ratio-unit problems. The template every time:
A:B = m:nIf A:B = 5:2 and B = 18:
There is no need to write equations like 5x = A and 2x = 18. That's the slow path. The ratio-unit method skips algebra entirely.
Componendo-Dividendo: If a/b = c/d, then (a+b)/(a-b) = (c+d)/(c-d). Useful in algebraic ratio questions but rarely tested at Agniveer level.
Duplicate ratio of a:b is a²:b². Triplicate ratio is a³:b³. Sub-duplicate ratio is √a:√b.
Whenever a ratio and one actual value are given, find "1 part" first. Example: if P:Q = 3:7 and Q = 49, then 7 parts = 49, so 1 part = 7, and P = 3 × 7 = 21. Standard equation method: set 7x = 49, x = 7, P = 3x = 21 — same result but you're writing more steps. Ratio-unit method: 2 steps vs 4 steps. Works for age problems, salary problems, partnership problems — any "ratio given, one value given, find the other" structure.
For any x:a = b:c question, the answer is always x = (a × b)/c. Don't think about it — just write the cross-product. For the question "x:7 = 56:40": x = (7 × 56)/40 = 392/40 = 9.8. Standard approach: write x/7 = 56/40, then cross-multiply and simplify — 3 written steps. This shortcut: mentally write numerator × numerator over denominator and divide — 1 step. Time saved: ~20 seconds on a question like this.
When finding a:c from a:b and b:c, write them as fractions and multiply: (a/b) × (b/c). Before you multiply out, check if the middle term (b) appears in both numerator and denominator — it almost always cancels cleanly. Example: a:b = 7:9, b:c = 5:7. Write (7/9) × (5/7). The 7s cancel: result is 5/9, so a:c = 5:9. Without cancellation, you'd compute 35/63 and then simplify — that's an extra simplification step. Cancellation-first saves that step and prevents arithmetic errors.
To merge A:B and B:C into A:B:C, find LCM of B's two values and scale. LCM tells you the multipliers instantly. A:B = 11:7 and B:C = 4:19. LCM(7,4) = 28. Multiply first ratio by 4 (28/7), second by 7 (28/4). Done: 44:28:133. This is systematic — no guessing, no trial and error. A candidate trying to "figure it out" without the LCM method takes 60-90 seconds. With LCM: under 30 seconds.
Before solving any proportion, simplify both ratios. In the question x:7 = 56:40, simplify 56:40 → 7:5 (divide by 8). Now x:7 = 7:5, so x = 7 × 7/5 = 49/5 = 9.8. Simplification reduces the numbers you work with, cutting multiplication and division errors. This matters most when the numbers look large — 56 and 40 are more intimidating than 7 and 5, even though they're the same ratio.
In the exam hall, run through this decision tree the moment you see a ratio/proportion question:
Step 1 — What's given?
Step 2 — Simplify before computing. Check if the ratio can be reduced. 56:40 → 7:5 takes 3 seconds and halves your arithmetic.
Step 3 — Solve in one line if possible. The answer to nearly every Agniveer ratio question fits in one multiplication and one division. If you're writing three lines of algebra, you've taken the slow path.
Step 4 — Verify by substituting back. For x:7 = 56:40, plug x = 9.8 back: 9.8/7 = 1.4 and 56/40 = 1.4. Match — confirmed in 5 seconds.
Why this question: Tests whether you can execute a basic proportion equation cleanly and handle a decimal answer without panicking.
Solving path: Write the proportion: x/7 = 56/40. Cross-multiply: x = 7 × 56/40 = 392/40 = 9.8. Alternatively, simplify 56:40 to 7:5 first, then x/7 = 7/5 → x = 49/5 = 9.8. Both paths give 9.8. The trap here is picking 10 (a nearby round number) — don't round until the final answer.
Why this question: Classic age-ratio pattern. Tests the ratio-unit method under time pressure.
Solving path: A:B = 5:2, B = 18. Find 1 part: 2 parts = 18, so 1 part = 9. A = 5 × 9 = 45. Answer: 45. The wrong answer 42 (option A) is exactly 18 + 24 — a common error when candidates try to add proportionally instead of scaling. Stick to the ratio-unit method.
Why this question: Simpler chain — just one ratio and one value. Tests if you can identify which part of the ratio corresponds to the known value.
Solving path: A:B = 1:3, B = 15. The "B-part" of the ratio is 3. So 3 parts = 15 → 1 part = 5 → A = 1 × 5 = 5 years. Or directly: A = (1/3) × 15 = 5. One multiplication. Answer: 5 years.
Why this question: The three-term chained ratio. This is the hardest structure on this topic at Agniveer level. Tests the LCM method.
Solving path: A:B = 11:7, B:C = 4:19. B is 7 in the first and 4 in the second. LCM(7, 4) = 28. Scale first ratio: multiply both by 4 → A:B = 44:28. Scale second ratio: multiply both by 7 → B:C = 28:133. B is now 28 in both. Read off: A:B:C = 44:28:133. Match with option D.
Why this question: Tests the fraction-multiplication shortcut for finding a:c from two given ratios. Clean cancellation makes this very fast if you see it.
Solving path: a/c = (a/b) × (b/c) = (7/9) × (5/7). The 7s cancel: = 5/9. So a:c = 5:9. Answer: option A. The wrong answer 7:15 comes from adding instead of multiplying — a common conceptual error when students don't know the fraction method.
Confusing which part maps to the known value. In A:B = 5:2 with B = 18, the "2" maps to B. Candidates who write A = (5/2) × 18 = 45 are right — but those who write A = (2/5) × 18 = 7.2 have swapped the ratio. Always identify: known quantity → its ratio part.
Not simplifying before computing. Working with 56:40 directly instead of reducing to 7:5 inflates the numbers and increases arithmetic errors. Simplify first, always.
Picking the nearest round number as a shortcut. In x:7 = 56:40, x = 9.8 — not 10. Agniveer questions deliberately place a round number as a distractor near the correct decimal answer. Do the actual computation.
Adding ratio parts instead of scaling. For A:B = 11:7 and B:C = 4:19, you cannot add the ratios. You must make B common using LCM. This is the single biggest error on three-term ratio questions.
In inverse proportion, using the direct formula. If 4 workers take 12 days and you need to find time for 6 workers, use 4 × 12 = 6 × x, not 4/12 = 6/x. More workers means fewer days — inverse, not direct.
Forgetting to verify the answer. A 5-second back-substitution catches arithmetic slips before you move on. For proportion questions, this is the fastest error-check available.