A ratio is a comparison of two quantities of the same kind. When you write a : b, you're saying "for every a units of the first thing, there are b units of the second." Nothing more.
Think of it like a recipe: if a dal recipe uses flour and water in the ratio 2 : 5, doubling the batch gives 4 : 10 — the ratio stays the same, only the scale changes. This is the single most important intuition for ratio problems.
A proportion is a statement that two ratios are equal: a : b = c : d, written as a/b = c/d. This gives you the cross-multiplication rule: ad = bc. Every "find the missing value" question in this topic is just one application of that rule.
Why it matters for CHSL: Ratio and proportion questions appear in almost every CHSL paper, often disguised as money-sharing, student-distribution, or mixture problems. The calculation is rarely hard — the trap is in how you set up the ratio. Get the setup right and you'll never struggle with the arithmetic.
One analogy that sticks: Ratio is like a photograph's aspect ratio — 16:9. You can resize the photo to any dimension, but the shape stays the same. The "shape" is the ratio; the "size" is the actual value.
Key vocabulary you must know:
a in a : b)b in a : b)a and b: √(ab) — the geometric meana and b: b²/a — because a : b = b : x gives x = b²/aa, b, c: bc/a — because a : b = c : x gives x = bc/aThis is the most tested operation in CHSL. You're given m:n and n:p and asked to find m:n:p. The method is straightforward — make the common term equal in both ratios.
Method: Find the LCM of the two values of the common term, then scale each ratio accordingly.
Example: m:n = 5:11 and n:p = 2:5.
n. It appears as 11 in the first ratio and 2 in the second.m:n = 10:22n:p = 22:55m:n:p = 10:22:55This exact question appeared in CHSL 2025.
When a ratio is given among groups and you need the percentage of a subset, do this:
Example: Ratio 11:5:6:7 (CS:EC:CE:ME). Total = 29 parts. CS + ME = 11 + 7 = 18 parts. Percentage = (18/29) × 100 = 62.07%.
No need to assume actual numbers. Work with parts directly.
This is a classic CHSL structure: "P gets ₹X more than R — find something else."
Method:
(ratio of P) − (ratio of R) = difference in partsExample: P:Q:R = 7:9:4. P gets ₹1500 more than R.
7 − 4 = 3 parts = ₹1500If a : b = b : x, then x = b²/a. This is the third proportional to a and b.
To find the third proportional to 25 and 45:
x = 45² / 25 = 2025 / 25 = 81
Mean proportional of a and b is the middle term when you write a : m = m : b, so m² = ab, giving m = √(ab).
When each ratio term is changed by a percentage, multiply each part by its multiplier.
Example: A:B = 13:17. A increased by 15%, B increased by 30%.
13 × 1.15 = 14.9517 × 1.30 = 22.114.95 : 22.1299 : 442299/13 = 23, 442/13 = 3423 : 34The trick here is choosing the right multiplier to get integers. If both numbers end in .5, multiply by 2. If they have one decimal place, multiply by 10.
When ratios are given as decimals or fractions (like 1.5:2 and 2:2.5), convert to integers first by multiplying each ratio by the LCM of denominators.
Seema:Komal = 1.5:2 → multiply by 2 → 3:4
Komal:Rita = 2:2.5 → multiply by 2 → 4:5
Now combine: Komal is already 4 in both. So Seema:Komal:Rita = 3:4:5.
When you have A as a fraction of B, B as a fraction of C, and so on, assume a convenient value for the last variable and work backwards.
If A = (3/4)B, B = (4/5)C, C = (3/8)D:
(3/8) × 80 = 30(4/5) × 30 = 24(3/4) × 24 = 18Check: 4A + 7D = 72 + 560 = 632. Average = 316. Confirmed.
Average of A, B, C, D = (18+24+30+80)/4 = 152/4 = 38.
When combining two ratios with a shared term, the shared term's values in both ratios must be made equal using LCM. Think of it as building a bridge — the bridge support (shared term) must be the same height on both sides.
Quick check: m:n = 5:11, n:p = 2:5. Shared term n appears as 11 and 2. LCM = 22. Scale first ratio ×2 (10:22), scale second ×11 (22:55). Answer: 10:22:55.
Standard method (trial and error): 45–60 seconds. LCM Bridge: under 20 seconds.
Whenever a ratio problem says "X gets ₹K more than Y" or "X has N more than Y", find (X's ratio part − Y's ratio part) = difference in parts. Then: 1 part = K ÷ difference. Use this single value to answer anything else.
Example: P:Q:R = 7:9:4. P − R = 3 parts = ₹1500. So 1 part = ₹500. Need Q's share? 9 × 500 = ₹4500. Total? 20 × 500 = ₹10000.
Standard method (assume total, solve equation): 4 steps. One-Part Rule: 2 steps.
Third proportional to a and b = b²/a. No need to set up the full proportion and cross-multiply — just square the second number and divide by the first.
Third proportional to 25 and 45: 45² / 25 = 2025 / 25 = 81. Done in one line.
Standard cross-multiplication setup: 3 steps. Direct formula: 1 step.
If a ratio contains decimals like 1.5 : 2, multiply the entire ratio by enough to get integers. For one decimal place, multiply by 10. For .5 endings, multiply by 2.
Seema:Komal = 1.5:2. Multiply by 2 → 3:4. Komal:Rita = 2:2.5. Multiply by 2 → 4:5. Now the Komal values match — combine directly to 3:4:5. No fractions at any stage.
Working with decimal ratios throughout: error-prone, 5+ steps. Clear-first method: 2 steps, zero decimal arithmetic.
Never assume actual numbers when a ratio is given and you need a percentage. Just use (target parts / total parts) × 100.
Ratio 11:5:6:7. Need CS+ME as percentage. CS+ME = 11+7 = 18. Total = 29. Answer = 18/29 × 100. For 18/29, note that 18/29 ≈ 0.6207 (since 18 × 1.61 ≈ 29). Multiply by 100 = 62.07%.
Assuming real numbers (say, 1100 students per part): inflates mental math. Ratio-parts method: immediate setup, one division.
Read the question and classify it immediately:
Is the ratio given and you need to find individual values? → Check if total is given (divide total by sum of parts) OR if a difference is given (use One-Part Rule).
Are two ratios with a shared term given? → Use the LCM Bridge. Find LCM of the shared term's two values, scale each ratio, combine.
Is "third proportional" or "mean proportional" mentioned?
→ Third proportional to a, b = b²/a. Mean proportional of a, b = √(ab). Apply directly.
Is a ratio changing after a percentage increase/decrease? → Multiply each part by its multiplier (1.15 for +15%, 0.90 for −10%), clear decimals, simplify.
Are decimals or fractions in the ratio? → Convert to integers first by multiplying the ratio by an appropriate factor. Never work with decimal ratios.
Is a percentage asked from a given ratio? → (Relevant parts / Total parts) × 100. Do not assume actual numbers.
In the exam hall: classify first, formula second, calculate last. Most errors happen when students skip the classification step and jump straight to arithmetic.
Why this question: This is the most direct test of chain ratio / combining ratios — the most commonly tested ratio operation in CHSL.
Solving path: Identify the shared term (n). Its values are 11 (in m:n) and 2 (in n:p). LCM(11,2) = 22. Scale m:n by 2 → 10:22. Scale n:p by 11 → 22:55. Combine: m:n:p = 10:22:55.
Why this question: Tests whether you can extract a percentage directly from a multi-part ratio without getting confused by actual numbers.
Solving path: Total parts = 11+5+6+7 = 29. CS + ME = 11+7 = 18. Percentage = (18/29) × 100. Compute: 18 ÷ 29 = 0.6207.... Multiply by 100 = 62.07%. Select option C.
Why this question: Third proportional is a definition-based question — if you know the formula, it is a 15-second question. If you don't, it wastes over a minute.
Solving path: Third proportional to 25 and 45 = 45² / 25 = 2025 / 25 = 81. Select option A.
Why this question: Tests the ratio-after-percentage-change technique, which trips up students who try to set up equations instead of using multipliers.
Solving path: New A = 13 × 1.15 = 14.95. New B = 17 × 1.30 = 22.1. Ratio = 14.95:22.1. Multiply both by 20 to clear decimals: 299:442. Divide both by 13: 23:34. Select option A.
Why this question: Combining ratios with decimal terms — a common variant that confuses students who forget to convert to integers first.
Solving path: Seema:Komal = 1.5:2. Multiply by 2 → 3:4. Komal:Rita = 2:2.5. Multiply by 2 → 4:5. Komal value is 4 in both — combine directly: Seema:Komal:Rita = 3:4:5. Total = 12 parts. Rita = (5/12) × 3600 = ₹1500. Select option D.
Forgetting to scale both ratios when combining. When making n equal using LCM, students often scale only one ratio and leave the other unchanged. Both ratios must be scaled. If n=11 in one and n=2 in the other, and LCM=22, the first ratio gets multiplied by 2 AND the second by 11.
Using the wrong formula for third vs. mean proportional. Third proportional to a, b is b²/a (three terms: a, b, x where a:b = b:x). Mean proportional is √(ab) (three terms: a, m, b where a:m = m:b). Swapping these costs a guaranteed wrong answer.
Dividing difference by the wrong number of parts. In "P gets ₹1500 more than R" with ratio 7:9:4 — the difference is P−R = 7−4 = 3 parts, not 7 or 4. Always subtract the ratio parts, not add them.
Working with decimal ratios without converting. Keeping 1.5:2 as-is and trying to do arithmetic leads to rounding errors. Always multiply through to get integers before doing any further ratio operations.
Treating ratio as actual quantity. A ratio of 3:4 does not mean the values are 3 and 4. They are 3k and 4k for some k. If no total or difference is given, you cannot find actual values — the question will always provide one anchor (total, difference, or one actual value).
Adding ratios that don't share a common term. You cannot directly combine A:B = 2:3 and C:D = 5:7 into a four-term ratio unless a connection between B and C is established. Students sometimes chain unrelated ratios — check that the shared middle term genuinely links the two ratios before combining.