Ratio and Proportion is one of the highest-scoring and most predictable chapters in CDS and OTA Elementary Mathematics. Almost every paper carries direct questions on dividing amounts, comparing quantities and solving proportions. If your basics are clean, these are guaranteed marks in under a minute. This Cavalier guide builds the concept from scratch, with formulas, shortcuts and exam-style solved questions.
Why Ratio and Proportion Matters in CDS
The CDS Elementary Mathematics paper (UPSC) and the OTA paper both lean heavily on arithmetic, and Ratio & Proportion sits at the heart of it. The chapter is not just a stand-alone topic — it powers Partnership, Mixtures, Ages, Speed-Time-Distance and Percentage questions too.
So mastering this one chapter quietly raises your accuracy across half a dozen others. Examiners love it because answers are exact, clean and quick — perfect for an objective paper of 100 questions in 120 minutes. There is no ambiguity, no approximation, and no lengthy calculation if your method is right.
For a defence aspirant short on time, this is exactly the kind of chapter to prioritise: the rules are few, they repeat across years, and a single hour of focused practice can lock in marks that you will never lose again. Many candidates also find that a strong grip on ratios makes the Data Sufficiency and word-problem sections far less intimidating.
On average, 4–6 marks every CDS Maths paper come directly or indirectly from ratios. Treat this chapter as a scoring banker, not an option.
What Is a Ratio?
A ratio compares two quantities of the same kind and same unit by division. The ratio of a to b is written as a : b and read as “a is to b”. It equals the fraction a⁄b.
- The first term a is the antecedent.
- The second term b is the consequent.
A ratio has no unit — it is a pure number. You can only form a ratio between like quantities: 5 kg to 2 kg gives 5 : 2, but 5 kg to 2 metres has no meaning.
Think of a ratio as a recipe. When we say two quantities are in the ratio 3 : 2, we mean that for every 3 portions of the first there are 2 portions of the second. The actual amounts could be 3 and 2, or 30 and 20, or 300 and 200 — the relationship stays the same. This “parts” way of thinking is the single most useful mental model for the whole chapter, because nearly every CDS question asks you to split a total into parts or scale parts up to a total.
Multiplying or dividing both terms of a ratio by the same non-zero number does not change it. 4 : 6 = 2 : 3 = 8 : 12. Always reduce to lowest terms before comparing.
Types and Properties of Ratios
CDS questions test whether you know how ratios behave. Keep these definitions ready.
- Duplicate ratio of a : b is a2 : b2.
- Triplicate ratio of a : b is a3 : b3.
- Sub-duplicate ratio of a : b is √a : √b.
- Sub-triplicate ratio of a : b is 3√a : 3√b.
- Inverse (reciprocal) ratio of a : b is b : a, i.e. 1⁄a : 1⁄b.
- Compound ratio of a : b and c : d is (a×c) : (b×d).
To compare ratios like 3 : 4 and 5 : 7, cross-multiply: 3×7 = 21 and 4×5 = 20. Since 21 > 20, 3 : 4 is the greater ratio. Faster than converting to decimals.
What Is a Proportion?
When two ratios are equal, the four quantities are said to be in proportion. We write a : b :: c : d, read as “a is to b as c is to d”, which means a⁄b = c⁄d.
- a and d are the extremes.
- b and c are the means.
In any proportion, Product of extremes = Product of means, i.e. a × d = b × c. This single rule (cross-multiplication) solves almost every proportion question.
So if a : b :: c : d and three terms are known, the fourth is found instantly. For 3 : 5 :: 12 : x, we get 3x = 5×12 = 60, so x = 20.
Mean, Third and Fourth Proportional
These three terms appear repeatedly in CDS papers, so memorise them precisely.
Fourth proportional
If a : b :: c : x, then x is the fourth proportional to a, b, c, and x = (b × c) ⁄ a.
Third proportional
If a : b :: b : x, then x is the third proportional to a and b, and x = b2 ⁄ a.
Mean (second) proportional
If a : x :: x : b, then x is the mean proportional between a and b, and x = √(a × b).
Students mix up third and mean proportional. In the mean proportional the middle term repeats (a : x :: x : b); in the third proportional the known term repeats (a : b :: b : x). Read the position carefully.
Continued Proportion and Three-Term Ratios
Quantities a, b, c are said to be in continued proportion if a : b = b : c, which gives b2 = a×c. Here b is the mean proportional between a and c, and c is the third proportional to a and b.
Often a CDS problem chains three ratios, e.g. A : B = 2 : 3 and B : C = 4 : 5. To combine, make B common:
Multiply each ratio to equalise B. A : B = 2 : 3 = 8 : 12 (×4) and B : C = 4 : 5 = 12 : 15 (×3). So A : B : C = 8 : 12 : 15. This LCM trick is a must-know.
Once you have A : B : C, dividing any total amount is simple: total parts = 8 + 12 + 15 = 35, and each share is that fraction of the total.
Componendo, Dividendo and Quick Properties
For higher-level CDS questions, a few algebraic properties of proportion save time.
If a⁄b = c⁄d, then:
- Invertendo: b⁄a = d⁄c.
- Alternendo: a⁄c = b⁄d.
- Componendo: (a+b)⁄b = (c+d)⁄d.
- Dividendo: (a−b)⁄b = (c−d)⁄d.
- Componendo & Dividendo: (a+b)⁄(a−b) = (c+d)⁄(c−d).
The combined Componendo & Dividendo rule is gold when a question gives you the ratio of a sum to a difference. It converts a messy equation into one clean cross-multiplication.
Partnership: Dividing Profit by Ratio
Partnership is a direct application of ratios and a recurring CDS favourite. When partners invest money, profit is shared in the ratio of (investment × time).
Profit share ratio = (Capital1 × Time1) : (Capital2 × Time2) : …
If all partners invest for the same period, time cancels out and profit splits simply in the ratio of capitals.
- Simple partnership: capitals invested for equal time.
- Compound partnership: capitals invested for different time periods — use the capital×time product.
A “working partner” may also take a fixed salary or commission off the top before the remaining profit is divided by ratio. Read the question for such deductions.
The logic is intuitive: a partner who keeps more money in the business for longer deserves a larger slice of the profit. By converting every investment into a single capital×time figure, you put all partners on the same footing and can compare them with one clean ratio. The same idea extends to a partner who joins or leaves mid-year — simply count the exact months their capital was active.
Worked Example
Three friends A, B and C start a business. A invests ₹12,000 for the whole year, B invests ₹16,000 for 9 months and C invests ₹15,000 for 8 months. If the annual profit is ₹38,000, find each one’s share.
Notice how reducing the products to small numbers (6 : 6 : 5) made the final division painless. Always simplify the ratio before splitting the money.
Common Mistakes to Avoid
Forming a ratio between different units. Convert first: a ratio of 2 hours to 30 minutes is 120 min : 30 min = 4 : 1, not 2 : 30.
- Adding ratios directly. 1 : 2 plus 1 : 3 is not 2 : 5 — bring to a common consequent first.
- Splitting a quantity in ratio 3 : 5 by giving 3 and 5 units — you must use fractions 3⁄8 and 5⁄8 of the total.
- Confusing “A is 50% more than B” (A : B = 3 : 2) with “A is half of B”. Translate words to ratios carefully.
- In partnership, forgetting to multiply capital by time when periods differ.
Previous-Year Style Question
Q. The monthly incomes of two persons are in the ratio 4 : 5 and their monthly expenditures are in the ratio 7 : 9. If each saves ₹50 per month, find the monthly income of each.
Answer: Let incomes be 4x and 5x, expenditures 7y and 9y. Then 4x − 7y = 50 and 5x − 9y = 50. Solving: multiply the first by 5 and second by 4 → 20x − 35y = 250 and 20x − 36y = 200; subtracting gives y = 50, so 4x = 50 + 7(50) = 400, x = 100. Incomes are 4x = ₹400 and 5x = ₹500.
Whenever a question gives two ratios and a common saving or difference, introduce two variables (x for income parts, y for expenditure parts) and form two linear equations. It works almost every time.
Quick Revision
- Ratio a : b = a⁄b; antecedent : consequent; no units; reduce to lowest terms.
- Proportion a : b :: c : d ⇒ product of extremes = product of means (a×d = b×c).
- Mean proportional between a and b = √(ab); third proportional = b2⁄a; fourth = bc⁄a.
- Combine chained ratios by equalising the common term via LCM.
- Partnership profit splits as (capital × time).
- Compare two ratios by cross-multiplication; never add ratios directly.
Practise 15–20 mixed PYQs from this chapter and you will rarely lose a ratio mark again. The Cavalier method: master the rule, simplify early, then compute.
Frequently asked questions
How many questions on Ratio and Proportion appear in CDS Maths?
Typically 2 to 4 direct questions per paper, plus several more in linked topics like partnership, mixtures and ages. It is among the most rewarding chapters for the time invested.
What is the difference between ratio and proportion?
A ratio compares two quantities (a : b), while a proportion states that two ratios are equal (a : b :: c : d). Proportion always involves four terms with the rule product of extremes equals product of means.
How do I find the mean proportional quickly?
The mean proportional between a and b is the square root of their product, √(a×b). For example, the mean proportional between 4 and 9 is √36 = 6.
Can a ratio be formed between quantities of different units?
No. Both quantities must be of the same kind and converted to the same unit first. Convert metres to centimetres or hours to minutes before writing the ratio, since a ratio is a pure unit-less number.
How is profit divided in a partnership?
Profit is shared in the ratio of each partner's capital multiplied by the time it was invested. If all partners invest for the same period, time cancels and profit splits simply in the ratio of their capitals.
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