Ratio and proportion is one of the highest-scoring and fastest topics in AFCAT Numerical Ability. A handful of clear rules unlock partnership, ages, mixtures, and a chunk of data-interpretation sums. In this Cavalier guide you will learn the core definitions, the cross-multiplication shortcut, and the exact traps that cost candidates easy marks.
Why Ratio and Proportion Matters in AFCAT
The AFCAT Numerical Ability section is short but time-pressured. Ratio and proportion is a favourite of paper-setters because it links to many other chapters — percentage, profit and loss, partnership, mixtures, ages and time-and-work all lean on ratios. Master this topic and you indirectly speed up half the arithmetic paper. Almost every comparison-based question, whether it talks about sharing money, mixing liquids or comparing speeds, can be reduced to a clean ratio statement.
Questions here are usually one or two steps. With the right method you can finish each in 30–45 seconds, banking time for tougher data-interpretation sets later. Because the underlying idea is so reusable, the hours you invest in ratios pay back across the whole arithmetic syllabus rather than in just one chapter.
Students at The Cavalier are trained to spot a ratio the moment they read words like “divided among”, “in the proportion of”, “mixed in” or “shared between”. Recognising the pattern early is half the battle, because it tells you exactly which formula to reach for before you even pick up the pen.
In AFCAT there is negative marking (typically −1 for a wrong answer against +3 for a correct one). Speed must never override accuracy — a clean ratio method gives you both.
What a Ratio Really Means
A ratio compares two quantities of the same kind by division. The ratio of a to b is written a : b and equals the fraction a/b, where b ≠ 0. Here a is the antecedent (first term) and b is the consequent (second term).
- Both quantities must be in the same unit before you form the ratio.
- A ratio has no unit — it is a pure number.
- Multiplying or dividing both terms by the same non-zero number does not change the ratio: 2 : 3 = 4 : 6 = 10 : 15.
To convert a ratio into simplest form, divide both terms by their HCF. Example: 18 : 24 → divide by 6 → 3 : 4.
Think of a ratio as a recipe rather than a count. When we say sand and cement are mixed in 3 : 1, we are not saying there are exactly three buckets and one bucket; we are saying that for every three measures of sand there is one measure of cement, whatever the actual size of the batch. That is why both terms can be scaled freely without disturbing the relationship, and it is the foundation of nearly every trick in this chapter.
Never form a ratio across different units. The ratio of 50 paise to 2 rupees is not 50 : 2. Convert first: 50 paise : 200 paise = 1 : 4. The same caution applies to mixing metres with centimetres or hours with minutes.
Types of Ratio You Must Know
AFCAT sometimes tests the vocabulary of ratios directly. Keep these handy:
- Duplicate ratio of a : b is a2 : b2.
- Sub-duplicate ratio of a : b is √a : √b.
- Triplicate ratio of a : b is a3 : b3.
- Sub-triplicate ratio of a : b is a1/3 : b1/3.
- 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).
If a question says “duplicate ratio of 3 : 4”, just square both: 9 : 16. “Sub-duplicate of 9 : 16” takes the root: 3 : 4. These appear as quick one-mark items.
Proportion and the Cross-Multiplication Rule
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 solves almost every “find x” question.
So if 4 : 6 :: 10 : x, then 4 × x = 6 × 10 → x = 60 ÷ 4 = 15.
Why does this rule work? Because a proportion is simply two equal fractions, a/b = c/d. If you cross-multiply both sides by b and by d to clear the denominators, you are left with a × d = b × c. So the extremes-and-means rule is not a separate fact to memorise — it is just the fraction equation tidied up. Once you see it that way, you will never set up the multiplication the wrong way round.
A quantity is said to be in continued proportion when a : b = b : c. Here b is the mean proportional between a and c, and c is the third proportional. Continued proportion appears often in geometry-flavoured AFCAT items, so keep the pattern a : b :: b : c at your fingertips.
Mean, Third and Fourth Proportional
These small definitions are pure marks if you know the formula.
- Mean proportional between a and b = √(a × b). (Here a : x :: x : b, so x2 = ab.)
- Third proportional to a and b is the value x where a : b :: b : x, so x = b2 ÷ a.
- Fourth proportional to a, b, c is x where a : b :: c : x, so x = (b × c) ÷ a.
Example: the mean proportional between 9 and 16 is √(9 × 16) = √144 = 12. The third proportional to 4 and 8 is 82 ÷ 4 = 64 ÷ 4 = 16.
Do not confuse third proportional (b2/a) with fourth proportional (bc/a). Read whether the question gives two numbers (third) or three numbers (fourth).
The Constant-k Method: Your Best Shortcut
When a ratio is given, replace each part with a multiple of a single variable k. If two numbers are in ratio 3 : 5, write them as 3k and 5k. This converts a wordy problem into one simple equation.
Two numbers are in the ratio 3 : 5. If their sum is 64, find the numbers.
The beauty of the constant-k method is that it works no matter how many parts the ratio has or whether the question gives you a sum, a difference or a product. You introduce a single unknown, build one equation from the condition in the problem, solve for k, and then read off every quantity you need. It removes guesswork and keeps your working short enough to fit in the margin of the question paper.
For a three-part ratio a : b : c with a known sum S, each share = (its part ÷ total parts) × S. For 2 : 3 : 5 of ₹500, the middle share = (3/10) × 500 = ₹150. This “part over total parts” idea is the single most-used shortcut in the entire chapter.
Combining and Comparing Ratios
AFCAT loves chained ratios like “A : B = 2 : 3 and B : C = 4 : 5, find A : B : C”. Make the common term (B) equal in both ratios.
- A : B = 2 : 3 → multiply by 4 → 8 : 12
- B : C = 4 : 5 → multiply by 3 → 12 : 15
- So A : B : C = 8 : 12 : 15.
To compare two ratios such as 3 : 4 and 5 : 7, cross-multiply: 3 × 7 = 21 and 4 × 5 = 20. Since 21 > 20, 3 : 4 > 5 : 7. Faster than converting to decimals.
To divide a ratio in a given proportion, scale the linking term to the LCM of the two consequent/antecedent values that must match.
Partnership: A Classic Application
In business partnership questions, profit is shared in the ratio of (capital × time) for each partner.
Profit ratio = (C1 × T1) : (C2 × T2) : …
If all partners invest for the same time, profit splits simply in the ratio of capitals.
The logic is fair: a partner who puts in more money, or keeps it in the business longer, deserves a larger slice. So we weight each partner by the product of how much they invested and for how long. Once you have these weights, the profit is divided in that ratio using the same “part over total parts” rule you already know. Partnership questions therefore reduce to two easy steps — build the capital×time ratio, then share.
A invests ₹6000 for 12 months, B invests ₹8000 for 9 months. Total profit is ₹3000. Find B’s share.
Mixtures, Ages and Ratio Changes
Many word problems add or remove a quantity and ask for the new ratio. Use the constant-k setup, then form an equation with the change.
The present ages of A and B are in the ratio 5 : 7. Four years later, the ratio becomes 3 : 4. Find their present ages.
When “4 years later” or “6 years ago” is given, add or subtract from each term before cross-multiplying. Forgetting one term is the No. 1 error in age sums.
Speed Tricks for the Exam Hall
These habits shave seconds off every ratio sum:
- Simplify first. Reduce big ratios by HCF before any arithmetic.
- Sum-of-parts shortcut. Each share = (its part / total parts) × total amount — no need to find k separately.
- Cross-multiply to compare instead of dividing.
- Same-time partnership → ignore time entirely, split by capital.
- Check units the instant you read the question.
If a ratio question gives a difference (not a sum), the difference of parts works the same way. For 7 : 4 with a difference of 18, one part = 18 / (7−4) = 6, so the numbers are 42 and 24.
Previous-Year Style Practice
Q. ₹7800 is divided among A, B and C in the ratio 2 : 3 : 8. How much more does C receive than A?
Answer: Total parts = 2 + 3 + 8 = 13. One part = 7800 ÷ 13 = ₹600. C gets 8 × 600 = ₹4800; A gets 2 × 600 = ₹1200. C receives ₹4800 − ₹1200 = ₹3600 more than A. (Quick route: difference in parts = 8 − 2 = 6, so 6 × 600 = ₹3600.)
Notice how the “one part” idea answers everything — individual shares and differences — from a single division.
Quick Recap and Revision
- Ratio compares same-unit quantities; simplify by HCF, no units attached.
- Proportion: product of extremes = product of means (a×d = b×c).
- Mean proportional = √(ab); third proportional = b2/a; fourth = bc/a.
- Use the constant-k method to turn ratios into one equation.
- Partnership profit = (capital × time) ratio.
- Each share = (part / total parts) × total amount.
Drill 15–20 mixed problems daily for a week and ratio sums will become near-instant in your AFCAT attempt. The Cavalier’s classroom sets build exactly this reflex.
Frequently asked questions
How many ratio and proportion questions appear in AFCAT?
Typically 2 to 4 direct questions appear in the Numerical Ability section, and the concept indirectly supports partnership, mixtures and data interpretation. It is a high-return topic relative to the time needed to learn it.
What is the fastest way to divide an amount in a given ratio?
Add all the parts to get the total parts, then each share equals (its own part divided by total parts) multiplied by the total amount. You skip solving for k separately, saving valuable seconds.
What is the difference between a ratio and a proportion?
A ratio compares two quantities of the same kind (a : b). A proportion is a statement that two ratios are equal (a : b :: c : d), which lets you use the extremes-and-means rule to find an unknown.
How do I find the mean proportional quickly?
The mean proportional between two numbers a and b is the square root of their product, that is, square root of (a times b). For 4 and 9 it is square root of 36, which equals 6.
Do I always need to convert units before forming a ratio?
Yes. Both quantities must be in the same unit before you write the ratio, otherwise the answer is wrong. For example, convert rupees to paise, or hours to minutes, first, then simplify.
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