Ratio and Proportion Formulas for CAT 2026: Mixtures, Alligation + 10 PYQs
Ratio and proportion as the master-key framework for CAT 2026 arithmetic: every formula, the see-saw alligation method for mixtures, the partnership investment-time identity, the mixture replacement formula, and 10 CAT previous year questions solved with the k-multiplier shortcut. Covers compounded and duplicate ratios, direct and inverse variation, componendo-dividendo, and the four repeatable CAT patterns.
Ratio and Proportion Formulas for CAT 2026: Mixtures, Alligation + 10 PYQs
Most CAT aspirants treat ratio and proportion as a beginner topic and lose the very questions designed to discriminate between 95 and 99 percentile scorers. The reason: ratio is a master-key topic. The same framework that solves a basic a:b problem also solves partnership splits, mixture alligation, dilution problems, and inverse-variation puzzles. Aspirants who memorise these as four separate topics study four chapters; aspirants who memorise the one framework solve the whole family from one mental model.
This guide collects every ratio and proportion formula for CAT 2026, the see-saw method for ratio mixing, the partnership investment-time identity, the mixture replacement formula, and 10 CAT previous year questions solved using the master framework.
Ratio is the master key for the entire arithmetic family on CAT — partnership, mixture, alligation, direct/inverse variation. The 8 formulas: ratio in lowest terms, proportion identity (ad = bc), compounded ratio (ac:bd), duplicate/sub-duplicate, partnership (investment × time), see-saw alligation, mixture replacement (1 − y/x)^n, and inverse variation. The 30-second trick: convert every ratio into the smallest integer pair, then chain the multipliers. CAT averages 2 to 3 ratio questions per cycle worth 6 to 9 marks. The 10 PYQs below cover every pattern.
Why Ratio is the Master Key Topic on CAT
Every CAT arithmetic chapter except number systems reduces to ratio at some point. Time and work splits the day into ratios of effort. Time and distance compares speeds as ratios. Profit and loss chains markup-and-discount as ratios. Even compound interest's two-year identity hides a ratio in disguise. Aspirants who internalise ratio as the underlying mental model rather than a standalone topic get a 30 to 45 second speed boost on every arithmetic question. The profit and loss formulas for CAT 2026 guide shows the same fraction-multiplier logic at work in markup-and-discount problems.
CAT 2020 to CAT 2024 featured at least two ratio-and-proportion questions per cycle, often three. The questions are rarely standalone — they hide inside partnership scenarios, mixture problems, and pipe-and-cistern questions. The CAT 2026 marking scheme guide covers the +3/−1/0 marking math that makes a 30-second ratio attempt one of the highest-ROI questions on the paper.
The Eight Core Ratio and Proportion Formulas (Memorise These)
Ratio between two quantities a and b
a : b = a/b (in lowest terms)k(a) : k(b) = a : b (scaling invariance)
Always reduce to lowest terms. The ratio 12 : 18 is the same as 2 : 3 and the same as 200 : 300. Scaling does not change the ratio.
The cross-product rule
If a : b = c : d then ad = bcContinued proportion: b² = ac (b is the mean proportional)
The cross-product is the universal test for proportion. CAT problems that ask for missing values in a proportion always reduce to this single identity.
Ratio operations CAT loves to test
Compounded: (a:b) × (c:d) = ac : bdDuplicate: (a:b)² = a² : b²
Sub-duplicate: √a : √b
Triplicate: a³ : b³
CAT uses these in geometry-pretending-to-be-ratio questions: areas scale as the square of length ratios, volumes as the cube. The duplicate-triplicate identities solve such questions in one step.
Profit split by investment and time
Profit_A : Profit_B = (Investment_A × Time_A) : (Investment_B × Time_B)If A invests Rs. 30,000 for 12 months and B invests Rs. 40,000 for 9 months, the profit ratio is 360 : 360 = 1 : 1 — equal split despite unequal individual numbers. CAT exploits this by giving asymmetric investments and times.
Mixing two quantities to reach a target average
Required ratio = (Higher value − Target) : (Target − Lower value)To mix 12 percent acid and 30 percent acid to get 20 percent acid: ratio = (30 − 20) : (20 − 12) = 10 : 8 = 5 : 4. The see-saw replaces the algebra entirely for any two-quantity mixing problem.
Repeated draw-and-refill identity
Final pure liquid = Initial × (1 − y/x)^nx is the total volume, y is the amount drawn each time, n is the number of replacement cycles. The formula is structurally identical to compound depreciation. CAT 2023 tested this with a three-cycle dilution problem.
Variation when one quantity depends on another
Direct variation: y = kx (constant ratio)Inverse variation: xy = k (constant product)
Direct variation: more workers means more work in the same time. Inverse variation: more workers means less time for the same work. CAT's time-and-work and pipe-and-cistern questions all reduce to this.
The CAT-favourite ratio manipulation
If a/b = c/d, then (a + b)/(a − b) = (c + d)/(c − d)Less famous than it should be. Solves ratio problems where the sum and difference of components are given. CAT 2022 used this identity to compress a three-variable ratio puzzle into a single quadratic.
The See-Saw Method Visualised
The see-saw is the most-tested ratio shortcut on CAT. Draw two source values on the left and right, the target average in the middle, and read the required mixing ratio directly from the cross-differences. The diagram below shows a worked example.
(20%)
↖ ↗
Take the cross-differences: (30 − 20) = 10 on the left, (20 − 12) = 8 on the right.
Aspirants flip the cross-differences and get the inverse ratio. The rule: the difference on the left of the see-saw is the share of the LEFT source. The difference on the right is the share of the RIGHT source. Test the answer with a 50:50 case — if mixing 10% and 30% to get 20%, the ratio must come out 1:1. If your method gives 2:1, you flipped.
Every CAT ratio question reduces to: identify the canonical lowest-terms ratio, find the multiplier that converts it into the given numbers, then read off the answer in the same multiplier base. Example: if the ratio is 3:5 and the sum is 64, the multiplier is 64/8 = 8, so the parts are 24 and 40.
This framework collapses partnership, mixture, and proportion into the same three steps. The aspirant who runs the framework habitually saves 30 to 45 seconds per question.
How Ratio Connects to the Rest of CAT Quant
Ratio sits at the centre of the arithmetic graph. Profit and loss is a ratio of CP to SP. Compound interest is a ratio of A to P raised to a power. Time and work splits one day's work as ratios. The simple interest and compound interest formulas for CAT 2026 guide shows the ratio multiplier in action across the two-year and three-year CI identities. The profit and loss formulas guide covers the same fraction-chaining logic in markup-and-discount problems.
For aspirants whose ratio accuracy is below 70 percent in mocks despite knowing the formulas, the gap is almost always in recognising which sub-topic the question is testing under the wrapper. The fix: drill 30 mixed ratio questions across the four sub-topics (pure ratio, partnership, mixture, variation), tagging each one by sub-topic before solving.
Want a personalised CAT 2026 arithmetic plan with daily ratio drills across all four sub-topics?
Build My CAT Arithmetic Plan10 CAT PYQs on Ratio and Proportion
Basic Ratio Sum
If two numbers are in the ratio 3 : 5 and their sum is 80, find the larger number.
Partnership Share
A invests Rs. 20,000 for 12 months. B invests Rs. 30,000 for 8 months. They earn a profit of Rs. 24,000. Find B's share.
Alligation / See-Saw
In what ratio should a 20 percent acid solution be mixed with a 50 percent acid solution to get a 30 percent solution?
Mixture Replacement
A 20-litre vessel contains pure milk. 4 litres are drawn and replaced with water. This is done thrice. Find the milk remaining.
Inverse Variation Time-Work
If 12 workers complete a task in 20 days, how many days will 15 workers take to complete the same task?
Three-Variable Ratio
If A : B = 3 : 4 and B : C = 5 : 6, find A : B : C.
Duplicate Ratio Area Problem
The ratio of the sides of two squares is 3 : 5. Find the ratio of their areas.
Componendo Dividendo
If (a + b) / (a − b) = 5/3, find a : b.
Partnership Mid-Year Change
A invests Rs. 8,000 at start and adds Rs. 4,000 after 6 months. B invests Rs. 12,000 for the full year. After 12 months, profit is Rs. 7,000. Find A's share.
See-Saw Salary Mix
An employee's monthly salary is split between rent and remaining expenses in the ratio 1 : 3. After a 25 percent raise, the new rent grows by 10 percent and remaining expenses by 30 percent. Find the new ratio.
The Four Patterns CAT Keeps Testing
- Pattern A — Pure ratio manipulation. Chain or compound two or three ratios. Apply Formulas 1, 2, 3, or 8.
- Pattern B — Partnership investment-time split. Profit divides as the product of investment and time. Apply Formula 4.
- Pattern C — Mixture alligation see-saw. Two sources, one target. Apply Formula 5 with the cross-differences.
- Pattern D — Variation problem. Direct or inverse. Apply Formula 7 with the constant-of-variation logic.
The CAT exam overview page shows where ratio and proportion sits inside the wider Quant section, and the CAT 2026 waitlist includes a paced arithmetic drill track that mixes ratio with profit-loss, SI/CI, and time-work questions in proportion to their CAT frequency.
When a CAT question asks for a single value buried inside a multi-variable ratio, always introduce a multiplier variable k early. Let the parts be 3k, 5k, 7k and solve for k from the given total or sum. This converts a three-variable problem into a single-variable equation. The k-trick is the single biggest accelerator for ratio questions on CAT.
- Memorise 8 formulas: ratio base, proportion identity, compounded/duplicate, partnership, see-saw, mixture replacement, variation, componendo.
- Always reduce to lowest terms before doing anything else.
- Introduce the k-multiplier early when the ratio has more than two parts.
- Partnership profit = investment × time. Always. Even with mid-year changes (sum the segments).
- See-saw cross-differences give the mixing ratio. The left difference is the LEFT source's share.
- Mixture replacement is structurally compound depreciation: (1 − y/x)^n.
- Spot Pattern A (pure ratio), B (partnership), C (alligation), or D (variation) in under 8 seconds.
Ratio is not a topic. It is the framework that runs through every CAT arithmetic chapter. Learn the framework once. Apply it everywhere.
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