Ratio and Proportion Formulas for CAT 2026: 20 + 15 PYQs
Ratio and proportion formulas for CAT 2026 are the master key of CAT Arithmetic. Time and Work efficiencies are ratios. Time Speed Distance comparisons are ratios. Profit Loss multipliers are ratio-driven. Mixtures and Alligation, Partnership, and Average problems are all explicit ratio applications. This cheatsheet pins 20 shortcuts across five recognition blocks, then closes with 15 CAT-level PYQs that mirror the trap patterns CAT setters reuse year after year.
The reason most online ratio content underperforms is that it stops at the basic a:b notation and a few proportion definitions. CAT 2026 tests componendo-dividendo, the alligation rule with reverse-subtraction traps, partnership splits with mid-year capital changes, and mean-proportional questions inside multi-step word problems. The 20 shortcuts below lead with the alligation criss-cross and the partnership capital-times-time formula because those two carry the bulk of CAT marks.
Why Ratio Acts as the Master Key of CAT Arithmetic
Ratio thinking compresses six other CAT Arithmetic topics. Time and Work splits payment in the ratio of efficiencies. Time Speed Distance compares speeds in a ratio that determines meeting points. Profit Loss converts markup and discount to multipliers, which are ratios on cost. Mixtures and Alligation is explicit ratio math. Partnership is ratio of capital times time. Average problems often reduce to weighted-ratio means. Master ratio first and every later topic compresses.
This is the gap most CAT preparation content misses. Aspirants treat ratio as a one-page beginner chapter and move on. The pattern: their downstream Arithmetic scores stay flat because the ratio layer underneath was never automated. The fix is to drill the 20 shortcuts until each one fires within two seconds, especially the alligation criss-cross and the capital-times-time partnership form, which together account for half of the topic's CAT marks.
The 20 Ratio and Proportion Formulas for CAT 2026
The cheatsheet groups all 20 ratio and proportion formulas for CAT 2026 into five blocks. Each block has a recognition cue describing the question type that triggers it. Working block by block builds the two-second drill that separates a 60-second solve from a three-minute setup.
Block 1 — Ratio Basics and Manipulations (5 formulas)
The basic block is the entry point. These five identities describe ratios, inverse ratios, duplicate and triplicate ratios, and the compound-ratio rule. Recognition cue: a direct ratio statement with simple manipulations.
| # | Formula | Use case |
|---|---|---|
| 1 | a:b = a/b; if a/b = c/d then ad = bc | Cross-multiplication. |
| 2 | Inverse ratio of a:b is b:a | Reciprocal-ratio shortcut. |
| 3 | Duplicate ratio of a:b is a2:b2; triplicate is a3:b3 | Squared and cubed-ratio cues. |
| 4 | Sub-duplicate of a:b is √a : √b; sub-triplicate is ∛a : ∛b | Square-root or cube-root ratio. |
| 5 | Compound ratio of a:b and c:d is ac:bd | Chained ratio statements. |
Block 2 — Proportion and Mean Proportional (4 formulas)
The proportion block extends to four-term proportions, mean and third proportional, and componendo-dividendo. Recognition cue: a:b = c:d statement, "find x such that" with proportion, or sum-and-difference of ratios.
| # | Formula | Recognition cue |
|---|---|---|
| 6 | Mean proportional of a and c: b = √(ac) | Three-term proportion a:b = b:c. |
| 7 | Third proportional of a and b: c = b2/a | "Find the third proportional". |
| 8 | Fourth proportional of a, b, c: d = bc/a | Four-term proportion. |
| 9 | Componendo-dividendo: if a/b = c/d, then (a + b)/(a − b) = (c + d)/(c − d) | Sum-and-difference cue. |
Block 3 — Mixtures and Alligation (4 formulas)
The alligation block is the highest-yield in CAT mixtures alligation CAT 2026. The criss-cross rule compresses what looks like a two-equation system into a one-line ratio. Recognition cue: mixture, blend, mix, average price, mean concentration, dilution, or replacement.
| # | Formula | Use case |
|---|---|---|
| 10 | Alligation: ratio of cheaper:dearer = (D − M) : (M − C) | Mixing two-grade ingredients. |
| 11 | Successive replacement: final amount = initial × (1 − replaced/total)n | Dilution problems with multiple replacements. |
| 12 | Three-ingredient alligation: apply criss-cross sequentially to pairs | Three-grade mixture. |
| 13 | Adding pure ingredient to dilute: new ratio = (original A) : (original B + added pure) | Strength-change dilution problem. |
Block 4 — Partnership and Share Problems (4 formulas)
The partnership block covers capital-times-time splits. CAT tests this in 1 of every 3 papers, usually with a mid-year capital change or a partner joining late. Recognition cue: partnership, share of profit, invested for, capital, withdraws, joins.
| # | Formula | Recognition cue |
|---|---|---|
| 14 | Profit share ratio = (C1 × t1) : (C2 × t2) : (C3 × t3) | Two or more partners. |
| 15 | If capitals are equal: profit ratio = t1 : t2 : t3 | Equal-capital shortcut. |
| 16 | If times are equal: profit ratio = C1 : C2 : C3 | Equal-time shortcut. |
| 17 | Working partner bonus: deduct fixed % from total profit first, then split remainder by capital-time ratio | Working-partner variant. |
Block 5 — Special Cases and Average Variants (3 formulas)
The special-cases block covers weighted average via alligation, ages problems, and the ratio-comparison shortcut. Recognition cue: weighted average, ages, ratio comparison, "which is greater".
| # | Formula | Use case |
|---|---|---|
| 18 | Weighted average = (q1v1 + q2v2) / (q1 + q2) | Weighted-mean problems. |
| 19 | Ages ratio shift: if A:B = m:n now, after t years (A+t):(B+t) gives new ratio equation | Ages and time problems. |
| 20 | Compare a:b and c:d: a:b > c:d iff ad > bc | Ratio comparison. |
Three Ratio Traps That Recur in CAT Papers
Three traps recur across the ratio proportion formulas for CAT 2026 question set. The first is reverse subtraction in the criss-cross. The ratio is (Dearer minus Mean) : (Mean minus Cheaper), not the reverse. Setting the cheaper-dearer convention explicitly before applying the rule eliminates this error. The second trap is splitting partnership profit by capital alone when partners invest for different durations. Always compute capital times time first, then take the ratio.
The third trap is the duplicate-ratio confusion in mixture problems. Doubling a mixture does not double the ratio; it preserves the ratio. The duplicate ratio identity applies to square-root or square-of-quantity questions, not to scaling. CAT setters embed this in dilution and concentration problems where the visual cue suggests duplication but the math demands replacement-formula thinking.
15 CAT-Level Ratio Proportion PYQs With Solutions
Divide 360 in the ratio 3:5:7.
Total parts = 15. Shares = 72, 120, 168. Answer: 72, 120, 168
Find the compound ratio of 2:3, 4:5, and 6:7.
(2 × 4 × 6) : (3 × 5 × 7) = 48 : 105 = 16:35. Answer: 16:35
Find the mean proportional between 12 and 27.
b = root(12 × 27) = root(324) = 18. Answer: 18
Find the third proportional to 6 and 12.
c = 122/6 = 144/6 = 24. Answer: 24
If (a + b)/(a − b) = 5/3, find a:b.
By componendo-dividendo applied in reverse, a:b = (5 + 3) : (5 − 3) = 8:2 = 4:1. Answer: 4:1
Two types of rice at 30 and 50 per kg are mixed to make a blend at 38 per kg. In what ratio?
Cheaper:Dearer = (50 − 38) : (38 − 30) = 12:8 = 3:2. Answer: 3:2
From 80 litres of pure milk, 8 litres are removed and replaced with water. This is repeated twice more. How much milk remains?
Remaining = 80 × (1 − 8/80)3 = 80 × (0.9)3 = 80 × 0.729 = 58.32 litres. Answer: 58.32 L
A 40-litre mixture has milk and water in ratio 5:3. How much water must be added so the ratio becomes 5:4?
Milk = 25, water initially = 15. New ratio 5:4 keeps milk at 25 and needs water = 20. Add 5 litres. Answer: 5 L
A and B invest 5000 and 8000 for the full year. Total profit 6500. A's share?
Ratio 5:8. A = 6500 × 5/13 = 2500. Answer: 2500
A invests 4000 for 12 months, B invests 6000 but joins 4 months later. Profit ratio?
Products: A = 4000 × 12 = 48000; B = 6000 × 8 = 48000. Ratio 1:1. Answer: 1:1
A and B invest 8000 and 12000 in a business. A is working partner taking 10% of total profit. Of the remaining, the rest is shared by capital. Profit 5000. Total received by A?
A's salary = 500. Remaining = 4500. Capital ratio 2:3. A's capital share = 4500 × 2/5 = 1800. Total A = 500 + 1800 = 2300. Answer: 2300
30 men have average wage 200 and 20 women have average wage 250. Average wage of group?
(30 × 200 + 20 × 250) / 50 = (6000 + 5000) / 50 = 220. Answer: 220
Present ages of A and B are in ratio 3:5. After 8 years the ratio becomes 1:1.4 (i.e., 5:7). Find present ages.
Let ages 3k and 5k. (3k + 8)/(5k + 8) = 5/7. 21k + 56 = 25k + 40, so 4k = 16, k = 4. Ages 12 and 20. Answer: 12 and 20
Which is greater: 7:9 or 8:11?
Cross-multiply: 7 × 11 = 77; 9 × 8 = 72. 77 > 72, so 7:9 > 8:11. Answer: 7:9 is greater
Three types of milk at 20, 24, 30 per litre. Find the ratio to mix them to get a 25 per litre blend.
Multiple valid answers. For ratio 1:1:2: average = (20 + 24 + 30 + 30) / 4 = 26 (close, not exact). For 1:2:3: (20 + 48 + 90) / 6 = 158/6 ≈ 26.33. For 1:1:1: 24.67. CAT typically gives a specific target; the solver uses sequential criss-cross. Answer: ratio depends on chosen pair-method
Anchor Ratio at the Centre of Your CAT 2026 Arithmetic Cluster
Ratio sits underneath six other CAT Arithmetic topics. A diagnostic-driven plan anchors this block in week four so every downstream topic, from Mixtures to Time and Work, compounds instead of stalling.
Anchor My Ratio Master-Key BlockWhere Ratio Sits in the CAT 2026 Arithmetic Cluster
Ratio is the connector topic in the CAT Arithmetic cluster sequence: Percentages first, Profit Loss second, Time and Work third, then Ratio and Proportion in week four. Ratio is the last foundational layer before Time Speed Distance, Averages, and Mixtures, all of which lean on the ratio reflex. A focused 3 to 4 day study block covers the topic; for working professionals with limited weekly hours, 6 to 8 days places it comfortably in month two.
The Optima Learn CAT exam guide sequences the rest of the Arithmetic cluster, and the CAT 2026 waitlist details page explains how the diagnostic-driven planner decides which Arithmetic topics each aspirant prioritises.
Three Reflexes That Compress Ratio Solves to Under 60 Seconds
Once 20 formulas are memorised, three reflexes separate aspirants who finish ratio questions in 60 seconds from those who take two minutes. Reflex one: criss-cross-first. For any mixture or weighted-mean question, draw the alligation cross before doing algebra. Reflex two: capital-times-time for partnership. Always compute C times T per partner before taking ratios. Reflex three: componendo-dividendo on sum-and-difference cues. Whenever the question shows a + b and a - b structure, reach for componendo-dividendo immediately. The CAT preparation blogs library has companion cheatsheets on Percentages, Profit Loss, and Time and Work.
Common Doubts About Ratio Proportion Preparation for CAT 2026
Is alligation only for mixture problems?
No. Alligation applies to any weighted-mean problem: average prices, salaries, ages, interest rates, concentrations, blended scores. Treating alligation as a mixture-only tool leaves marks on the table in Average and Partnership questions where the same criss-cross works.
How tricky are the recent CAT 2024 and CAT 2025 ratio questions?
Recent papers lean on mid-year-change partnership and three-ingredient alligation. CAT 2024 had one mid-year partnership question and CAT 2025 had a three-grade mixture problem. Both rewarded the criss-cross-first and capital-times-time reflexes from this cheatsheet.
How do I revise the cheatsheet one week before CAT 2026?
A one-week revision plan: day one, re-read the 20-formula cheatsheet. Day two, drill alligation and criss-cross variants. Day three, drill componendo-dividendo and proportion identities. Day four, drill partnership including mid-year changes. Day five, attempt 15 mixed-block PYQs under timed conditions. Day six, review every error. Day seven, scan the cheatsheet for 15 minutes only before the exam.
Final note. Ratio proportion formulas CAT 2026 reduce to 20 shortcuts across five blocks, with alligation and partnership carrying the heaviest CAT weight. The topic rewards block-recognition over direct algebra. Drill block by block, build the three reflexes, and the CAT score predictor alongside mocks will track the lift across Mixtures, Average, and Partnership downstream.
