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Time, Speed & Distance
Every CAT Time, Speed & Distance formula on one page — relative speed, trains crossing poles and platforms, boats and streams, and delay problems — each with a worked example.
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Ratio, Proportion & Variation
CAT'26 QUANT CHEATSHEET
Every ratio, proportion & variation formula you need for CAT 2026 — on one page.
Ratio, Proportion and Variation is the quiet backbone of CAT arithmetic — it rarely gets its own standalone question, but Time & Work, Speed, Mixtures and partnership problems all lean on the same handful of ratio moves. The core habit is simple: assume ratio parts as 3x and 2x instead of real numbers, and equalise common terms before combining two ratios. This sheet lays out every formula you need for the CAT quant section: dividing a quantity by a ratio, combining chained ratios, proportion and cross-multiplication, direct, inverse and joint variation, and how a percentage change shifts a ratio, each with a worked example in real numbers. Keep it open while you practise, and after a mock check where you stand on the CAT score predictor to see which idea is costing you marks.
Dev note: the canonical route
/cheatsheets/optima-learn-ratio-proportion-variation-cheatsheet is not live yet
(no /cheatsheets hub in the current sitemap). Built production-ready as a drop-in once it ships.Ratio, Proportion & Variation: every formula you need
1Core Idea
A ratio compares two quantities of the same kind.
A : B = A/B
Example: Boys:Girls = 3:2, total 100 students → Boys = 60, Girls = 40.
CAT Hack: Assume ratio parts as 3x and 2x instead of plugging in actual numbers — it keeps the algebra clean.
2Division in a Ratio
Split a total amount according to a given ratio.
Share = (Part ÷ Total Parts) × Total Amount
Example: ₹1000 divided 2:3 → ₹400 and ₹600.
CAT Favourite: Extremely common in partnership questions — profit is almost always split the same way as investment ratio.
3Combining Ratios
Chain two ratios together by equalising the shared term.
A:B and B:C → scale both so the B terms match, then combine
Example: A:B=2:3, B:C=4:5 → equalise B to 12 → A:B=8:12, B:C=12:15 → A:B:C = 8:12:15.
CAT Favourite: A CAT favourite — it shows up constantly in Data Interpretation and Arithmetic sets.
4Proportion
Four quantities in proportion cross-multiply cleanly.
a : b = c : d ⇒ ad = bc
Example: 3:5 = 12:x → 3x = 5·12 = 60 → x = 20.
CAT Insight: Cross-multiply first, then solve for the single unknown — don't try to scale the ratio by eye.
5Direct Variation
Both quantities move the same way: more gives more.
y ∝ x ⇒ y = kx
Example: workers up, output up — double the workers and you double the output.
6Inverse Variation
The quantities move opposite ways: more gives less.
y ∝ 1/x ⇒ xy = k
Example: 10 workers take 12 days; 20 workers (double) take 6 days (half the time).
Common Mistake: Don't treat inverse variation like direct variation — check whether the relationship truly moves opposite ways first.
7Joint Variation
One quantity depends on two (or more) others simultaneously.
z ∝ xy ⇒ z = kxy
Example: Area = Length × Breadth → area varies jointly with both dimensions.
8Ratio Change After % Change
A percentage change on one part shifts the whole ratio.
convert the % change to a fraction, then rescale that part of the ratio
Example: A:B=4:5, A up 25% (i.e. ×5/4) → new A = 5 → new ratio = 5:5 = 1:1.
CAT Favourite: Frequently mixed with percentages on CAT — a 25% increase is the same as multiplying by 5/4.
9CAT Power Shortcuts
Split a total directly, without solving for x first.
A = (a/(a+b))×Total, B = (b/(a+b))×Total | direct: y₁/y₂=x₁/x₂; inverse: x₁y₁=x₂y₂
Example: ratio a:b splits a total directly into a/(a+b) and b/(a+b) of it — no intermediate x needed.
CAT Hack: Must-know for Time & Work and Speed questions — both topics lean on these two variation identities constantly.
CAT traps & revision
10
Most Common CAT Traps
- Adding ratios directly: 2:3 + 4:5 is not 6:8.
- Forgetting to equalise the common term before combining two ratios.
- Treating inverse variation as if it were direct variation.
- Using actual values when assuming ratio variables (like 3x, 2x) would be faster.
- Missing that a ratio changes after one part undergoes a percentage change.
11
30-Second Revision Box
- Ratio = comparison; A:B = A/B
- Proportion ⇒ cross multiplication (ad = bc)
- Direct variation ⇒ same direction (y=kx)
- Inverse variation ⇒ opposite direction (xy=k)
- Combine ratios by equalising the common term
- Assume variables (x) wherever possible; master this before Mixtures & Time-Work
This topic rewards setting up the ratio correctly over grinding through arithmetic — once the parts are assumed as x-multiples and the common terms are equalised, most questions collapse to one line. Drill this sheet until combining ratios and spotting direct versus inverse variation are reflex, then test them on full sets and track progress with the CAT score predictor. It pairs directly with the Mixtures & Alligations cheat sheet, since both lean on the same ratio reasoning. For more guides, browse the Optima Learn blog or explore every study guide, work through the CAT exam hub, and when you want mentor-led prep, book a free CAT 2026 call.
