CAT Quant Formula Sheet: Arithmetic, Algebra, Geometry and Modern Maths
Every formula you need for CAT Quantitative Ability, grouped by topic, with 8 worked examples and the speed rules that save marks.
14 min read · updated 2026-11
How to use this sheet
Quantitative Ability is the third of three CAT sections. Each section is timed at exactly 40 minutes, and you cannot move between sections. A correct answer earns 3 marks, a wrong MCQ costs 1 mark, and a skipped question scores zero. Non-MCQ (type-in) questions have no negative marking.
The official CAT 2026 pages do not state how many QA questions there are, so plan around minutes, not a fixed count. The rest of this sheet is teaching content: formulas, shortcuts and worked examples.
Arithmetic: percentages, profit, mixtures, interest
| Idea | Formula |
|---|---|
| Successive change of a% then b% | Net % = a + b + ab/100 (a fall is negative) |
| Selling price from cost | SP = CP × (1 + p/100) |
| Selling price from marked price | SP = MP × (1 - d/100) |
| Same SP, one item at x% profit and one at x% loss | Overall loss = x²/100 % |
| Alligation (cheaper and dearer into a mean) | Cheaper : Dearer = (Dearer - Mean) : (Mean - Cheaper) |
| Repeated replacement (x litres of mixture drawn and replaced by water each time, vessel holds V) | Original liquid left after n rounds = V × (1 - x/V)^n |
| Combined average of two groups | (n1 × a1 + n2 × a2) / (n1 + n2) |
| Simple interest | SI = P × R × T / 100 |
| Compound amount | A = P × (1 + r/100)^n |
| CI minus SI for 2 years | P × (r/100)² |
- Take cost price = 100. Marked price = 140.
- Selling price = 140 × 0.85 = 119.
- Profit = 119 - 100 = 19.
- Check with successive change: 40 - 15 + (40 × -15)/100 = 25 - 6 = 19.
Answer: 19% profit
- Cheaper = 42, dearer = 66, mean = 50.
- Cheaper : dearer = (66 - 50) : (50 - 42) = 16 : 8 = 2 : 1.
- Check: (2 × 42 + 1 × 66) / 3 = 150 / 3 = 50.
Answer: 2 : 1 (cheaper : dearer)
Time, speed, distance and work
| Idea | Formula |
|---|---|
| Average speed, equal distances at a and b | 2ab / (a + b) |
| Unit change | km/h × 5/18 = m/s |
| Relative speed | Same direction: a - b. Opposite: a + b |
| Two trains crossing | Time = (L1 + L2) / relative speed |
| Boats | Downstream = b + c, upstream = b - c, so b = (D + U)/2 and c = (D - U)/2 |
| Work rate | Rate = 1 / days; rates add |
| Two workers together | ab / (a + b) days |
| Pipes | Inlet rates add, outlet rates subtract |
- Distances are equal, so use 2ab/(a + b) = 2 × 50 × 75 / 125 = 60.
- Check with totals: time = 150/50 + 150/75 = 3 + 2 = 5 hours; distance = 300 km; 300 / 5 = 60.
Answer: 60 km/h (not 62.5, which is the plain average)
- Rates: A = 1/12, B = 1/18. Together = 3/36 + 2/36 = 5/36 per day.
- In 4 days: 4 × 5/36 = 5/9 done, so 4/9 remains.
- B alone: (4/9) ÷ (1/18) = 8 days.
- Check with 36 units of work: A does 3 a day, B does 2. Together 5 a day, so 20 units in 4 days. 16 units left at 2 a day = 8 days.
Answer: 8 days
Number system
| Idea | Formula |
|---|---|
| Number of factors of N = p^a × q^b × r^c | (a + 1)(b + 1)(c + 1) |
| Sum of factors | Product of (p^(a+1) - 1)/(p - 1) over each prime |
| Trailing zeros in n! | floor(n/5) + floor(n/25) + floor(n/125) + ... |
| HCF and LCM of two numbers | HCF × LCM = product of the two numbers |
| Euler's totient | φ(N) = N × (1 - 1/p) over each prime p of N |
| Euler's theorem | a^φ(m) leaves remainder 1 mod m when HCF(a, m) = 1 |
| Divisible by 11 | Alternating digit sum is divisible by 11 |
| Divisible by 9 | Digit sum is divisible by 9 |
For last digits, use the cycle: 2, 3, 7 and 8 repeat every 4 powers; 4 and 9 repeat every 2; 0, 1, 5 and 6 never change.
- 360 = 2³ × 3² × 5.
- Total factors = 4 × 3 × 2 = 24.
- Odd factors use no 2: 3 × 2 = 6.
- Even factors = 24 - 6 = 18.
Answer: 18
Algebra, progressions and logarithms
- x² + 1/x² = 5² - 2 = 23.
- x³ + 1/x³ = 5³ - 3 × 5 = 125 - 15 = 110.
Answer: 23 and 110
Geometry and mensuration
| Idea | Formula |
|---|---|
| Heron's area | √(s(s - a)(s - b)(s - c)), s = (a + b + c)/2 |
| Inradius and circumradius | r = Area/s; R = abc/(4 × Area) |
| Right triangle | Circumradius = hypotenuse / 2 |
| Similar figures | Area ratio = (side ratio)²; volume ratio = (side ratio)³ |
| Polygon angle sum | (n - 2) × 180°; diagonals = n(n - 3)/2 |
| Circle | Arc = θ/360 × 2πr; sector = θ/360 × πr²; angle at centre = 2 × angle at circumference |
| Cylinder | V = πr²h; curved area = 2πrh |
| Cone | V = ⅓πr²h; slant l = √(r² + h²); curved area = πrl |
| Sphere | V = 4/3 πr³; surface = 4πr² |
| Cuboid diagonal | √(l² + b² + h²) |
| Triangle from coordinates | ½ |x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)| |
Learn these Pythagorean triples with their multiples: 3-4-5, 5-12-13, 8-15-17, 7-24-25.
- s = 21. Area = √(21 × 8 × 7 × 6) = √7056 = 84.
- Inradius = 84 / 21 = 4.
- Circumradius = (13 × 14 × 15) / (4 × 84) = 2730 / 336 = 8.125 = 65/8.
Answer: Area 84 cm², inradius 4 cm, circumradius 8.125 cm
Modern maths: counting, probability, sets
| Idea | Formula |
|---|---|
| Arrangements and selections | nPr = n!/(n - r)!; nCr = n!/(r!(n - r)!) |
| Arrangements with repeats | n!/(p! q! r!) |
| Circular arrangements | (n - 1)! |
| Identical items into r distinct boxes | (n + r - 1)C(r - 1); each box at least one: (n - 1)C(r - 1) |
| Probability of A or B | P(A) + P(B) - P(A and B) |
| At least one | 1 - P(none) |
| Two sets | n(A ∪ B) = n(A) + n(B) - n(A ∩ B) |
- Total arrangements = 6! = 720.
- A and B together: treat them as one block. 5! × 2 = 240.
- Not together = 720 - 240 = 480.
Answer: 480
Speed rules for the 40 minutes
- ☐ Read the last line of the question first. Know what is being asked before you start computing.
- ☐ Plug in options when the algebra looks long. Start with the middle value.
- ☐ Round when the options are far apart. Exact arithmetic is wasted effort.
- ☐ Give a new question about two minutes to show a path. If none appears, mark it and move on.
- ☐ Never leave a non-MCQ blank if you have any answer. There is no negative marking for them.
- ☐ On a wrong-prone MCQ, guess only after removing at least one option. Each wrong MCQ costs 1 mark against 3 for a right one.
Common questions
Which CAT quant formulas should I memorise first?
Start with successive percentage change, alligation, average speed, work rates, factor counting, quadratic roots and Heron's formula. They are short and fit many question types.
Is a formula sheet enough to crack CAT quant?
No. A sheet helps you recall a method; speed comes from solving timed sets and reviewing every mistake. Use this page for recall, then practise.
How many quant questions are there in CAT 2026?
The official CAT 2026 pages do not state the number. They state 40 minutes for QA, 3 marks for a correct answer and -1 for a wrong MCQ. Check the official notice.
Is there negative marking for type-in answers in CAT?
No. The official CAT FAQ says non-MCQ questions carry no negative marks. A wrong MCQ loses 1 mark and a skipped question scores zero.