Quant Formulae

01

Number System

High ▼

📐 HCF & LCM

HCF × LCM = Product of two numbers
LCM of fractions = LCM of Numerators / HCF of Denominators
HCF of fractions = HCF of Numerators / LCM of Denominators
HCF of co-prime numbers = 1
LCM of co-prime numbers = Product of numbers

📐 Factors & Divisors

If N = a^p × b^q × c^r, then:
→ Total factors = (p+1)(q+1)(r+1)
→ Sum of factors = [(a^p+1−1)/(a−1)] × [(b^q+1−1)/(b−1)] × [(c^r+1−1)/(c−1)]
→ Product of all factors = N^{(total factors / 2)}
→ Number of odd factors: Ignore power of 2
→ Number of even factors: Total − Odd factors

📐 Divisibility Rules

Div. by	Rule
2	Last digit is even (0, 2, 4, 6, 8)
3	Sum of all digits divisible by 3
4	Last two digits divisible by 4
5	Last digit is 0 or 5
6	Divisible by both 2 and 3
7	Double last digit, subtract from rest → check if divisible by 7
8	Last three digits divisible by 8
9	Sum of all digits divisible by 9
11	Difference of sum of digits at odd & even places = 0 or multiple of 11
12	Divisible by both 3 and 4

📐 Unit Digit Cyclicity

Digit	Cycle	Period
0, 1, 5, 6	Same always	1
2	2, 4, 8, 6	4
3	3, 9, 7, 1	4
4	4, 6	2
7	7, 9, 3, 1	4
8	8, 4, 2, 6	4
9	9, 1	2

📐 Sum Formulas

Sum of first n natural numbers = n(n+1)/2
Sum of squares = n(n+1)(2n+1)/6
Sum of cubes = [n(n+1)/2]²
Sum of first n odd numbers = n²
Sum of first n even numbers = n(n+1)
AP: Sum = n/2 × [2a + (n−1)d] | nth term = a + (n−1)d
GP: Sum = a(rⁿ−1)/(r−1) | nth term = a × r⁽ⁿ⁻¹⁾

📐 Remainder Theorem

(a × b) mod n = [(a mod n) × (b mod n)] mod n
(a + b) mod n = [(a mod n) + (b mod n)] mod n
(a − b) mod n = [(a mod n) − (b mod n)] mod n

02

Simplification & Approximation

High ▼

📐 Core Rules

BODMAS → Brackets → Orders (powers/roots) → Division → Multiplication → Addition → Subtraction
Vinculum (bar) is solved first inside brackets
Brackets order: () → {} → []

📐 Useful Approximations

√2 ≈ 1.414 √3 ≈ 1.732 √5 ≈ 2.236
√6 ≈ 2.449 √7 ≈ 2.646 √8 ≈ 2.828
π ≈ 22/7 ≈ 3.14159
Trick: For approximation questions, round to nearest 5 or 10, then calculate

📐 Algebraic Identities (Frequently Used)

(a + b)² = a² + 2ab + b²
(a − b)² = a² − 2ab + b²
a² − b² = (a + b)(a − b)
(a + b)³ = a³ + 3a²b + 3ab² + b³
(a − b)³ = a³ − 3a²b + 3ab² − b³
a³ + b³ = (a + b)(a² − ab + b²)
a³ − b³ = (a − b)(a² + ab + b²)
(a + b + c)² = a² + b² + c² + 2(ab + bc + ca)

03

Percentage

High ▼

📐 Basic Formulas

Percentage = (Part / Whole) × 100
Increase by x% → New = Original × (1 + x/100)
Decrease by x% → New = Original × (1 − x/100)
If A is x% more than B → B is less than A by [x/(100+x)] × 100
If A is x% less than B → B is more than A by [x/(100−x)] × 100

📐 Successive Percentage Change

Net effect of a% then b% = a + b + (ab/100) %
General: [(1 ± p/100)(1 ± q/100) − 1] × 100
Use + for increase, − for decrease

📐 Population/Depreciation

After n years (growth r%) = P × (1 + r/100)ⁿ
n years ago = P / (1 + r/100)ⁿ
Depreciation after n years = P × (1 − r/100)ⁿ

📐 Fraction ↔ Percentage Reference

Fraction	%	Fraction	%	Fraction	%
1/2	50%	1/7	14.28%	1/12	8.33%
1/3	33.33%	1/8	12.5%	1/13	7.69%
1/4	25%	1/9	11.11%	1/14	7.14%
1/5	20%	1/10	10%	1/15	6.67%
1/6	16.67%	1/11	9.09%	1/20	5%

04

Profit, Loss & Discount

High ▼

📐 Basic P&L Formulas

Profit = SP − CP
Loss = CP − SP
Profit % = (Profit / CP) × 100
Loss % = (Loss / CP) × 100
SP = CP × (1 + Profit%/100)
SP = CP × (1 − Loss%/100)
CP = SP / (1 + P%/100) or CP = SP / (1 − L%/100)

📐 Marked Price & Discount

SP = MP × (1 − Discount%/100)
Discount = MP − SP
Discount % = [(MP − SP) / MP] × 100
MP = CP × (1 + Profit%/100) / (1 − Discount%/100)

📐 Special Tricks

If CP of x articles = SP of y articles → Profit% = [(x−y)/y] × 100
Successive discounts a% and b% → Equivalent = a + b − (ab/100) %
Dishonest shopkeeper (false weight) → Profit% = [(True − False)/False] × 100
Two items sold at same SP, one at x% profit & other at x% loss → Always a Net Loss = x²/100 %

05

Simple Interest (SI)

Medium ▼

SI = (P × R × T) / 100
Amount = P + SI = P(1 + RT/100)
P = (SI × 100) / (R × T)
R = (SI × 100) / (P × T)
T = (SI × 100) / (P × R)
Different rates: SI = P × [(r₁t₁ + r₂t₂ + r₃t₃)/100]
If SI = Principal → T = 100/R years
If amount doubles → RT = 100 → T = 100/R
If amount triples → RT = 200 → T = 200/R

💡 Shortcut

If SI on a sum for 2 years at 5% = ₹100, then Principal = (100 × 100)/(5 × 2) = ₹1000

06

Compound Interest (CI)

Medium ▼

📐 Core Formulas

Amount = P × (1 + R/100)^T
CI = Amount − P = P × [(1 + R/100)^T − 1]
Half-yearly: A = P × (1 + R/200)^2T
Quarterly: A = P × (1 + R/400)^4T

📐 CI vs SI Difference

For 2 years: CI − SI = P × (R/100)²
For 3 years: CI − SI = P × (R²/10⁴) × (3 + R/100)

📐 Doubling & Rule of 72

If amount doubles in n years → 4× in 2n years → 8× in 3n years
Rule of 72: Doubling time ≈ 72 / Rate%
Example: At 8%, money doubles in ≈ 72/8 = 9 years

07

Ratio & Proportion

High ▼

a : b = a/b
If a:b = c:d → ad = bc (cross multiplication)
Duplicate ratio of a:b = a² : b²
Sub-duplicate ratio of a:b = √a : √b
Triplicate ratio of a:b = a³ : b³
Componendo: (a+b)/b = (c+d)/d
Dividendo: (a−b)/b = (c−d)/d
Componendo-Dividendo: (a+b)/(a−b) = (c+d)/(c−d)

📐 Division in Ratio

Amount A divided in ratio x:y:
→ First part = A × x/(x+y)
→ Second part = A × y/(x+y)

08

Average

Medium ▼

Average = Sum of observations / Number of observations
Sum = Average × Count
Weighted Average = Σ(value × weight) / Σ(weights)
If 1 number added: New Avg = (nA + x) / (n+1)
If 1 number removed: New Avg = (nA − x) / (n−1)
If 1 number replaced (x by y): New Avg = A + (y−x)/n
Average of first n natural numbers = (n+1)/2
Average of first n even numbers = (n+1)
Average of first n odd numbers = n
Average of consecutive numbers a to b = (a+b)/2
Average speed (same distance) = 2S₁S₂/(S₁+S₂)

09

Ages

Medium ▼

"x years ago" → Subtract x from present age
"x years hence/later" → Add x to present age
Age difference ALWAYS remains constant
If present ages ratio = a:b → Let ages = ax and bx
Use linear equations to solve

💡 Tip

In ratio-based problems, the difference of ratio terms × multiplier = actual age difference. This helps find the multiplier quickly.

10

Partnership

High ▼

Simple Partnership (same time):
Profit Ratio = Investment Ratio = A : B
Compound Partnership (different time):
Profit Ratio = (A × T₁) : (B × T₂)
A's Profit = Total Profit × [A's ratio / Sum of ratios]
If A invests ₹x for t₁ months, B invests ₹y for t₂ months:
A's share : B's share = xt₁ : yt₂

11

Time & Work

Medium ▼

If A does work in x days → A's 1 day work = 1/x
If B does work in y days → B's 1 day work = 1/y
Together → 1 day work = 1/x + 1/y
Time together = xy / (x + y)
If A is n times as efficient as B, and B takes T days → A takes T/n days
Efficiency ratio a:b → Time ratio = b:a (inverse)

📐 LCM Method (Best for Exams!)

Total Work = LCM of individual days
Efficiency of each = Total Work / Individual days
Combined time = Total Work / Sum of efficiencies

💡 Example

A does work in 10 days, B in 15 days.
LCM(10,15) = 30 units total work.
A's efficiency = 30/10 = 3 units/day. B's efficiency = 30/15 = 2 units/day.
Together = 5 units/day. Time = 30/5 = 6 days ✓

12

Pipes & Cisterns

Medium ▼

Same concept as Time & Work
Filling pipe → Positive (+) work
Draining/leak pipe → Negative (−) work
If pipe fills in x hrs, drains in y hrs (y > x):
Net per hr = 1/x − 1/y
Time to fill = xy / (y − x)
If pipe fills in x hrs, drains in y hrs (x > y):
Tank will never fill (empties in xy/(x−y) hrs)

13

Time, Speed & Distance

Medium ▼

📐 Basic Formulas

Distance = Speed × Time
Speed = Distance / Time
Time = Distance / Speed

📐 Conversions

km/h to m/s → Multiply by 5/18
m/s to km/h → Multiply by 18/5

📐 Average & Relative Speed

Average Speed (same distance) = 2S₁S₂ / (S₁ + S₂)
Average Speed (same time) = (S₁ + S₂) / 2
Relative Speed (same direction) = S₁ − S₂
Relative Speed (opposite direction) = S₁ + S₂

14

Problems on Trains

Medium ▼

Time to cross a pole/person = Length of train / Speed
Time to cross a platform/bridge = (L_train + L_platform) / Speed
Two trains, same direction:
Time = (L₁ + L₂) / (S₁ − S₂)
Two trains, opposite direction:
Time = (L₁ + L₂) / (S₁ + S₂)
If a train passes a man on another train:
Distance = Length of the passing train only

15

Boats & Streams

Medium ▼

Let B = speed of boat in still water, S = speed of stream
Downstream speed = B + S
Upstream speed = B − S
B = (Downstream + Upstream) / 2
S = (Downstream − Upstream) / 2
Time = Distance / Speed
Average speed (equal dist. up & down) = (B² − S²) / B
If speed of current = S and B = S → cannot go upstream

16

Mixture & Alligation

Medium ▼

📐 Alligation Rule

Two items of price A and B mixed to get price M:
Ratio = (B − M) : (M − A)
Cheaper quantity : Dearer quantity

📐 Repeated Dilution / Replacement

Container has x litres. y litres replaced n times:
Remaining original = x × (1 − y/x)ⁿ
Final concentration = Initial × (1 − y/x)ⁿ

📐 Weighted Average of Mixtures

Mixture concentration = (C₁Q₁ + C₂Q₂) / (Q₁ + Q₂)
C = concentration, Q = quantity

17

Number Series

High ▼

Common Patterns to Identify:
Arithmetic: +c, +c, +c... (constant difference)
Geometric: ×r, ×r, ×r... (constant ratio)
Square-based: +1², +2², +3², +4²... or 1², 2², 3²...
Cube-based: +1³, +2³, +3³... or 1³, 2³, 3³...
Prime numbers: 2, 3, 5, 7, 11, 13...
Fibonacci: Each term = sum of previous two
Mixed: Alternating +, ×, or odd/even position patterns
Difference of differences: Find pattern in gaps of gaps
×n then ±c: ×2+1, ×2+3, ×2+5...

💡 Strategy

Always compute the differences first. If differences don't show a pattern, compute differences of differences. Also check ratios between consecutive terms.

18

Quadratic Equations

High ▼

📐 Core Formulas

Standard form: ax² + bx + c = 0
Roots: x = [−b ± √(b² − 4ac)] / 2a
Sum of roots (α + β) = −b/a
Product of roots (α × β) = c/a

📐 Nature of Roots (Discriminant D)

D = b² − 4ac
D > 0 → Two distinct real roots
D = 0 → Two equal real roots
D < 0 → No real roots (imaginary)

📐 Sign Method (Exam Shortcut)

Both roots positive → c positive, b negative
Both roots negative → both b and c positive
Roots of opposite signs → c is negative
Compare x and y values to determine the relationship (x > y, x < y, etc.)

19

Permutation & Combination

Low ▼

📐 Fundamental Formulas

Factorial: n! = n × (n−1) × (n−2) × ... × 1
0! = 1 1! = 1
Permutation (arrangement, order matters):
ⁿPᵣ = n! / (n−r)!
Combination (selection, order doesn't matter):
ⁿCᵣ = n! / [r! × (n−r)!]

📐 Key Identities

ⁿC₀ = ⁿCₙ = 1
ⁿCᵣ = ⁿCₙ₋ᵣ
ⁿPᵣ = ⁿCᵣ × r!
Circular permutation = (n−1)!
Identical items: n! / (p! × q! × r!)
Fundamental Counting: If event A can be done in m ways and event B in n ways → Total = m × n

20

Probability

Low ▼

📐 Core Formulas

P(E) = Favourable outcomes / Total outcomes
0 ≤ P(E) ≤ 1
P(E) + P(not E) = 1
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
If A & B are mutually exclusive: P(A ∪ B) = P(A) + P(B)
If A & B are independent: P(A ∩ B) = P(A) × P(B)

📐 Quick References

Deck of Cards: Total = 52 | 4 Suits × 13 each | Face cards = 12 | Aces = 4
Single Die: 6 outcomes
Two Dice: 36 outcomes | P(sum=7) = 6/36 = 1/6
Coin: P(Head) = P(Tail) = 1/2 | n coins → 2ⁿ outcomes

21

Mensuration (2D & 3D)

Low ▼

📐 2D Shapes

Shape	Area	Perimeter
Square (side a)	a²	4a
Rectangle (l × b)	l × b	2(l + b)
Triangle	½ × base × height	a + b + c
Equilateral △ (a)	(√3/4) × a²	3a
Right Triangle (b, h)	½ × b × h	a + b + √(a²+b²)
Circle (r)	πr²	2πr
Semi-circle (r)	πr²/2	πr + 2r
Parallelogram (b, h)	b × h	2(a + b)
Rhombus (d₁, d₂)	½ × d₁ × d₂	4 × side
Trapezium (a, b, h)	½(a+b) × h	a + b + c + d

Heron's Formula: Area = √[s(s−a)(s−b)(s−c)] where s = (a+b+c)/2
Diagonal of square = a√2
Diagonal of rectangle = √(l² + b²)
Area of sector = (θ/360) × πr²
Arc length = (θ/360) × 2πr

📐 3D Shapes

Shape	Volume	CSA	TSA
Cube (a)	a³	4a²	6a²
Cuboid (l,b,h)	lbh	2h(l+b)	2(lb+bh+lh)
Cylinder (r, h)	πr²h	2πrh	2πr(r+h)
Cone (r, h, l)	⅓πr²h	πrl	πr(r+l)
Sphere (r)	(4/3)πr³	4πr²	4πr²
Hemisphere (r)	(2/3)πr³	2πr²	3πr²

Slant height of cone: l = √(r² + h²)
Diagonal of cube = a√3
Diagonal of cuboid = √(l² + b² + h²)
Frustum of cone: V = (πh/3)(R² + r² + Rr)

⚡

Quick Reference — Squares, Cubes & Primes

Must Memorize ▼

📐 Squares (11² to 30²)

11² = 121

12² = 144

13² = 169

14² = 196

15² = 225

16² = 256

17² = 289

18² = 324

19² = 361

20² = 400

21² = 441

22² = 484

23² = 529

24² = 576

25² = 625

26² = 676

27² = 729

28² = 784

29² = 841

30² = 900

📐 Cubes (1³ to 15³)

1³ = 1

2³ = 8

3³ = 27

4³ = 64

5³ = 125

6³ = 216

7³ = 343

8³ = 512

9³ = 729

10³ = 1000

11³ = 1331

12³ = 1728

13³ = 2197

14³ = 2744

15³ = 3375

📐 Prime Numbers (1 to 100) — Total: 25

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97

📐 Percentage ↔ Fraction Quick Reference

Fraction	%	Fraction	%	Fraction	%
1/2	50%	1/8	12.5%	1/14	7.14%
1/3	33.33%	1/9	11.11%	1/15	6.67%
1/4	25%	1/10	10%	1/16	6.25%
1/5	20%	1/11	9.09%	1/20	5%
1/6	16.67%	1/12	8.33%	1/25	4%
1/7	14.28%	1/13	7.69%	1/50	2%

Table of Contents (Click to Jump)

Number System

📐 HCF & LCM

📐 Factors & Divisors

📐 Divisibility Rules

📐 Unit Digit Cyclicity

📐 Sum Formulas

📐 Remainder Theorem

Simplification & Approximation

📐 Core Rules

📐 Useful Approximations

📐 Algebraic Identities (Frequently Used)

Percentage

📐 Basic Formulas

📐 Successive Percentage Change

📐 Population/Depreciation

📐 Fraction ↔ Percentage Reference

Profit, Loss & Discount

📐 Basic P&L Formulas

📐 Marked Price & Discount

📐 Special Tricks

Simple Interest (SI)

Compound Interest (CI)

📐 Core Formulas

📐 CI vs SI Difference

📐 Doubling & Rule of 72

Ratio & Proportion

📐 Division in Ratio

Average

Ages

Partnership

Time & Work

📐 LCM Method (Best for Exams!)

Pipes & Cisterns

Time, Speed & Distance

📐 Basic Formulas

📐 Conversions

📐 Average & Relative Speed

Problems on Trains

Boats & Streams

Mixture & Alligation

📐 Alligation Rule

📐 Repeated Dilution / Replacement

📐 Weighted Average of Mixtures

Number Series

Quadratic Equations

📐 Core Formulas

📐 Nature of Roots (Discriminant D)

📐 Sign Method (Exam Shortcut)

Permutation & Combination

📐 Fundamental Formulas

📐 Key Identities

Probability

📐 Core Formulas

📐 Quick References

Mensuration (2D & 3D)

📐 2D Shapes

📐 3D Shapes

Quick Reference — Squares, Cubes & Primes

📐 Squares (11² to 30²)

📐 Cubes (1³ to 15³)

📐 Prime Numbers (1 to 100) — Total: 25

📐 Percentage ↔ Fraction Quick Reference