📐 Arithmetic — Complete Formula Sheet for Banking Exams

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📌 Arithmetic in Banking Exams: 15–20 marks in Prelims + Mains | Covers 19 core topics
Strategy: Memorise all formulas → practice shortcuts → attempt DI-linked arithmetic last

Formula	Expression	Use Case
Percentage	(Value / Total) × 100	Find % of a number
% Change	[(New − Old) / Old] × 100	Increase / Decrease
A is x% of B	A = (x/100) × B	Find value
x% of A = y% of B	A/B = y/x	Ratio comparison
Successive % change	a + b + (ab/100) %	Two consecutive changes

Basic: a:b = a/b  |  Compound Ratio: (a:b) × (c:d) = ac:bd  |  Duplicate Ratio: a²:b²  |  Triplicate: a³:b³

Proportion: a:b = c:d → ad = bc (Product of extremes = Product of means)

Mean Proportion between a and b = √(ab)

Third Proportion of a, b → b²/a  |  Fourth Proportion of a,b,c → bc/a

Dividing X in ratio a:b:c: First part = X × a/(a+b+c), and so on

Average = Sum of observations / Number of observations

Weighted Average = (w₁x₁ + w₂x₂ + … ) / (w₁ + w₂ + …)

Situation	Formula
New avg when one value added	New Avg = (Old Sum + New Value) / (n+1)
If avg increases by d after adding x	x = Old Avg + (n+1)×d
Average of first n natural numbers	(n+1)/2
Average of n odd numbers	n (the nth odd number)
Average speed (equal distances)	2s₁s₂/(s₁+s₂)

Term	Formula
Profit %	[(SP − CP) / CP] × 100
Loss %	[(CP − SP) / CP] × 100
SP from CP + Profit%	SP = CP × (100 + P%) / 100
CP from SP + Profit%	CP = SP × 100 / (100 + P%)
Dishonest shopkeeper profit%	[(True wt − False wt) / False wt] × 100
Marked Price & Discount	SP = MP × (100 − D%) / 100
Two articles sold — one at x% profit, one at x% loss	Overall Loss% = x²/100

Discount = MP − SP  |  Discount % = (Discount / MP) × 100

Two successive discounts d₁% and d₂%: Net Discount = d₁ + d₂ − (d₁×d₂)/100

Equivalent single discount for d₁, d₂, d₃: apply successively

SP after n discounts d%: SP = MP × [(100−d)/100]ⁿ

Find	Formula
Principal P	P = (SI × 100) / (R × T)
Rate R	R = (SI × 100) / (P × T)
Time T	T = (SI × 100) / (P × R)
When SI = P (doubles)	T = 100/R years
Sum doubles at R%	Time = 100/R years

Compounding	Formula
Half-yearly	A = P × (1 + R/200)^(2n)
Quarterly	A = P × (1 + R/400)^(4n)
CI − SI for 2 years	CI − SI = P(R/100)²
CI − SI for 3 years	CI − SI = P(R/100)² × (3 + R/100)
Effective rate for 2 yrs	2R + R²/100

If A can do work in x days, A's 1-day work = 1/x

Situation	Formula
A and B together	Time = xy/(x+y) days
A, B, C together	Time = 1/(1/x + 1/y + 1/z)
A twice as fast as B	A takes half B's time
Wages ∝ Work done	Wage of A/B = Work ratio of A/B
Men × Days = Work	M₁D₁H₁ = M₂D₂H₂ (same work)

Inlet fills in x hrs, Outlet empties in y hrs. Net rate = 1/x − 1/y per hour

Situation	Formula
Pipe A fills in x hrs, B fills in y hrs	Together: xy/(x+y) hrs
A fills, B empties; A faster	Together: xy/(y−x) hrs
Tank full, both open	Empties in: xy/(x−y) hrs (if x>y)
Partial fill + leak	Net = (fill rate) − (leak rate)

Note: Treat inlet as positive work and outlet as negative work — exactly like Time & Work.

Concept	Formula
km/h → m/s conversion	× (5/18)
m/s → km/h conversion	× (18/5)
Average speed (same distance)	2s₁s₂/(s₁+s₂)
Trains passing each other (opposite)	Time = (L₁+L₂)/(S₁+S₂)
Trains passing (same direction)	Time = (L₁+L₂)/(S₁−S₂)
Train crossing a pole/person	Time = Train Length / Speed
Meeting time (towards each other)	T = D / (S₁+S₂)
Catch-up time (same direction)	T = Gap / (S₁−S₂)

Find	Formula
Boat speed (u)	(Downstream + Upstream) / 2
Stream speed (v)	(Downstream − Upstream) / 2
Time ratio (DS:US)	US time / DS time = (u+v)/(u−v) inverted
Same distance: round trip time	T = D/(u+v) + D/(u−v)

Alligation Rule: Ratio of mixing = (C − Mean) : (Mean − C') where C = dearer price, C' = cheaper price, Mean = desired price

Removal & replacement: If from V litres of mixture, x litres removed and replaced with water n times:
Final pure liquid = V × (1 − x/V)ⁿ

Profit sharing ∝ Capital × Time

Type	Formula
Same time	Profit ∝ Capital → P₁:P₂ = C₁:C₂
Different time	P₁:P₂ = C₁T₁ : C₂T₂
Working + Sleeping partners	Deduct working partner's salary first, then split remaining profit
A:B:C ratio and total profit	A's share = Total × A/(A+B+C)

Let present age = x. Set up linear equation from the given condition.

Translation	Algebra
Age n years ago	x − n
Age n years hence	x + n
A is twice B	A = 2B
Ratio of ages a:b now; ratio c:d after n yrs	Set up two equations; solve simultaneously
Sum of ages	Sum = sum of individual ages

Tip: Always define variables clearly. For ratio problems, let ages be ax and bx, then form equations from conditions.

Rule	Formula
Complementary	P(E') = 1 − P(E)
Addition (mutually exclusive)	P(A∪B) = P(A) + P(B)
Addition (general)	P(A∪B) = P(A) + P(B) − P(A∩B)
Multiplication (independent)	P(A∩B) = P(A) × P(B)
Cards: Total 52 → 4 suits × 13 cards	Face cards = 12, Aces = 4, Red = 26, Black = 26
Dice: 1 die has 6 faces	P(sum=7 with 2 dice) = 6/36 = 1/6

Situation	Formula
Arrange n things in n places	n!
Select r from n (order doesn't matter)	nCr
Circular arrangement of n	(n−1)!
Necklace / bracelet (flips allowed)	(n−1)! / 2
Words from letters with repetition	n! / (p! × q! × r! …) for repeated letters p,q,r
nCr + nCr-1 = (n+1)Cr	Pascal's triangle rule

Shape	Area	Perimeter / Volume
Rectangle	l × b	P = 2(l+b)
Square	a²	P = 4a; Diagonal = a√2
Triangle	½ × b × h  |  √[s(s−a)(s−b)(s−c)]	P = a+b+c; s = P/2
Circle	πr²	Circumference = 2πr
Trapezium	½ × (a+b) × h	Sum of all sides
Parallelogram	b × h	2(a+b)
Equilateral △	(√3/4)a²	3a; Height = (√3/2)a
Cube	Surface = 6a²	Volume = a³; Diagonal = a√3
Cuboid	2(lb + bh + lh)	V = l×b×h; Diagonal = √(l²+b²+h²)
Cylinder	CSA = 2πrh; TSA = 2πr(h+r)	V = πr²h
Cone	CSA = πrl; TSA = πr(l+r); l=√(r²+h²)	V = (1/3)πr²h
Sphere	4πr²	V = (4/3)πr³
Hemisphere	CSA = 2πr²; TSA = 3πr²	V = (2/3)πr³

π ≈ 22/7 = 3.14159  |  Use π = 3.14 for approximation unless specified

Concept	Key Facts
Divisibility by 2	Last digit even (0,2,4,6,8)
Divisibility by 3	Sum of digits divisible by 3
Divisibility by 4	Last 2 digits divisible by 4
Divisibility by 7	Double last digit, subtract from rest; repeat if needed
Divisibility by 8	Last 3 digits divisible by 8
Divisibility by 9	Sum of digits divisible by 9
Divisibility by 11	(Sum of odd-position digits) − (Sum of even-position digits) = 0 or divisible by 11
Sum of first n natural numbers	n(n+1)/2
Sum of first n odd numbers	n²
Sum of first n even numbers	n(n+1)
Sum of squares 1²+2²+…+n²	n(n+1)(2n+1)/6
Sum of cubes 1³+2³+…+n³	[n(n+1)/2]²

Unit digit cycles: 2→2,4,8,6; 3→3,9,7,1; 4→4,6; 7→7,9,3,1; 8→8,4,2,6; 9→9,1; 5 always 5; 6 always 6

Concept	Rule
HCF of fractions	HCF of numerators / LCM of denominators
LCM of fractions	LCM of numerators / HCF of denominators
Largest number dividing a,b,c leaving same remainder r	HCF of (a−r), (b−r), (c−r)
Smallest number divisible by a,b,c	LCM of a, b, c
Bells ringing together after t min; ring again together	LCM of their intervals
Largest square tile for a floor a × b	HCF of a and b

Methods: Prime Factorisation (best for exams) | Division Method (faster for HCF)

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