Percentagex% of y = xy/100

Percentage change((New - Old) / Old) x 100

Successive changea% and b% = a + b + ab/100

Value after changeNew = Old x (1 +/- r/100)

Population after n yearsP(1 +/- r/100)ⁿ

Profit %((SP - CP) / CP) x 100

Loss %((CP - SP) / CP) x 100

SP from profitSP = CP x (100 + profit%)/100

SP from lossSP = CP x (100 - loss%)/100

Discount %((MP - SP) / MP) x 100

Successive discountsNet discount = a + b - ab/100

Simple interestSI = PRT/100

Simple amountA = P + SI

Compound amountA = P(1 + r/100)ⁿ

Compound interestCI = A - P

Half-yearly compoundingA = P(1 + r/200)²ⁿ

Quarterly compoundingA = P(1 + r/400)⁴ⁿ

Proportiona:b = c:d means ad = bc

Fourth proportionalIf a:b = c:x, then x = bc/a

Mean proportionalx = sqrt(ab)

Direct variationy = kx

Inverse variationxy = k

Partnership shareProfit ratio = capital x time

AverageSum of values / number of values

Weighted average(w1x1 + w2x2 + ...)/(w1 + w2 + ...)

Combined average(n1a1 + n2a2)/(n1 + n2)

Alligation ratioCheaper : Dearer = (Dearer - Mean) : (Mean - Cheaper)

ReplacementFinal quantity = Initial x (1 - removed/total)ⁿ

Work rateIf A finishes in x days, A's rate = 1/x

Combined rate1/T = 1/a + 1/b + 1/c

EfficiencyEfficiency is inversely proportional to time

Man-daysWork = people x days x hours x efficiency

Pipes and cisternsNet rate = filling rates - emptying rates

Basic relationDistance = Speed x Time

Average speedTotal distance / total time

Equal distancesAverage speed = 2xy/(x + y)

Relative speed same direction|x - y|

Relative speed opposite directionx + y

Boat downstreamSpeed = boat speed + stream speed

Boat upstreamSpeed = boat speed - stream speed

Number of factorsIf N = p^aq^br^c, factors = (a + 1)(b + 1)(c + 1)

Sum of factors(p^a+1 - 1)/(p - 1) x (q^b+1 - 1)/(q - 1) ...

Product of factorsN^{number of factors / 2}

HCF x LCMFor two numbers, HCF x LCM = product of numbers

Remaindera = bq + r, where 0 <= r < b

Euler ideaIf a and n are coprime, a^phi(n) leaves remainder 1 modulo n

Square identity(a + b)² = a² + 2ab + b²

Difference squarea² - b² = (a - b)(a + b)

Cube identitya³ + b³ = (a + b)(a² - ab + b²)

Quadratic rootsx = [-b +/- sqrt(b² - 4ac)] / 2a

AP nth terma_n = a + (n - 1)d

AP sumS_n = n/2[2a + (n - 1)d]

GP nth terma_n = ar^n-1

GP sumS_n = a(rⁿ - 1)/(r - 1), r != 1

Triangle area1/2 x base x height

Heron's formulaArea = sqrt[s(s-a)(s-b)(s-c)], s = (a+b+c)/2

Equilateral triangle area(sqrt3/4)a²

PythagorasHypotenuse² = base² + height²

Circle areapi r²

Circle circumference2 pi r

Arc length(theta/360) x 2 pi r

Sector area(theta/360) x pi r²

Polygon interior sum(n - 2) x 180 degrees

Regular polygon angleInterior angle = (n - 2) x 180/n degrees

Exterior angleEach exterior angle of a regular n-gon = 360/n degrees

Triangle inradiusr = Area/s, where s is semiperimeter

Triangle circumradiusR = abc/(4 x Area)

Median lengthm_a = 1/2 sqrt(2b² + 2c² - a²)

Two-side triangle areaArea = 1/2 ab sin C

Brahmagupta formulaArea = sqrt[(s-a)(s-b)(s-c)(s-d)] for cyclic quadrilateral

Rhombus area1/2 x d₁ x d₂

Chord length2r sin(theta/2)

Tangent powerPT² = PA x PB

Distance formulasqrt[(x2 - x1)² + (y2 - y1)²]

Slope(y2 - y1)/(x2 - x1)

Point-line distance|Ax1 + By1 + C| / sqrt(A² + B²)

Section formula((mx2 + nx1)/(m+n), (my2 + ny1)/(m+n))

Centroid((x1+x2+x3)/3, (y1+y2+y3)/3)

Cuboid volumelbh

Cuboid surface area2(lb + bh + hl)

Cube volumea³

Cube surface area6a²

Cylinder volumepi r²h

Cylinder curved surface area2 pi rh

Cone volume1/3 pi r²h

Sphere volume4/3 pi r³

Sphere surface area4 pi r²

Factorialn! = n x (n-1) x ... x 1

PermutationnPr = n!/(n-r)!

CombinationnCr = n!/[r!(n-r)!]

Circular arrangement(n - 1)!

ProbabilityFavourable outcomes / total outcomes

ComplementP(not A) = 1 - P(A)

Independent eventsP(A and B) = P(A)P(B)

Product of powersa^m x aⁿ = a^m+n

Quotient of powersa^m / aⁿ = a^m-n

Power of power(a^m)ⁿ = a^mn

Log productlog(ab) = log a + log b

Log quotientlog(a/b) = log a - log b

Log powerlog(aⁿ) = n log a

Change of baselog_ab = log b / log a