Math Calculations
Percentages, ratios, series and geometry — the arithmetic behind everyday numbers, shown step by step.
17 calculations published in this category.
Percentages
How To Calculate A Percentage
Every percentage question is one of three sentences wearing different clothes. Once you can identify which one you are looking at, the arithmetic is a single multiplication or division, and the common mistakes all come from picking the wrong one.
Part = Whole x Percentage / 100
How To Calculate Percent Decrease
Percent decrease is a drop divided by the level it fell from. The denominator decides everything: 120 to 90 is a fall of 25.00%, and the same 30 measured against 90 gives 33.3333%, which answers a different question entirely.
Percent Decrease = (Original Value - New Value) / Original Value x 100
How To Calculate Percent Increase
Percent increase is a gap divided by where you started. Get the denominator right and the rest is arithmetic; get it wrong and every figure downstream is quietly answering a different question.
Percent Increase = (New Value - Original Value) / Original Value x 100
How To Calculate Percentage Change
Percentage change is a difference divided by a starting point. That denominator is where almost every mistake lives — including the one that makes a 90% collapse look recoverable.
Percentage Change = (New Value - Original Value) / Original Value x 100
Statistics
How To Calculate A Weighted Average
A weighted average answers "what is the typical value?" when some values count more than others. Grades, portfolio returns, inventory costs and survey results are almost never plain averages, which is why the distinction matters.
Weighted Average = Sum of (Value x Weight) / Sum of Weights
How To Calculate A Z-Score
A z-score restates a value as how many standard deviations it sits from the mean. That rescaling removes the original units, which is exactly what lets you compare things that were never commensurable.
z = (x - mean) / standard deviation
How To Calculate Standard Deviation
Standard deviation is the typical distance of observations from their mean, measured in the original units. The squaring is what makes it algebraically useful — and the n-1 is what makes it honest for a sample.
SD = sqrt(Sum of (xi - Mean)^2 / (n - 1))
Algebra
How To Calculate Slope
Slope is how much a quantity changes vertically for one unit of horizontal movement. It turns two data points into a rate, and once you have it, the whole line follows.
Slope = (y2 - y1) / (x2 - x1)
How To Solve A Quadratic Equation
Every quadratic equation has a closed-form solution. The discriminant tells you what kind of answer to expect before you compute anything, which is the part people skip.
x = (-b +/- sqrt(b^2 - 4ac)) / (2a)
Geometry
How To Calculate The Area Of A Circle
Circle area is pi times the radius squared. Everything difficult about the topic sits in two decisions: which length is the radius, and how much precision you carry.
Area = pi x r^2
How To Calculate The Area Of A Triangle
A triangle covers exactly half of whatever parallelogram shares its base and height. Nearly every wrong answer comes from one of two slips: a height that was never perpendicular, or a base and a height borrowed from two different sides.
Area = 1/2 x base x height
How To Calculate The Volume Of A Cylinder
Cylinder volume is the area of the circular base multiplied by the height. Nearly every wrong answer comes from one of two places: a diameter used where a radius belongs, or units that were never made consistent before being cubed.
V = pi x r^2 x h
How To Calculate The Volume Of A Rectangular Prism
Volume of a rectangular prism is length times width times height. Nearly every error on this page comes from somewhere else: mismatched units, external instead of internal measurements, or a cubic answer read as a liquid one.
V = l x w x h
How To Use The Pythagorean Theorem
For right triangles, the squares of the two shorter sides add exactly to the square of the longest. That single relation turns any two known sides into the third.
c^2 = a^2 + b^2, so c = sqrt(a^2 + b^2)
Sequences
How To Find The Sum Of A Geometric Series
In a geometric series each term is the previous one multiplied by a fixed ratio. That single change turns polynomial growth into exponential growth — and turns the sum formula into something with a constraint worth respecting.
S = a1 (1 - r^n) / (1 - r), and S-infinity = a1 / (1 - r) when |r| < 1
How To Find The Sum Of An Arithmetic Series
An arithmetic series adds the terms of a sequence that changes by a fixed amount each step. The sum has a closed form, which means adding a thousand terms costs the same effort as adding five.
S = n/2 x (a1 + an)