Math Formula Library

Every formula here is written out, its symbols named, and then put through a worked example with real numbers — because a formula you have never seen used is a picture rather than a tool. Search by name, by symbol or by what you are trying to do: try quadratic, pythagoras, npr or reverse vat. Where a calculator on this site does the same job, the card links to it.

Showing all 82 formulas

Every Formula, Written Out in Full

• Quadratic Formula

  Algebra
  x = (−b ± √(b² − 4ac)) / 2a

  Symbols

  a, b and c are the coefficients of ax² + bx + c = 0.

  When to use it

  Solves any quadratic, including the ones that do not factorise. Rearrange to the form ax² + bx + c = 0 first, or the signs will be wrong.

  Worked example

  For 2x² + 5x − 3 = 0: b² − 4ac = 25 + 24 = 49, √49 = 7, so x = (−5 ± 7) / 4, giving x = 0.5 and x = −3.

• Discriminant

  Algebra
  Δ = b² − 4ac

  Symbols

  The part of the quadratic formula under the square root.

  When to use it

  Tells you how many real solutions a quadratic has before you solve it. Positive gives two, zero gives one, negative gives none.

  Worked example

  For x² + 2x + 5: Δ = 4 − 20 = −16. Negative, so the curve never crosses the x-axis and there are no real roots.

• Difference of Two Squares

  Algebra
  a² − b² = (a + b)(a − b)

  Symbols

  a and b are any two terms.

  When to use it

  Factorises instantly whenever you recognise two perfect squares with a minus between them. Also a quick mental multiplication trick.

  Worked example

  97 × 103 is (100 − 3)(100 + 3) = 100² − 3² = 10,000 − 9 = 9,991.

• Square of a Binomial

  Algebra
  (a + b)² = a² + 2ab + b²

  Symbols

  a and b are the two terms being squared together.

  When to use it

  Expands a bracket squared. The middle term is the one people drop — (a + b)² is never a² + b².

  Worked example

  (x + 4)² = x² + 8x + 16, since 2ab = 2 × x × 4 = 8x.

• Gradient of a Line

  Algebra
  m = (y₂ − y₁) / (x₂ − x₁)

  Symbols

  (x₁, y₁) and (x₂, y₂) are any two points on the line.

  When to use it

  Measures steepness. Take the points in the same order top and bottom, or the sign flips.

  Worked example

  Between (2, 3) and (6, 11): m = (11 − 3) / (6 − 2) = 8 / 4 = 2. The line rises two units for every one across.

• Equation of a Straight Line

  Algebra
  y = mx + c

  Symbols

  m is the gradient, c is where the line crosses the y-axis.

  When to use it

  Describes any straight line except a vertical one. Written y = mx + b in American textbooks — same formula.

  Worked example

  A line of gradient 2 through (0, −5) is y = 2x − 5. At x = 4 that gives y = 3.

• Distance Between Two Points

  Algebra
  d = √((x₂ − x₁)² + (y₂ − y₁)²)

  Symbols

  The two points are (x₁, y₁) and (x₂, y₂).

  When to use it

  Pythagoras applied to coordinates. Squaring removes the sign, so the order of the points does not matter here.

  Worked example

  From (1, 2) to (4, 6): d = √(3² + 4²) = √25 = 5.

• Laws of Indices

  Algebra
  aᵐ × aⁿ = aᵐ⁺ⁿ and aᵐ ÷ aⁿ = aᵐ⁻ⁿ

  Symbols

  a is the base, m and n the powers.

  When to use it

  Only applies when the bases match. You cannot combine 2³ × 3³ this way, though you can write it as 6³.

  Worked example

  2⁵ × 2³ = 2⁸ = 256. And 2⁵ ÷ 2³ = 2² = 4.

• Logarithm Rules

  Algebra
  log(xy) = log x + log y and log(xⁿ) = n log x

  Symbols

  x and y are positive; the base must be the same throughout.

  When to use it

  Turns multiplication into addition, which is why logs are how you solve for an exponent.

  Worked example

  To solve 3ˣ = 81: take logs to get x log 3 = log 81, so x = log 81 / log 3 = 4.

• Arithmetic Sequence, nth Term

  Algebra
  aₙ = a₁ + (n − 1)d

  Symbols

  a₁ is the first term, d the common difference, n the position.

  When to use it

  For sequences that go up by a fixed amount each time. The (n − 1) catches people out — the first term has had no steps added yet.

  Worked example

  For 5, 8, 11, 14…: a₁ = 5 and d = 3, so the 20th term is 5 + 19 × 3 = 62.

• Sum of a Geometric Series

  Algebra
  Sₙ = a(1 − rⁿ) / (1 − r)

  Symbols

  a is the first term, r the common ratio, n the number of terms.

  When to use it

  For sequences that multiply by a fixed factor. Where compound interest and repayment schedules come from.

  Worked example

  Sum of 2 + 6 + 18 + 54: a = 2, r = 3, n = 4, so S = 2(1 − 81) / (1 − 3) = 2 × (−80) / (−2) = 80.

• Midpoint of a Line

  Algebra
  M = ((x₁ + x₂) / 2, (y₁ + y₂) / 2)

  Symbols

  The two endpoints are (x₁, y₁) and (x₂, y₂).

  When to use it

  Average the x values and the y values separately. The answer is a point, so it has two coordinates rather than one number.

  Worked example

  Between (2, 3) and (8, 7): M = (5, 5).

• Sum of an Arithmetic Series

  Algebra
  Sₙ = n/2 × (a₁ + aₙ)

  Symbols

  n terms, running from a₁ up to aₙ.

  When to use it

  Pairs the first term with the last, the second with the second from last, and so on — every pair has the same total.

  Worked example

  Adding 1 to 100: S = 50 × 101 = 5,050.

• Factorial

  Algebra
  n! = n × (n − 1) × … × 2 × 1

  Symbols

  Defined for whole numbers, with 0! equal to 1 by convention.

  When to use it

  Counts the ways n things can be put in order. It grows faster than almost anything else in ordinary maths.

  Worked example

  5! = 120, while 10! = 3,628,800 — doubling n multiplied the answer by thirty thousand.

• Area of a Rectangle

  Geometry
  A = l × w

  Symbols

  l is length, w is width, both in the same unit.

  When to use it

  The starting point for almost every material calculation. Mixing feet and inches here is the most common source of a wrong order.

  Worked example

  A patio 6.4 m by 3.5 m has an area of 22.4 m².

  Gravel Calculator

• Area of a Triangle

  Geometry
  A = ½ × b × h

  Symbols

  b is the base, h the perpendicular height — not the slanted side.

  When to use it

  Works for any triangle as long as the height is measured at right angles to the base you chose.

  Worked example

  Base 12 cm, perpendicular height 5 cm: A = 0.5 × 12 × 5 = 30 cm².

• Area of a Circle

  Geometry
  A = πr²

  Symbols

  r is the radius — half the width, not the width.

  When to use it

  Using the diameter instead of the radius is the classic error and makes the answer four times too big.

  Worked example

  A circle of diameter 3 m has radius 1.5 m, so A = π × 2.25 = 7.07 m².

• Circumference of a Circle

  Geometry
  C = 2πr = πd

  Symbols

  r is the radius, d the diameter.

  When to use it

  The distance around the edge. Both versions are the same formula, so use whichever measurement you actually have.

  Worked example

  A wheel of diameter 0.7 m travels π × 0.7 = 2.20 m per revolution.

• Pythagorean Theorem

  Geometry
  a² + b² = c²

  Symbols

  c is the hypotenuse, the side opposite the right angle.

  When to use it

  Right-angled triangles only. Also how builders check a corner is square without a set square.

  Worked example

  Legs of 3 m and 4 m give c = √(9 + 16) = 5 m. Measure 3 along one wall, 4 along the other, and if the diagonal is 5 the corner is true.

• Area of a Trapezium

  Geometry
  A = ½(a + b) × h

  Symbols

  a and b are the two parallel sides, h the distance between them.

  When to use it

  Average the parallel sides, then treat it as a rectangle. Called a trapezoid in American usage.

  Worked example

  Parallel sides 8 m and 5 m, 4 m apart: A = 0.5 × 13 × 4 = 26 m².

• Heron's Formula

  Geometry
  A = √(s(s − a)(s − b)(s − c)), s = (a + b + c) / 2

  Symbols

  a, b and c are the three sides; s is the semi-perimeter.

  When to use it

  Gives the area of any triangle from the three sides alone, when no height has been measured.

  Worked example

  Sides 5, 6 and 7: s = 9, so A = √(9 × 4 × 3 × 2) = √216 = 14.70.

• Volume of a Cuboid

  Geometry
  V = l × w × h

  Symbols

  Length, width and depth, all in the same unit.

  When to use it

  Any rectangular box or layer of material. For a layer, the depth is usually the one quoted in different units from the other two.

  Worked example

  A bed 6 m by 2 m covered 0.1 m deep holds 6 × 2 × 0.1 = 1.2 m³ of mulch.

  Mulch Calculator

• Volume of a Cylinder

  Geometry
  V = πr²h

  Symbols

  r is the radius of the circular face, h the height.

  When to use it

  The area of the circle, extended along its length. Works for pipes, tanks and post holes.

  Worked example

  A tank 0.5 m across and 1.2 m tall: V = π × 0.25² × 1.2 = 0.236 m³, or about 236 litres.

• Volume of a Sphere

  Geometry
  V = (4/3)πr³

  Symbols

  r is the radius.

  When to use it

  The cube on the radius is why a small increase in size makes such a large difference to volume.

  Worked example

  Doubling a ball's radius from 5 cm to 10 cm multiplies its volume by 2³ = 8, not by 2.

• Volume of a Cone

  Geometry
  V = (1/3)πr²h

  Symbols

  r is the base radius, h the vertical height.

  When to use it

  Exactly one third of the cylinder that would enclose it — a useful check on any answer.

  Worked example

  A conical heap 2 m across and 1.5 m high: V = (1/3) × π × 1 × 1.5 = 1.57 m³.

• Surface Area of a Sphere

  Geometry
  A = 4πr²

  Symbols

  r is the radius.

  When to use it

  Exactly four times the area of the circle you would see looking at it — which is not obvious, and worth remembering.

  Worked example

  A sphere of radius 0.3 m has surface area 4π × 0.09 = 1.13 m².

• Perimeter of a Rectangle

  Geometry
  P = 2(l + w)

  Symbols

  l is length and w is width.

  When to use it

  The distance around the edge, for fencing, skirting or edging. It is not the area, and the two are easy to order by mistake.

  Worked example

  A patio 6.4 m by 3.5 m needs 2 × 9.9 = 19.8 m of edging.

• Area of a Parallelogram

  Geometry
  A = b × h

  Symbols

  b is the base and h the perpendicular height between the parallel sides.

  When to use it

  The same as a rectangle, because sliding the top across changes no area. The slanted side is not the height.

  Worked example

  Base 7 m with a perpendicular height of 4 m: A = 28 m².

• Interior Angles of a Polygon

  Geometry
  sum = (n − 2) × 180°

  Symbols

  n is the number of sides.

  When to use it

  Any polygon divides into n − 2 triangles and each holds 180°. Divide the total by n for one angle of a regular shape.

  Worked example

  A hexagon gives (6 − 2) × 180 = 720°, so each angle of a regular hexagon is 120°.

• Area of a Sector

  Geometry
  A = (θ / 360) × πr²

  Symbols

  θ is the angle at the centre, measured in degrees.

  When to use it

  A slice of a circle. The fraction θ/360 is simply how much of the whole circle the slice covers.

  Worked example

  A quarter circle of radius 4: (90/360) × π × 16 = 12.57.

• Arc Length

  Geometry
  L = (θ / 360) × 2πr

  Symbols

  θ in degrees, r the radius.

  When to use it

  The curved edge of a sector — the same fraction as its area, applied to the circumference instead.

  Worked example

  A quarter of a circle of radius 4: 0.25 × 2π × 4 = 6.28.

• Surface Area of a Cylinder

  Geometry
  A = 2πr² + 2πrh

  Symbols

  Two circular ends, plus the curved side unrolled into a rectangle.

  When to use it

  For painting, wrapping or lagging. Drop the first term for each end that is open.

  Worked example

  With r = 0.5 m and h = 1.2 m: 1.571 + 3.770 = 5.34 m².

• Volume of a Pyramid

  Geometry
  V = (1/3) × base area × h

  Symbols

  h is the vertical height from the base to the apex.

  When to use it

  One third of the prism with the same base and height, exactly as a cone is one third of its cylinder.

  Worked example

  A square base 4 m by 4 m, 9 m tall: (1/3) × 16 × 9 = 48 m³.

• SOH CAH TOA

  Trigonometry
  sin θ = opp/hyp, cos θ = adj/hyp, tan θ = opp/adj

  Symbols

  Opposite, adjacent and hypotenuse are named relative to the angle θ you are working with.

  When to use it

  Right-angled triangles only. Which side counts as opposite changes when you switch angles, which is where mistakes creep in.

  Worked example

  A ladder 4 m long at 70° to the ground reaches 4 × sin 70° = 3.76 m up the wall.

• Sine Rule

  Trigonometry
  a / sin A = b / sin B = c / sin C

  Symbols

  Each side is paired with the angle directly opposite it.

  When to use it

  For triangles with no right angle, when you know two angles and a side, or two sides and a non-included angle.

  Worked example

  With A = 30°, a = 5 and B = 60°: b = 5 × sin 60° / sin 30° = 5 × 0.866 / 0.5 = 8.66.

• Cosine Rule

  Trigonometry
  c² = a² + b² − 2ab cos C

  Symbols

  C is the angle between sides a and b.

  When to use it

  When you know two sides and the angle between them, or all three sides. Pythagoras is this formula with C = 90°, where cos C = 0.

  Worked example

  Sides 7 and 9 with 40° between: c² = 49 + 81 − 2 × 63 × 0.766 = 33.5, so c = 5.79.

• Pythagorean Identity

  Trigonometry
  sin²θ + cos²θ = 1

  Symbols

  True for every angle θ.

  When to use it

  Converts between sine and cosine when only one is known. Pythagoras on a circle of radius 1.

  Worked example

  If sin θ = 0.6 then cos²θ = 1 − 0.36 = 0.64, so cos θ = 0.8 for an angle in the first quadrant.

• Tangent as a Ratio

  Trigonometry
  tan θ = sin θ / cos θ

  Symbols

  Undefined wherever cos θ = 0, which is at 90° and 270°.

  When to use it

  Explains why tan has no value at a right angle: the formula divides by zero there.

  Worked example

  At 45°, sin and cos are both 0.7071, so tan 45° = 1 exactly.

• Area From Two Sides and an Angle

  Trigonometry
  A = ½ab sin C

  Symbols

  a and b are two sides, C the angle between them.

  When to use it

  Gives the area without needing a perpendicular height, which is rarely the thing you can measure on site.

  Worked example

  Sides 8 m and 5 m with 30° between: A = 0.5 × 8 × 5 × 0.5 = 10 m².

• Degrees and Radians

  Trigonometry
  radians = degrees × π / 180

  Symbols

  A full turn is 360° or 2π radians.

  When to use it

  Spreadsheets and programming languages expect radians. A trig answer that looks wildly wrong is usually a calculator in the other mode.

  Worked example

  60° = 60 × π / 180 = π/3 ≈ 1.047 radians.

  Unit Conversion Calculator

• Angle Sum of a Triangle

  Trigonometry
  A + B + C = 180°

  Symbols

  The three interior angles of any flat triangle.

  When to use it

  Finds the third angle from two known ones, which is often the step that makes the sine rule usable.

  Worked example

  Given angles of 40° and 75°, the third is 180 − 115 = 65°.

• Double Angle Formulae

  Trigonometry
  sin 2θ = 2 sin θ cos θ, cos 2θ = cos²θ − sin²θ

  Symbols

  θ is any angle.

  When to use it

  Rewrites a doubled angle in terms of the single one. sin 2θ is not 2 sin θ, and these exist to stop exactly that.

  Worked example

  At θ = 30°: sin 60° = 0.866, and 2 × 0.5 × 0.866 = 0.866. Writing 2 sin 30° would have given 1.

• Mean (Average)

  Statistics
  x̄ = Σx / n

  Symbols

  Σx is the total of all values, n how many there are.

  When to use it

  The everyday average. One extreme value drags it a long way, which is why income is usually reported as a median.

  Worked example

  For 4, 8, 9, 11: total 32 ÷ 4 = 8.

  Mean, Median and Mode Calculator

• Median

  Statistics
  position = (n + 1) / 2, after sorting

  Symbols

  n is how many values there are.

  When to use it

  The middle value once sorted. With an even count the position lands between two, and you average them.

  Worked example

  For 3, 7, 9, 15: (4 + 1) / 2 = 2.5, so average the 2nd and 3rd values — (7 + 9) / 2 = 8.

  Mean, Median and Mode Calculator

• Range

  Statistics
  range = maximum − minimum

  Symbols

  The two extreme values in the set.

  When to use it

  The crudest measure of spread, and easily distorted, since it depends on exactly two numbers however large the set.

  Worked example

  For 12, 15, 19, 40: range = 40 − 12 = 28.

• Variance

  Statistics
  σ² = Σ(x − μ)² / N

  Symbols

  μ is the mean and N the count. For a sample rather than a whole population, divide by n − 1.

  When to use it

  Averages the squared distances from the mean. Squaring stops positive and negative gaps cancelling out.

  Worked example

  For 2, 4, 6 the mean is 4, the squared gaps are 4, 0 and 4, so σ² = 8 / 3 = 2.67.

• Standard Deviation

  Statistics
  σ = √variance

  Symbols

  The square root of the variance.

  When to use it

  Returns the spread to the original units, which variance cannot report because it is in units squared.

  Worked example

  Variance of 2.67 gives σ = 1.63, in the same units as the data.

  Mean, Median and Mode Calculator

• Z-Score

  Statistics
  z = (x − μ) / σ

  Symbols

  x is the value, μ the mean, σ the standard deviation.

  When to use it

  Expresses a value as a number of standard deviations from the mean, so results on different scales can be compared.

  Worked example

  A score of 78 where the mean is 65 and σ is 10 gives z = 1.3 — one and a bit deviations above average.

• Basic Probability

  Statistics
  P(A) = favourable outcomes / total outcomes

  Symbols

  All outcomes must be equally likely for this to hold.

  When to use it

  The definition, and only valid when the outcomes really are equally likely — which a shuffled deck satisfies and a football match does not.

  Worked example

  Drawing a heart from a full deck: 13 / 52 = 0.25, a one in four chance.

• Combinations

  Statistics
  C(n, r) = n! / (r!(n − r)!)

  Symbols

  n items, choosing r of them, order not mattering.

  When to use it

  Counts selections where order is irrelevant — a lottery ticket, a committee, a hand of cards.

  Worked example

  Choosing 3 from 5: 120 / (6 × 2) = 10 possible groups.

• Permutations

  Statistics
  P(n, r) = n! / (n − r)!

  Symbols

  n items, arranging r of them, order mattering.

  When to use it

  Use when the sequence counts — a podium, a PIN, a running order. Always larger than the equivalent combination.

  Worked example

  Arranging 3 of 5: 120 / 2 = 60, six times as many as the 10 combinations, because each group of 3 can be ordered 3! = 6 ways.

• Percentage Change

  Statistics
  change = (new − old) / old × 100

  Symbols

  Always divide by the original value, not the new one.

  When to use it

  A rise then an equal-looking fall does not return you to the start, because the base changes between the two.

  Worked example

  £80 rising to £100 is a 25% rise. £100 falling to £80 is a 20% fall. Same two numbers, different percentages.

  Percentage Calculator

• Interquartile Range

  Statistics
  IQR = Q₃ − Q₁

  Symbols

  Q₁ and Q₃ sit a quarter and three quarters of the way through the sorted data.

  When to use it

  The spread of the middle half, which survives the outliers that make the plain range useless.

  Worked example

  With Q₁ = 12 and Q₃ = 28, the IQR is 16 — the span holding the central 50%.

• Weighted Mean

  Statistics
  x̄ = Σ(w × x) / Σw

  Symbols

  Each value x carries a weight w — credits, hours or quantities.

  When to use it

  For when the values do not count equally. Averaging the values and ignoring the weights is the classic grade-point mistake.

  Worked example

  Grades of 3.7 over 4 credits, 3.0 over 3 and 4.0 over 2: (14.8 + 9 + 8) / 9 = 3.53.

  GPA Calculator

• Expected Value

  Statistics
  E(X) = Σ x × P(x)

  Symbols

  Each outcome multiplied by its probability, then totalled.

  When to use it

  The long-run average of a random process. It need not be a result that can actually occur.

  Worked example

  A fair die: (1 + 2 + 3 + 4 + 5 + 6) / 6 = 3.5, a face the die does not have.

• Complement Rule

  Statistics
  P(not A) = 1 − P(A)

  Symbols

  A is any event.

  When to use it

  Usually far easier to find the chance of something not happening and subtract, especially for questions asking about at least one.

  Worked example

  At least one six in two rolls: 1 − (5/6)² = 1 − 25/36 = 11/36, about 30.6%.

• Independent Events

  Statistics
  P(A and B) = P(A) × P(B)

  Symbols

  Valid only when one event has no effect on the other.

  When to use it

  Multiplying requires independence. Drawing a second card without replacing the first breaks it, and the answer comes out wrong.

  Worked example

  Two heads in a row: 0.5 × 0.5 = 0.25.

• Simple Interest

  Finance
  I = P × r × t

  Symbols

  P is the principal, r the rate as a decimal, t the time in years.

  When to use it

  Interest on the original sum only. Rare in practice, but the baseline that shows what compounding adds.

  Worked example

  £2,000 at 4% for 3 years: I = 2000 × 0.04 × 3 = £240.

• Compound Interest

  Finance
  A = P(1 + r/n)^(nt)

  Symbols

  P principal, r annual rate as a decimal, n compounding periods per year, t years. A is the final amount, not the interest.

  When to use it

  Subtract P from A to get the interest alone. More frequent compounding raises the total, but by less than most people expect.

  Worked example

  £2,000 at 4% compounded monthly for 3 years: A = 2000 × (1 + 0.04/12)^36 = £2,254.54, so £254.54 of interest against £240 simple.

  Loan Calculator

• Loan Repayment (EMI)

  Finance
  EMI = P × i × (1 + i)ⁿ / ((1 + i)ⁿ − 1)

  Symbols

  P is the amount borrowed, i the monthly rate (annual ÷ 12, as a decimal), n the number of monthly payments.

  When to use it

  The fixed monthly payment that clears a loan exactly. Using the annual rate for i is the mistake that makes the answer nonsense.

  Worked example

  £10,000 over 5 years at 6%: i = 0.005 and n = 60, giving EMI = £193.33 a month, £11,599.80 in total.

  EMI Calculator

• Percentage of an Amount

  Finance
  part = whole × percentage / 100

  Symbols

  The percentage as a plain number, not already divided.

  When to use it

  The most-used calculation of all. Reversible: if you know the part and the whole, percentage = part / whole × 100.

  Worked example

  15% of £240 = 240 × 15 / 100 = £36.

  Percentage Calculator

• Adding VAT

  Finance
  gross = net × (1 + rate)

  Symbols

  rate is the VAT rate as a decimal — 0.2 for 20%.

  When to use it

  Goes from a price excluding tax to the price a customer pays.

  Worked example

  £250 net at 20% VAT: 250 × 1.2 = £300 gross, of which £50 is VAT.

  VAT Calculator

• Removing VAT

  Finance
  net = gross / (1 + rate)

  Symbols

  rate as a decimal, the same as when adding.

  When to use it

  Going backwards is division, not subtraction. Taking 20% off a gross price gives the wrong answer.

  Worked example

  £300 gross at 20%: 300 / 1.2 = £250 net. Subtracting 20% would have given £240, which is £10 out.

  VAT Calculator

• Compound Annual Growth Rate

  Finance
  CAGR = (end / start)^(1/n) − 1

  Symbols

  n is the number of years between the two figures.

  When to use it

  The steady annual rate that would have produced the same result. Smooths over what actually happened in between.

  Worked example

  £5,000 growing to £8,000 over 4 years: (1.6)^0.25 − 1 = 0.1247, so 12.5% a year.

• Rule of 72

  Finance
  years to double ≈ 72 / annual percentage rate

  Symbols

  The rate as a whole number, so 6 rather than 0.06.

  When to use it

  A mental approximation, accurate to within a few months for rates between about 4% and 12%.

  Worked example

  At 6%, money doubles in roughly 72 / 6 = 12 years. The exact answer is 11.9, which is close enough to do in your head.

• Break-Even Point

  Finance
  units = fixed costs / (price − variable cost)

  Symbols

  The bracket is the contribution each sale makes towards the fixed costs.

  When to use it

  How many you must sell before anything is earned. If the price sits below the variable cost, no volume ever breaks even.

  Worked example

  £12,000 of fixed costs, selling at £25 against £10 of cost per unit: 12,000 / 15 = 800 units.

• Profit Margin

  Finance
  margin = (revenue − cost) / revenue × 100

  Symbols

  Divided by revenue, meaning the selling price.

  When to use it

  The share of the selling price that is profit, so it can never reach 100%.

  Worked example

  An item costing £180 and sold for £250: 70 / 250 = 28%.

• Markup, and Why It Is Not Margin

  Finance
  markup = (revenue − cost) / cost × 100

  Symbols

  Divided by cost, meaning what you paid.

  When to use it

  The same profit over a different denominator, so the two figures never agree. Quoting a markup as a margin overstates what you earn.

  Worked example

  That same £180 item sold at £250 is a 28% margin but a 38.9% markup — 70/250 against 70/180.

• Straight-Line Depreciation

  Finance
  annual charge = (cost − salvage value) / useful life

  Symbols

  Salvage value is what the asset is worth at the end of that life.

  When to use it

  Spreads the loss in value evenly across the years. Real assets rarely lose value evenly, which is why other methods exist.

  Worked example

  A van costing £18,000 and worth £3,000 after five years: 15,000 / 5 = £3,000 a year.

• Price After a Discount

  Finance
  final = original × (1 − discount)

  Symbols

  discount as a decimal, so 0.3 for 30% off.

  When to use it

  Two discounts do not add up. 20% then a further 10% is 0.8 × 0.9 = 0.72, which is 28% off rather than 30%.

  Worked example

  £80 with 30% off: 80 × 0.7 = £56.

  Discount Calculator

• Effective Annual Rate

  Finance
  EAR = (1 + r/n)ⁿ − 1

  Symbols

  r is the nominal annual rate, n the number of compounding periods a year.

  When to use it

  Makes two rates with different compounding frequencies comparable, which the advertised rate alone cannot do.

  Worked example

  12% compounded monthly: (1.01)¹² − 1 = 12.68%, not 12%.

• Celsius to Fahrenheit

  Measurement
  °F = °C × 9/5 + 32

  Symbols

  The 32 is an offset, not a scaling factor.

  When to use it

  Temperature is the one conversion where you cannot just multiply, because the two scales do not share a zero.

  Worked example

  20°C = 20 × 1.8 + 32 = 68°F.

  Unit Conversion Calculator

• Fahrenheit to Celsius

  Measurement
  °C = (°F − 32) × 5/9

  Symbols

  Subtract the offset first, then scale.

  When to use it

  Doing the two steps in the wrong order is the usual error, and it gets worse the further you are from freezing.

  Worked example

  95°F = (95 − 32) × 5/9 = 35°C.

  Unit Conversion Calculator

• Speed, Distance and Time

  Measurement
  speed = distance / time

  Symbols

  Units must match — miles and hours, or metres and seconds.

  When to use it

  Rearranges to distance = speed × time and time = distance / speed. Mixing minutes into an hourly speed is the usual slip.

  Worked example

  150 miles in 2 hours 30 minutes is 150 / 2.5 = 60 mph. Using 2.30 instead of 2.5 would give 65.2, which is wrong.

  Time Calculator

• Density

  Measurement
  density = mass / volume

  Symbols

  Rearranges to mass = density × volume.

  When to use it

  Converts a volume you calculated into the weight a supplier will actually quote, which is how aggregate is sold.

  Worked example

  2.5 m³ of gravel at 1,600 kg/m³ weighs 4,000 kg — four tonnes.

  Gravel Calculator

• Cubic Feet to Cubic Yards

  Measurement
  cubic yards = cubic feet / 27

  Symbols

  A yard is 3 feet, and 3³ = 27.

  When to use it

  Dividing by 3 instead of 27 is the error that turns a small order into a very large one.

  Worked example

  540 cubic feet of topsoil is 540 / 27 = 20 cubic yards.

  Gravel Calculator

• Mixed Number to Decimal

  Measurement
  a b/c = a + (b ÷ c)

  Symbols

  a is the whole part, b/c the fraction.

  When to use it

  Tape measures read in fractions while calculators want decimals, so this sits between measuring and ordering.

  Worked example

  3 5/8 inches = 3 + 0.625 = 3.625 inches.

  Mixed Number Calculator

• Miles and Kilometres

  Measurement
  km = miles × 1.609344

  Symbols

  Exact by definition rather than an approximation.

  When to use it

  Reverse it by dividing. The mental version — multiply by 1.6, or take five eighths coming back — is close enough for a road sign.

  Worked example

  60 miles is 96.56 km, and 100 km is 62.14 miles.

  Unit Conversion Calculator

• Kilograms and Pounds

  Measurement
  lb = kg × 2.20462

  Symbols

  One pound is exactly 0.45359237 kg.

  When to use it

  Doubling is the usual shortcut in your head, and it runs about 10% light every time.

  Worked example

  70 kg is 154.32 lb. Doubling would have said 140, which is 14 lb short.

  Unit Conversion Calculator

• Litres and Gallons

  Measurement
  US gallons = litres / 3.785411784

  Symbols

  An imperial gallon is a different size again, at 4.54609 litres.

  When to use it

  The two gallons differ by about a fifth, so a figure quoted without saying which one is not usable.

  Worked example

  50 litres is 13.21 US gallons, or 11.00 imperial gallons.

  Unit Conversion Calculator

• Square Metres and Square Feet

  Measurement
  ft² = m² × 10.7639

  Symbols

  It is the length factor squared: 3.28084² = 10.7639.

  When to use it

  Area factors are always the length factor squared, which is why using 3.28 here leaves the answer more than three times too small.

  Worked example

  25 m² is 269.10 ft².

  Unit Conversion Calculator

• MPG and Litres per 100 km

  Measurement
  L/100km = 282.481 / mpg (imperial)

  Symbols

  Use 235.215 instead for US mpg, whose gallon is smaller.

  When to use it

  These are reciprocals rather than a scale, so the same step in mpg saves very different amounts of fuel at different ends of the range.

  Worked example

  40 mpg is 7.06 L/100km. Moving from 20 to 25 mpg saves 2.82 L/100km, while 40 to 45 saves only 0.78.

  Fuel Cost Calculator