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Formulas & Calculation Guide

Complete mathematical reference with every formula and calculation method rendered as displayed LaTeX equations.

Fractions Statistics Probability Algebra & Quadratic Matrices Percentages Geometry Physics Financial Trig & Logarithms
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1. Fraction Formulas

Formulas for adding, subtracting, multiplying, dividing, and simplifying fractions.

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Fraction Addition

$$ \frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd} $$

How to Calculate:

1. Cross-multiply numerators by opposite denominators:

$$ \text{Numerator} = (a \times d) + (b \times c) $$

2. Multiply denominators together:

$$ \text{Denominator} = b \times d $$

3. Reduce by dividing both by $\gcd(ad + bc, bd)$.

Fraction Subtraction

$$ \frac{a}{b} - \frac{c}{d} = \frac{ad - bc}{bd} $$

How to Calculate:

1. Cross-multiply and subtract products:

$$ \text{Numerator} = (a \times d) - (b \times c) $$

2. Common denominator:

$$ \text{Denominator} = b \times d $$

Fraction Multiplication

$$ \frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d} $$

How to Calculate:

$$ \text{Result} = \frac{ac}{\gcd(ac, bd)} \div \frac{bd}{\gcd(ac, bd)} $$

Fraction Division

$$ \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{a \times d}{b \times c} $$

How to Calculate:

1. Multiply by the reciprocal of the divisor:

$$ \left(\frac{c}{d}\right)^{-1} = \frac{d}{c} \implies \text{Result} = \frac{ad}{bc} $$
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2. Statistics Formulas

Formulas for central tendency, variance, standard deviation, and quartiles.

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Arithmetic Mean (Average)

$$ \bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i = \frac{x_1 + x_2 + \dots + x_n}{n} $$

How to Calculate:

$$ \text{Sum} = \sum_{i=1}^{n} x_i, \quad \bar{x} = \frac{\text{Sum}}{n} $$

Sample Standard Deviation ($s$)

$$ s = \sqrt{\frac{1}{n - 1} \sum_{i=1}^{n} (x_i - \bar{x})^2} $$

How to Calculate:

$$ s^2 = \frac{\sum (x_i - \bar{x})^2}{n - 1}, \quad s = \sqrt{s^2} $$

Population Standard Deviation ($\sigma$)

$$ \sigma = \sqrt{\frac{1}{N} \sum_{i=1}^{N} (x_i - \mu)^2} $$

How to Calculate:

$$ \sigma^2 = \frac{\sum (x_i - \mu)^2}{N}, \quad \sigma = \sqrt{\sigma^2} $$

Interquartile Range & MAD

$$ IQR = Q_3 - Q_1 $$ $$ MAD = \frac{1}{n} \sum_{i=1}^{n} |x_i - \bar{x}| $$

How to Calculate:

$$ Q_1 = \text{25th Percentile}, \quad Q_3 = \text{75th Percentile} $$
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3. Probability & Combinatorics

Permutations, combinations, factorials, and binomial distribution formulas.

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Permutations ($nPr$)

$$ P(n, r) = \frac{n!}{(n - r)!} = \prod_{i=0}^{r-1} (n - i) $$

How to Calculate:

$$ P(n, r) = n \times (n - 1) \times \dots \times (n - r + 1) $$

Combinations ($nCr$)

$$ \binom{n}{r} = C(n, r) = \frac{n!}{r!(n - r)!} $$

How to Calculate:

$$ C(n, r) = \frac{P(n, r)}{r!} = \frac{\prod_{i=1}^{r}(n - i + 1)}{\prod_{i=1}^{r} i} $$

Binomial Distribution Formulas

Exact Probability PMF $$ P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k} $$
Cumulative Probability CDF $$ P(X \le k) = \sum_{i=0}^{k} \binom{n}{i} p^i (1 - p)^{n - i} $$

Mean and Variance:

$$ \mu = np, \quad \sigma^2 = np(1 - p), \quad \sigma = \sqrt{np(1 - p)} $$
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4. Algebra & Quadratic Equations

Quadratic formula, discriminant, vertex, and linear systems.

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The Quadratic Formula

$$ ax^2 + bx + c = 0 \implies x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$

How to Calculate Discriminant:

$$ \Delta = b^2 - 4ac $$

If $\Delta > 0$: two real roots; $\Delta = 0$: one real double root; $\Delta < 0$: two complex conjugate roots.

Parabola Vertex & Axis of Symmetry

$$ x_{\text{vertex}} = -\frac{b}{2a}, \quad y_{\text{vertex}} = c - \frac{b^2}{4a} $$

Axis of Symmetry:

$$ x = -\frac{b}{2a} $$

Simultaneous Equations (Cramer's Rule $2 \times 2$)

$$ \begin{cases} a_1 x + b_1 y = c_1 \\ a_2 x + b_2 y = c_2 \end{cases} \implies x = \frac{c_1 b_2 - c_2 b_1}{a_1 b_2 - a_2 b_1}, \quad y = \frac{a_1 c_2 - a_2 c_1}{a_1 b_2 - a_2 b_1} $$

How to Calculate Determinants:

$$ D = a_1 b_2 - a_2 b_1, \quad D_x = c_1 b_2 - c_2 b_1, \quad D_y = a_1 c_2 - a_2 c_1 \implies x = \frac{D_x}{D}, \; y = \frac{D_y}{D} $$
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5. Matrix Operations

Determinants, inverses, products, transposes, and traces.

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Determinant of $2 \times 2$ Matrix

$$ \det(A) = \begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc $$

How to Calculate:

Subtract secondary diagonal from main diagonal product.

Inverse of $2 \times 2$ Matrix

$$ A^{-1} = \frac{1}{ad - bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix} $$

How to Calculate:

Swap diagonal entries $(a \leftrightarrow d)$, negate off-diagonals $(-b, -c)$, divide by $\det(A)$.

Determinant of $3 \times 3$ Matrix

$$ \det(A) = a_{11}(a_{22}a_{33} - a_{23}a_{32}) - a_{12}(a_{21}a_{33} - a_{23}a_{31}) + a_{13}(a_{21}a_{32} - a_{22}a_{31}) $$

Matrix Multiplication ($C = AB$):

$$ C_{ij} = \sum_{k=1}^{p} A_{ik} B_{kj} $$
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6. Percentage Formulas

Percentage change, discount, markup, and tax formulas.

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Percentage of a Number

$$ \text{Value} = \left(\frac{P}{100}\right) \times X $$

What % is $X$ of $Y$:

$$ P = \left(\frac{X}{Y}\right) \times 100\% $$

Percentage Change (Increase / Decrease)

$$ \Delta\% = \left(\frac{V_{\text{new}} - V_{\text{old}}}{|V_{\text{old}}|}\right) \times 100\% $$

Discount Price:

$$ P_{\text{final}} = P_{\text{original}} \times \left(1 - \frac{\text{Discount}\%}{100}\right) $$
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7. Geometry Formulas (2D & 3D)

Area, perimeter, surface area, and volume formulas.

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Circle

$$ A = \pi r^2 $$ $$ C = 2\pi r = \pi d $$

Area is pi times radius squared; circumference is 2 times pi times radius.

Triangle

$$ A = \frac{1}{2} b h $$ $$ P = a + b + c $$

Half base times height; perimeter is sum of all three sides.

Rectangle

$$ A = l \times w, \quad P = 2(l + w) $$ $$ d = \sqrt{l^2 + w^2} $$

Cylinder

$$ V = \pi r^2 h $$ $$ A_{\text{total}} = 2\pi r(r + h) $$

Sphere

$$ V = \frac{4}{3} \pi r^3 $$ $$ A = 4 \pi r^2 $$

Cone

$$ V = \frac{1}{3} \pi r^2 h $$ $$ A = \pi r (r + \sqrt{r^2 + h^2}) $$
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8. Physics Formulas

Mechanics, energy, dynamics, and electricity equations.

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Kinematics

$$ v = \frac{d}{t}, \quad a = \frac{v - v_0}{t} $$ $$ d = v_0 t + \frac{1}{2} a t^2 $$

Dynamics & Force

$$ F = m \times a $$ $$ p = m \times v $$

Energy & Work

$$ E_k = \frac{1}{2} m v^2, \quad E_p = m g h $$ $$ W = F \times d \times \cos(\theta) $$

Ohm's Law & Power

$$ V = I \times R $$ $$ P = V \times I = I^2 R = \frac{V^2}{R} $$

Density & Pressure

$$ \rho = \frac{m}{V}, \quad P = \frac{F}{A} $$

Wave & Frequency

$$ v = f \times \lambda, \quad T = \frac{1}{f} $$
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9. Financial Formulas

Compound interest, loan EMI payments, and CAGR growth rates.

Compound Interest

$$ A = P \left(1 + \frac{r}{n}\right)^{nt} $$ $$ I = A - P $$

How to Calculate:

Divide annual rate $r$ by compounding frequency $n$, raise $(1 + r/n)$ to $nt$, and multiply by principal $P$.

Loan EMI Formula

$$ \text{EMI} = \frac{P \times r \times (1 + r)^n}{(1 + r)^n - 1} $$

How to Calculate:

Here $r$ is monthly interest rate and $n$ is total months. Total payment is $\text{EMI} \times n$.

CAGR Growth Rate

$$ \text{CAGR} = \left(\frac{V_{\text{final}}}{V_{\text{initial}}}\right)^{\frac{1}{t}} - 1 $$

How to Calculate:

Divide ending value by starting value, raise to exponent $1/t$, subtract 1, and multiply by 100%.

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10. Trigonometry & Advanced Scientific Math

Circular functions, hyperbolic identities, and logarithm bases.

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Angle Modes

$$ \text{rad} = \text{deg} \times \frac{\pi}{180^\circ} $$ $$ \text{deg} = \text{rad} \times \frac{180^\circ}{\pi} $$

Hyperbolic Identities

$$ \sinh(x) = \frac{e^x - e^{-x}}{2} $$ $$ \cosh(x) = \frac{e^x + e^{-x}}{2} $$

Logarithm Change of Base

$$ \log_b(x) = \frac{\ln(x)}{\ln(b)} = \frac{\log_{10}(x)}{\log_{10}(b)} $$