Math Formulas: Complete List by Topic, With Examples

Grades 6 to 8

area and perimeter of rectangles, triangles and circles, simple interest, percent change, laws of exponents, $(a + b)^{2}$ and $(a - b)^{2}$

every later chapter reuses them

Grades 9 to 10

Pythagorean theorem, quadratic formula, distance and midpoint formulas, surface area and volume, AP $n$th term and sum, basic trig ratios, mean of grouped data

board exams test them directly

Grades 11 to 12

trig identities, binomial theorem, $^{n}C_{r}$, limits and derivatives, standard integrals, conic sections, probability rules

entrance exams such as JEE and the SAT build on them

Brackets or parentheses first

$2 \times (3 + 4) = 2 \times 7 = 14$

Simplify whatever is inside brackets before anything else, starting with the innermost pair and working outward.

Nested brackets

$\lbrace 2 + [3 \times (5 - 1)]\rbrace = \lbrace 2 + 12\rbrace = 14$

Clear round brackets ( ) first, then square brackets [ ], then curly braces { }, from the inside out.

Orders or exponents

$3 + 2^{3} = 3 + 8 = 11$

Powers and roots come next. BODMAS calls them Orders and PEMDAS calls them Exponents.

"Of" means multiply

$\frac{1}{2} \text{ of } 10 = \frac{1}{2} \times 10 = 5$

The word "of" between a fraction or percent and a number means multiplication, as in half of 10.

Division and multiplication, left to right

$12 \div 3 \times 2 = 4 \times 2 = 8$

These two share the same rank, so work them in the order they appear from left to right, not division first.

Addition and subtraction, left to right

$10 - 4 + 3 = 6 + 3 = 9$

These also share one rank and are done last, again strictly from left to right.

Fraction bar as a bracket

$\frac{6 + 4}{7 - 2} = \frac{10}{5} = 2$

A fraction bar groups its numerator and its denominator, so simplify each one fully before dividing.

Minus sign and powers

$-3^{2} = -9$

The exponent binds only to the 3, so square first and apply the minus after. Compare $(-3)^{2} = 9$.

Adding like fractions

$\frac{a}{c} + \frac{b}{c} = \frac{a + b}{c}$

When the denominators are equal, add the numerators and keep the common denominator $c$, where $c \neq 0$.

Subtracting like fractions

$\frac{a}{c} - \frac{b}{c} = \frac{a - b}{c}$

Subtract the numerators and keep the common denominator $c$, with $c \neq 0$.

Adding unlike fractions

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

Rewrite both over the common denominator $bd$, with $b, d \neq 0$. Using the LCM of $b$ and $d$ keeps numbers smaller.

Subtracting unlike fractions

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

Same method as addition, subtract the cross products over $bd$, then reduce to lowest terms.

Multiplying fractions

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

Multiply numerator by numerator and denominator by denominator, with $b, d \neq 0$. Cancel common factors first to save work.

Dividing fractions

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

Keep the first fraction, flip the second and multiply. Needs $b, c, d \neq 0$ because you cannot divide by zero.

Reciprocal of a fraction

$\frac{a}{b} \to \frac{b}{a}$

Swap numerator and denominator, valid when $a \neq 0$. A number times its reciprocal always equals 1.

Mixed number to improper fraction

$a\frac{b}{c} = \frac{ac + b}{c}$

Multiply the whole number $a$ by the denominator $c$, add the numerator $b$ and keep $c$ as the denominator.

Improper fraction to mixed number

$\frac{p}{q} = Q\frac{R}{q}$

Divide $p$ by $q$. The quotient $Q$ is the whole part and the remainder $R$ is the new numerator over $q$.

Equivalent fractions

$\frac{a}{b} = \frac{a \times k}{b \times k}$

Multiplying or dividing top and bottom by the same nonzero number $k$ gives the same value in a different form.

Simplest form

$\frac{a}{b} = \frac{a \div h}{b \div h}$

Divide numerator and denominator by $h$, the HCF of $a$ and $b$, to reduce the fraction to lowest terms.

Comparing fractions

$\frac{a}{b} > \frac{c}{d} \iff ad > bc$

Cross-multiply to compare two fractions without finding a common denominator. Valid when $b$ and $d$ are both positive.

Fraction of a quantity

$\frac{a}{b} \text{ of } N = \frac{a \times N}{b}$

To find a fraction of an amount $N$, multiply $N$ by the numerator and divide by the denominator.

Percent as a fraction

$p$

Percent means "per hundred", so $p$ percent is $p$ parts out of 100. For example $35$.

Fraction to percent

$\frac{a}{b} = \left(\frac{a}{b} \times 100\right)$

Multiply a fraction by 100 to express it as a percent, with $b \neq 0$.

Decimal to percent

$d = (100d)$

Move the decimal point two places to the right and add the percent sign, so $0.075 = 7.5$.

Terminating decimal to fraction

$0.d_{1}d_{2}\ldots d_{k} = \frac{d_{1}d_{2}\ldots d_{k}}{10^{k}}$

Write the $k$ digits after the point over $10^{k}$, then simplify. For example $0.125 = \frac{125}{1000} = \frac{1}{8}$.

Pure recurring decimal to fraction

$0.\overline{d_{1}d_{2}\ldots d_{k}} = \frac{d_{1}d_{2}\ldots d_{k}}{10^{k} - 1}$

A block of $k$ digits that repeats forever equals that block over $k$ nines, so $0.\overline{7} = \frac{7}{9}$.

Mixed recurring decimal to fraction

$0.a\overline{b} = \frac{ab - a}{90}$

Here $ab$ is the two-digit number formed by the digits. One non-repeating digit $a$ is followed by a repeating digit $b$, so $0.1\overline{6} = \frac{15}{90} = \frac{1}{6}$.

Multiplying by a power of 10

$x \times 10^{n}$

Shift the decimal point of $x$ by $n$ places to the right. Dividing by $10^{n}$ shifts it $n$ places to the left.

Decimal places in a product

$\text{places in } (x \times y) = \text{places in } x + \text{places in } y$

Multiply as whole numbers, then place the point so the answer has as many decimal places as both factors together.

Scientific notation

$N = a \times 10^{n}$

Write any number with $1 \leq a < 10$ and integer $n$. Large numbers get positive $n$ and small numbers negative $n$.

Percent of a number

$p$

Convert the percent to a fraction of 100 and multiply by the amount $N$.

What percent A is of B

$\frac{A}{B} \times 100$

Gives $A$ as a percent of the whole $B$, with $B \neq 0$. Used for marks, shares and proportions of a total.

Percentage change

$\frac{\text{new} - \text{old}}{\text{old}} \times 100$

A positive result is an increase and a negative result is a decrease. Always divide by the original (old) value.

Percentage increase

$\frac{\text{increase}}{\text{original}} \times 100$

The rise in value as a percent of the starting value, where increase equals new minus original.

Percentage decrease

$\frac{\text{decrease}}{\text{original}} \times 100$

The fall in value as a percent of the starting value, where decrease equals original minus new.

Increasing by p%

$N \times \left(1 + \frac{p}{100}\right)$

Multiplies $N$ by a single factor to give the value after a $p$ percent rise, so a 20 percent rise means times 1.2.

Decreasing by p%

$N \times \left(1 - \frac{p}{100}\right)$

Gives the value after a $p$ percent fall, so a 15 percent fall means multiplying by 0.85.

Reverse percentage after an increase

$\text{original} = \frac{\text{final}}{1 + \frac{p}{100}}$

Finds the starting value when you know the value after a $p$ percent increase. Divide, do not subtract $p$ percent of the final value.

Reverse percentage after a decrease

$\text{original} = \frac{\text{final}}{1 - \frac{p}{100}}$

Finds the starting value from the value after a $p$ percent decrease, with $p < 100$.

Two successive percentage changes

$\text{net change} = a + b + \frac{ab}{100}$

Net percent change after changes of $a$ percent then $b$ percent. Write a decrease as a negative number.

Percent error

$\frac{\lvert \text{measured} - \text{actual} \rvert}{\text{actual}} \times 100$

How far a measured or estimated value is from the true value, as a percent of the true value.

Ratio formula

$a : b = \frac{a}{b}$

Compares two quantities in the same unit by division, read "a to b", with $b \neq 0$.

Simplest form of a ratio

$a : b = \frac{a}{h} : \frac{b}{h}$

Divide both terms by $h$, their HCF, so the terms have no common factor other than 1.

Proportion and cross-multiplication

$a : b = c : d \iff ad = bc$

Two ratios are equal when the product of the extremes $a, d$ equals the product of the means $b, c$.

Fourth proportional

$x = \frac{bc}{a}$

The value $x$ that completes $a : b = c : x$, found by cross-multiplication with $a \neq 0$.

Third proportional

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

The value $x$ in $a : b = b : x$, where the middle term is repeated.

Mean proportional

$x = \sqrt{ab}$

The value $x$ in $a : x = x : b$ for positive $a$ and $b$. It is the geometric mean of the two numbers.

Direct proportion

$\frac{y_{1}}{x_{1}} = \frac{y_{2}}{x_{2}}$

When $y = kx$, both quantities rise or fall together and their ratio stays fixed at the constant $k$.

Inverse proportion

$x_{1}y_{1} = x_{2}y_{2}$

When $y = \frac{k}{x}$, one quantity rises as the other falls and their product stays fixed. Typical case: workers and days.

Dividing N in the ratio a : b

$\frac{a}{a + b} \times N$ and $\frac{b}{a + b} \times N$

Each share is its ratio part over the total number of parts, multiplied by the amount $N$ being shared.

Dividing N in the ratio a : b : c

$\frac{a}{a + b + c} \times N$

The first share of three. Replace the numerator by $b$ or $c$ for the other two shares.

Combining two ratios

$a : b : c = pr : qr : qs$

Joins $a : b = p : q$ and $b : c = r : s$ by making the shared term $b$ equal in both.

Compound ratio

$(a : b) \times (c : d) = ac : bd$

Multiply the first terms together and the second terms together to combine two ratios into one.

Componendo and dividendo

$\frac{a + b}{a - b} = \frac{c + d}{c - d}$

Follows from $\frac{a}{b} = \frac{c}{d}$ when $a \neq b$ and $c \neq d$. Useful for solving ratio equations quickly.

Average formula (arithmetic mean)

$\bar{x} = \frac{x_{1} + x_{2} + \cdots + x_{n}}{n}$

Add all $n$ values and divide by how many there are. This is the everyday meaning of "average".

Total from the average

$\text{sum} = \bar{x} \times n$

Rearranged mean formula. Most average word problems are solved by converting averages back to totals first.

Weighted average

$\bar{x}{w} = \frac{\sum{i=1}^{n} w_{i}x_{i}}{\sum_{i=1}^{n} w_{i}}$

Each value $x_{i}$ counts $w_{i}$ times, for example credits or marks weightage. Weights must not sum to zero.

Combined average of two groups

$\bar{x} = \frac{n_{1}\bar{x}{1} + n{2}\bar{x}{2}}{n{1} + n_{2}}$

Merges groups of sizes $n_{1}$ and $n_{2}$ with means $\bar{x}{1}$and$\bar{x}{2}$. Do not simply average the two means.

Average of equally spaced numbers

$\bar{x} = \frac{\text{first} + \text{last}}{2}$

For any list with a constant gap, such as consecutive integers or an AP, the mean is the midpoint of the ends.

Average of the first n natural numbers

$\frac{n + 1}{2}$

The mean of $1, 2, \ldots, n$. For 1 to 100 it is 50.5.

Average of the first n even numbers

$n + 1$

The mean of $2, 4, \ldots, 2n$. For the first 10 even numbers it is 11.

Average of the first n odd numbers

$n$

The mean of the odd numbers $1, 3, \ldots, 2n - 1$ is simply $n$, so the first 8 odd numbers average 8.

New average after adding one value

$\bar{x}_{\text{new}} = \frac{n\bar{x} + a}{n + 1}$

When a value $a$ joins $n$ values with mean $\bar{x}$, rebuild the total and divide by the new count.

New average after replacing one value

$\bar{x}_{\text{new}} = \bar{x} + \frac{\text{new} - \text{old}}{n}$

Swapping one of $n$ values shifts the mean by the difference spread over all $n$ values.

HCF by prime factorisation

$\text{HCF} = \prod p_{i}^{\min(a_{i}, b_{i})}$

Write both numbers as products of primes $p_{i}$ and multiply each common prime raised to its lowest power.

LCM by prime factorisation

$\text{LCM} = \prod p_{i}^{\max(a_{i}, b_{i})}$

Multiply every prime that appears in either number, each raised to its highest power.

HCF times LCM

$\text{HCF}(a, b) \times \text{LCM}(a, b) = a \times b$

True for any pair of positive integers $a$ and $b$. It does not hold for three or more numbers in general.

LCM from the HCF

$\text{LCM}(a, b) = \frac{a \times b}{\text{HCF}(a, b)}$

The quickest way to find an LCM when the HCF is known or easy to find.

Co-prime numbers

$\text{HCF}(a, b) = 1 \Rightarrow \text{LCM}(a, b) = ab$

Two numbers with no common factor except 1 have their product as their LCM, for example 8 and 15.

Euclid's division algorithm

$\text{HCF}(a, b) = \text{HCF}(b, r)$

Here $r$ is the remainder when $a$ is divided by $b$. Repeat until the remainder is 0, and the last divisor is the HCF.

HCF of fractions

$\text{HCF}\left(\frac{a}{b}, \frac{c}{d}\right) = \frac{\text{HCF}(a, c)}{\text{LCM}(b, d)}$

For fractions in lowest terms, take the HCF of the numerators over the LCM of the denominators.

LCM of fractions

$\text{LCM}\left(\frac{a}{b}, \frac{c}{d}\right) = \frac{\text{LCM}(a, c)}{\text{HCF}(b, d)}$

For fractions in lowest terms, take the LCM of the numerators over the HCF of the denominators.

Greatest common factor divides the difference

$\text{HCF}(a, b) \mid (a - b)$

Any common factor of two numbers also divides their difference, which helps narrow down the HCF quickly.

Prime factorisation

$N = p_{1}^{a_{1}} p_{2}^{a_{2}} \cdots p_{k}^{a_{k}}$

Every integer greater than 1 can be written as a product of prime powers in exactly one way, apart from order.

Number of factors

$d(N) = (a_{1} + 1)(a_{2} + 1)\cdots(a_{k} + 1)$

Add 1 to each exponent and multiply. The count includes 1 and $N$ itself.

Sum of factors

$\sigma(N) = \frac{p_{1}^{a_{1}+1} - 1}{p_{1} - 1} \times \cdots \times \frac{p_{k}^{a_{k}+1} - 1}{p_{k} - 1}$

Multiply one geometric-series sum per prime. The total includes 1 and $N$.

Product of all factors

$N^{d(N)/2}$

Factors pair up to multiply to $N$, so the product of all $d(N)$ factors is $N$ raised to half that count.

Number of odd factors

$(a_{2} + 1)(a_{3} + 1)\cdots(a_{k} + 1)$

For even $N$, take $p_{1} = 2$ and ignore its exponent, since odd factors cannot contain the prime 2. If $N$ is odd, every factor is odd.

Number of even factors

$d(N) - \text{odd factors}$

All factors minus the odd ones. Equivalently $a_{1}(a_{2} + 1)\cdots(a_{k} + 1)$ when $p_{1} = 2$.

Factor count of a perfect square

$d(N)$ is odd

A number has an odd number of factors exactly when it is a perfect square, because its square root pairs with itself.

Divisibility by 2

$2 \mid N \iff d_{0} \in \lbrace 0, 2, 4, 6, 8\rbrace$

A number is divisible by 2 when its last digit is even.

Divisibility by 3

$3 \mid N \iff 3 \mid (\text{sum of digits})$

Add all the digits. If that sum is a multiple of 3, so is the number.

Divisibility by 4

$4 \mid N \iff 4 \mid (\text{last two digits})$

Only the number formed by the tens and units digits matters, because 100 is a multiple of 4.

Divisibility by 5

$5 \mid N \iff d_{0} \in \lbrace 0, 5\rbrace$

A number is divisible by 5 when it ends in 0 or 5.

Divisibility by 6

$6 \mid N \iff 2 \mid N \text{ and } 3 \mid N$

The number must pass both the test for 2 and the test for 3, since 2 and 3 are co-prime.

Divisibility by 7

$7 \mid N \iff 7 \mid (\text{rest} - 2d_{0})$

Remove the units digit, double it and subtract from the remaining number. Repeat until the result is small.

Divisibility by 8

$8 \mid N \iff 8 \mid (\text{last three digits})$

Only the last three digits matter, because 1000 is a multiple of 8.

Divisibility by 9

$9 \mid N \iff 9 \mid (\text{sum of digits})$

If the digit sum is a multiple of 9, the number is too. The remainder on division by 9 equals the digit-sum remainder.

Divisibility by 10

$10 \mid N \iff d_{0} = 0$

A number is divisible by 10 when its last digit is 0.

Divisibility by 11

$11 \mid N \iff 11 \mid (S_{\text{odd}} - S_{\text{even}})$

Sum the digits in odd places and in even places counting from the right. A difference of 0 or a multiple of 11 passes.

Divisibility by 12

$12 \mid N \iff 3 \mid N \text{ and } 4 \mid N$

The number must pass the tests for both 3 and 4.

Divisibility by 25

$25 \mid N \iff \text{last two digits} \in \lbrace 00, 25, 50, 75\rbrace$

Only the last two digits matter, because 100 is a multiple of 25.

Co-prime divisors rule

$a \mid N \text{ and } b \mid N \Rightarrow ab \mid N$

Holds when $\text{HCF}(a, b) = 1$. This is why tests for 6, 12, 15 and 18 combine two simpler tests.

Square of a number

$n^{2} = n \times n$

A number multiplied by itself. The square of any real number is never negative.

Perfect squares

$N = k^{2}$

$N$ is a perfect square when $k$ is an integer. Perfect squares end in 0, 1, 4, 5, 6 or 9, never in 2, 3, 7 or 8.

Cube of a number

$n^{3} = n \times n \times n$

A number used three times as a factor. The cube keeps the sign of the number, so $(-2)^{3} = -8$.

Perfect cube

$N = k^{3}$

$N$ is a perfect cube when $k$ is an integer, as in 1, 8, 27, 64 and 125.

Square root

$\sqrt{N} = k \iff k^{2} = N$

The principal square root is the non-negative value $k$, defined for real numbers only when $N \geq 0$.

Cube root

$\sqrt[3]{N} = k \iff k^{3} = N$

Defined for every real $N$, including negatives, so $\sqrt[3]{-27} = -3$.

Square root of a square

$\sqrt{x^{2}} = \lvert x \rvert$

The result is the absolute value of $x$, since a principal square root is never negative.

Product rule for roots

$\sqrt{ab} = \sqrt{a} \times \sqrt{b}$

Valid for $a \geq 0$ and $b \geq 0$. Used to simplify roots by pulling out perfect-square factors.

Quotient rule for roots

$\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$

Valid for $a \geq 0$ and $b > 0$. The root of a fraction is the root of the top over the root of the bottom.

Simplifying a square root

$\sqrt{a^{2}b} = a\sqrt{b}$

For $a \geq 0$ and $b \geq 0$, move a perfect-square factor $a^{2}$ outside the root as $a$.

Squaring a number ending in 5

$(10a + 5)^{2} = 100a(a + 1) + 25$

Multiply the leading part $a$ by the next integer and write 25 after it, so $35^{2} = 1225$.

Gap between consecutive squares

$(n + 1)^{2} - n^{2} = 2n + 1$

Consecutive squares differ by consecutive odd numbers, so $n^{2}$ to $(n + 1)^{2}$ always steps by $2n + 1$.

Pythagorean triples

$(m^{2} - n^{2}, \ 2mn, \ m^{2} + n^{2})$

For integers $m > n > 0$ these three numbers always satisfy $a^{2} + b^{2} = c^{2}$. $m = 2, n = 1$ gives 3, 4, 5.

Sum of natural numbers formula

$1 + 2 + 3 + \cdots + n = \frac{n(n + 1)}{2}$

Adds the first $n$ counting numbers. Pairing the first and last terms gives $n$ halves of $n + 1$.

Sum of even numbers formula

$2 + 4 + 6 + \cdots + 2n = n(n + 1)$

Adds the first $n$ even numbers. It is twice the sum of the first $n$ natural numbers.

Sum of odd numbers

$1 + 3 + 5 + \cdots + (2n - 1) = n^{2}$

The first $n$ odd numbers always add to a perfect square, for example $1 + 3 + 5 = 9$.

Sum of squares

$1^{2} + 2^{2} + \cdots + n^{2} = \frac{n(n + 1)(2n + 1)}{6}$

Adds the squares of the first $n$ natural numbers. The answer is always a whole number.

Sum of cubes of n natural numbers

$1^{3} + 2^{3} + \cdots + n^{3} = \left[\frac{n(n + 1)}{2}\right]^{2}$

The sum of the first $n$ cubes equals the square of the sum of the first $n$ natural numbers.

Sum of squares of the first n even numbers

$2^{2} + 4^{2} + \cdots + (2n)^{2} = \frac{2n(n + 1)(2n + 1)}{3}$

Four times the sum of the first $n$ squares, because each term is $4k^{2}$.

Sum of squares of the first n odd numbers

$1^{2} + 3^{2} + \cdots + (2n - 1)^{2} = \frac{n(2n - 1)(2n + 1)}{3}$

Adds the squares of the first $n$ odd numbers, for example $1 + 9 + 25 = 35$ when $n = 3$.

Sum of consecutive integers from a to b

$a + (a + 1) + \cdots + b = \frac{(b - a + 1)(a + b)}{2}$

Number of terms times the average of the first and last terms, for integers $a \leq b$.

Sum of multiples of k up to N

$k \times \frac{m(m + 1)}{2}$

Here $m = \left\lfloor \frac{N}{k} \right\rfloor$ is the number of multiples of $k$ from 1 to $N$.

Dividend, divisor, quotient and remainder

$a = bq + r, \quad 0 \leq r < b$

Dividend $a$ equals divisor $b$ times quotient $q$ plus remainder $r$, for integers with $b > 0$. The remainder is always smaller than the divisor.

Finding the divisor

$b = \frac{a - r}{q}$

Rearranges the division algorithm when the dividend, quotient and remainder are known, with $q \neq 0$.

Finding the quotient

$q = \left\lfloor \frac{a}{b} \right\rfloor$

The quotient is the whole-number part of $a \div b$, found by rounding down, for positive $a$ and $b$.

Exact division

$r = 0 \iff b \mid a$

A zero remainder means $b$ is a factor of $a$ and $a$ is a multiple of $b$.

Same remainder test

$a \equiv c \pmod{m} \iff m \mid (a - c)$

Two integers leave the same remainder on division by $m$ exactly when $m$ divides their difference.

Remainder of a sum

$(a + c) \bmod m = [(a \bmod m) + (c \bmod m)] \bmod m$

Find each remainder first, add them, then take the remainder again. Saves work with large numbers.

Remainder of a product

$(a \times c) \bmod m = [(a \bmod m) \times (c \bmod m)] \bmod m$

Multiply the separate remainders and reduce again. Used to find remainders of large products and powers.

Division involving zero

$\frac{0}{a} = 0$

Zero divided by any nonzero $a$ is 0, but $\frac{a}{0}$ is undefined for every number $a$.

Integers from a to b, both included

$b - a + 1$

Counts every integer in the closed range from $a$ to $b$, where $a \leq b$. From 15 to 40 there are 26.

Integers from a to b, both excluded

$b - a - 1$

Counts only the integers strictly between $a$ and $b$, where $a < b$.

Integers from a to b, one end included

$b - a$

Counts the integers when exactly one of the two endpoints is part of the range.

Multiples of k from 1 to N

$\left\lfloor \frac{N}{k} \right\rfloor$

Divide $N$ by $k$ and drop the decimal part, for positive integers $N$ and $k$.

Multiples of k from a to b

$\left\lfloor \frac{b}{k} \right\rfloor - \left\lfloor \frac{a - 1}{k} \right\rfloor$

Multiples up to $b$ minus multiples below $a$, for positive integers $a \leq b$.

Terms in an equally spaced list

$\frac{\text{last} - \text{first}}{d} + 1$

Counts the numbers in a list with constant gap $d$, such as $7, 11, 15, \ldots, 99$.

Even number

$2k$

Any integer of the form $2k$, where $k$ is an integer, is even. Zero is even.

Odd numbers

$2k + 1$

Any integer of the form $2k + 1$, where $k$ is an integer, is odd. It leaves remainder 1 when divided by 2.

Sum of two evens or two odds

$2m + 2n = 2(m + n)$

Even plus even is even, and odd plus odd is also even, since $(2m + 1) + (2n + 1) = 2(m + n + 1)$.

Sum of an even and an odd

$2m + (2n + 1) = 2(m + n) + 1$

Adding or subtracting an even and an odd number always gives an odd number.

Products of evens and odds

$(2m + 1)(2n + 1) = 2(2mn + m + n) + 1$

Odd times odd is odd. If at least one factor is even, the product is even.

Consecutive integers

$n, \ n + 1, \ n + 2, \ \ldots$

Integers one apart. Consecutive even or odd integers step by 2, as in $n, \ n + 2, \ n + 4$.

Product of consecutive integers

$n(n + 1)$ is even

Any two consecutive integers include an even number, and any three consecutive integers have a product divisible by 6.

Simple interest

$SI = \frac{P \times R \times T}{100}$

$P$ is the principal, $R$ the rate in percent per year and $T$ the time in years. Interest is charged only on the original principal.

Amount

$A = P + SI$

The total repaid or received at the end of the period is the principal plus the interest.

Amount in one step

$A = P\left(1 + \frac{RT}{100}\right)$

Gives the final amount directly without finding the interest first. $P$, $R$ and $T$ mean the same as above.

Principal

$P = \frac{100 \times SI}{R \times T}$

Rearranged form used when the interest, rate and time are known and the sum lent is unknown.

Rate

$R = \frac{100 \times SI}{P \times T}$

Finds the yearly rate in percent from the interest, the principal and the time in years.

Time period

$T = \frac{100 \times SI}{P \times R}$

Finds the number of years needed to earn a given interest on principal $P$ at $R$ per year.

Interest with a decimal rate

$I = Prt$

The same formula with $r$ written as a decimal (8% becomes 0.08) and $t$ in years, as in US textbooks.

Time in months or days

$T = \frac{m}{12}$ or $T = \frac{d}{365}$

Change $m$ months or $d$ days into years before putting $T$ into the simple interest formula.

Time for a sum to become k times

$T = \frac{100(k - 1)}{R}$

Years needed at simple interest for the amount to reach $k$ times the principal at a rate of $R$ per year.

Amount, compounded annually

$A = P\left(1 + \frac{R}{100}\right)^{n}$

$P$ is the principal. Interest is added once a year, and each year's interest also earns interest after that.

Compound interest

$CI = A - P$

The compound interest earned is the final amount minus the principal that was invested.

Compounded half-yearly

$A = P\left(1 + \frac{R}{200}\right)^{2n}$

Interest is added every six months, so the rate per period is halved and the number of periods doubles.

Compounded quarterly

$A = P\left(1 + \frac{R}{400}\right)^{4n}$

Interest is added four times a year, so each quarter uses $\frac{R}{4}$ and there are $4n$ periods.

Compounded n times a year

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

General form. Here $r$ is the annual rate as a decimal, $n$ the compounding periods per year and $t$ the years. Monthly uses $n = 12$, daily $n = 365$.

Compounded continuously

$A = Pe^{rt}$

$e \approx 2.71828$, $r$ is the annual rate as a decimal and $t$ the time in years. It is the limit of compounding ever more often.

Different rates in successive years

$A = P\left(1 + \frac{R_{1}}{100}\right)\left(1 + \frac{R_{2}}{100}\right)\left(1 + \frac{R_{3}}{100}\right)$

Used when the rate changes each year, with $R_{1}$ in the first year, $R_{2}$ in the second and so on.

Fractional number of years

$A = P\left(1 + \frac{R}{100}\right)^{n}\left(1 + \frac{fR}{100}\right)$

For $n$ whole years plus a fraction $f$ of a year, with simple interest on the part year. For 2.5 years, $n = 2$ and $f = \frac{1}{2}$.