GMAT formula sheet

Every formula you need for the GMAT Quantitative Reasoning section, organized by topic. Each formula comes with a one-sentence explanation of when to use it. Click a topic to practice it.

263 formulas · 13 sections · PDF: 171 formulas, two-sided, US Letter

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Fractions and decimals

20 formulas

Course: Fractions Course: Decimals Practice: Arithmetic

Fractions

A fraction represents a part of a whole, and simplifying divides both the numerator and denominator by a common factor.

\frac{a}{b}

Lesson: Comparing and combining fractions

Adding or subtracting two fractions with different denominators

\frac{a}{b} \pm \frac{c}{d} = \frac{a d \pm b c}{b d}

Lesson: Comparing and combining fractions

The product of two fractions multiplies straight across.

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

Lesson: Comparing and combining fractions

Comparing two fractions with positive denominators

\frac{a}{b} > \frac{c}{d}

⟺ a d > b c

Lesson: Comparing and combining fractions

The part equals the fraction multiplied by the whole.

part = fraction \times whole

Lesson: Fractions of a remainder

The original whole equals the final leftover amount divided by the product of the remaining shares.

W = \frac{L}{( 1 - f _{1} ) ( 1 - f _{2} ) \dots ( 1 - f _{n} )}

Lesson: Fractions of a remainder

For positive numbers, increasing the numerator or decreasing the denominator makes the fraction larger.

If a > b > 0

and c > 0,

then \frac{a}{c} > \frac{b}{c}

and \frac{c}{b} > \frac{c}{a}

Lesson: How fractions change in size

Adding the same amount $k$ to both numerator and denominator moves a positive fraction toward 1.

\frac{a + k}{b + k}

Lesson: How fractions change in size

Multiplying the numerator by $m$ and the denominator by $n$ multiplies the fraction by $\frac{m}{n}$ .

\frac{a \times m}{b \times n} = \frac{a}{b} \times \frac{m}{n}

Lesson: How fractions change in size

For positive fractions, the larger fraction has the smaller reciprocal.

\frac{a}{b} > \frac{c}{d}

⟹

\frac{b}{a} < \frac{d}{c}

(a, b, c, d > 0)

Lesson: How fractions change in size

The decimal form of a fraction

decimal = \frac{numerator}{denominator}

Lesson: Converting fractions and decimals

The number of decimal places a terminating fraction has, given its reduced denominator

decimal places = max (a, b)

for a denominator of 2^{a} \times 5^{b}

Lesson: Converting fractions and decimals

The value of a digit in the kth place after the decimal point

value = digit \times 1 0^{- k}

Lesson: Decimal place value and rounding

The value of a digit in the kth place after the decimal point

d \times 1 0^{- k}

Lesson: Decimal place value and rounding

A decimal $0. d_{1} d_{2} \dots d_{n}$ with unknown digits, expressed as a linear combination of those digits

0. d_{1} d_{2} \dots d_{n} = \frac{1 0 ^{n - 1} d _{1} + 1 0 ^{n - 2} d _{2} + \dots + d _{n}}{1 0 ^{n}}

Lesson: Decimal place value and rounding

The decimal-point shift from multiplying or dividing by a power of ten

\times 1 0^{n}

\to n places right;

\div 1 0^{n}

\to n places left

Lesson: Decimal arithmetic with powers of ten

Multiplying decimals

The product of two decimals equals the product of their integer forms divided by ten to the power of the total decimal places.

a \times b = \frac{A \times B}{1 0 ^{d_{a} + d_{b}}}

Lesson: Decimal arithmetic with powers of ten

Multiplying two numbers in scientific notation

(a \times 1 0^{m}) \times (b \times 1 0^{n}) = (a \times b) \times 1 0^{m + n}

Lesson: Decimal arithmetic with powers of ten

Estimating magnitude

Estimate the magnitude of a product or quotient by rounding each factor to one significant digit and tracking powers of ten separately.

Estimate = (d_{1} \times d_{2} \times \dots \times d_{n}) \times 1 0^{k_{1} + k_{2} + \dots + k_{n}}

Lesson: Decimal arithmetic with powers of ten

Mixed scales

Combine coefficients and exponents separately, adding exponents for multiplication and subtracting them for division.

(a \times 1 0^{m}) (b \times 1 0^{n}) = ab \times 1 0^{m + n},

\frac{a \times 1 0 ^{m}}{b \times 1 0 ^{n}} = \frac{a}{b} \times 1 0^{m - n}

Lesson: Decimal arithmetic with powers of ten

Number properties

26 formulas

Course: Integers Course: Even and odd integers Course: Consecutive integers Course: Factors, multiples, and primes Course: Divisibility and remainders Practice: Number properties

Divisibility by 3

A number is divisible by 3 if the sum of its digits is.

Sum of digits divisible by 3

Divisibility by 9

Same rule as 3 but with 9.

Sum of digits divisible by 9

GCD and LCM

When you know one, you know the other.

GCD (a, b) \times LCM (a, b) = a \times b

The sign of a quotient matches the sign of its corresponding product

sign (\frac{x}{y}) = sign (x \cdot y)

Lesson: Sign rules for integers

The sign of a power depends on whether the exponent is even or odd.

If n is even,

x^{n} > 0 (for x \neq = 0). If n is odd,

x^{n} has the same sign as x .

Lesson: Sign rules for integers

A two-digit number with tens digit a and units digit b

10 a + b

Lesson: Place value and digit problems

Difference between a two-digit number and its reverse

(10 a + b) - (10 b + a) = 9 (a - b)

Lesson: Place value and digit problems

Difference between a three-digit number and its reverse

(100 a + 10 b + c) - (100 c + 10 b + a) = 99 (a - c)

Lesson: Place value and digit problems

A condition relating the tens digit to the units digit

a = k \cdot b

Lesson: Place value and digit problems

Definition of divisibility and the digit-sum test for 3

n is divisible by d

⟺ n = d \cdot k

for some integer k

n is divisible by 3

⟺ digit sum of n is divisible by 3

Lesson: Divisibility rules

A total divided into equal groups leaves a remainder smaller than the divisor.

n = d q + r

where 0 \leq r < d

Lesson: Remainders

When the divisor exceeds the dividend, the quotient is zero and the remainder equals the dividend.

n = d \cdot q + r

⟹ q = 0

and r = n

Lesson: Remainders

The remainder from a decimal quotient

r = (\frac{n}{d} - q) \times d

Lesson: Remainders

The remainder of a sum or product equals the remainder of the sum or product of the individual remainders.

rem (a \pm b, d) = rem (rem (a, d) \pm rem (b, d), d)

rem (a \cdot b, d) = rem (rem (a, d) \cdot rem (b, d), d)

Lesson: Remainders of sums, products, and powers

Powers

The remainder of a high power equals the remainder of the exponent divided by the cycle length, mapped back to the cycle.

E = k \cdot L + r

⟹ a^{E} \equiv c_{r} (mod d)

Lesson: Remainders of sums, products, and powers

A is a factor of b when b equals a times an integer k

b = a \cdot k

Lesson: Factors and Multiples

An integer n is prime when its count of positive divisors is exactly 2.

d (n) = 2

Lesson: Prime numbers

One less than a square factors into two consecutive integers

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

Lesson: Prime numbers

The prime factorization of an integer greater than 1

n = p^{a} q^{b}

Lesson: Prime factorization

The number of positive factors of an integer from its prime factorization

Number of factors = (a + 1) (b + 1)

Lesson: Prime factorization

The power of a prime p in the factorial of n

⌊ \frac{n}{p} ⌋ + ⌊ \frac{n}{p ^{2}} ⌋ + ⌊ \frac{n}{p ^{3}} ⌋ + \dots

Lesson: Prime factorization

If gcd(a, b) = g, write a = gm and b = gn with m and n coprime.

a = g m,

b = g n,

g cd (m, n) = 1

Lesson: GCD and LCM from prime factorizations

Every even integer is a multiple of 2 and every odd integer is one more than a multiple of 2

even = 2 k,

odd = 2 k + 1

Lesson: Even and odd rules

The number of integers in an inclusive range from a first value to a last value

count = last - first + 1

Lesson: Counting integers in a range

For any evenly spaced set, the sum equals the average of the first and last terms times the count of terms.

sum = \frac{first + last}{2} \times count

Lesson: Consecutive and evenly spaced integers

The algebraic representation of $k$ consecutive even or odd integers starting from $n$

n,

n + 2,

n + 4,

\dots,

n + 2 (k - 1)

Lesson: Consecutive and evenly spaced integers

Exponents and roots

20 formulas

Course: Exponents and roots Practice: Exponents

Product rule

Add exponents when multiplying same base.

a^{m} \times a^{n} = a^{m + n}

Quotient rule

Subtract exponents when dividing same base.

\frac{a ^{m}}{a ^{n}} = a^{m - n}

Power rule

Multiply exponents when raising a power to a power.

(a^{m})^{n} = a^{mn}

Exponent rules

When you multiply or divide powers with the same base, add or subtract the exponents. When you raise a power to a power, multiply the exponents.

a^{m} \times a^{n} = a^{m + n}

and

\frac{a ^{m}}{a ^{n}} = a^{m - n}

and

(a^{m})^{n} = a^{mn}

Lesson: Exponent rules

A power of a product splits across the factors, and a shared exponent combines into a single base.

(ab)^{n} = a^{n} \times b^{n}

a^{n} \times b^{n} = (ab)^{n}

Lesson: Exponent rules

Rewrite a composite base as a power of its prime factor before applying exponent rules.

a = p^{k}

⟹ a^{m} = (p^{k})^{m} = p^{k m}

Lesson: Exponent rules

Factoring a sum of powers with the same base

a^{m} + a^{m + n} = a^{m} (1 + a^{n})

Lesson: Exponent rules

The three exponent extensions

a^{0} = 1

and

a^{- n} = \frac{1}{a ^{n}}

and

a^{m / n} = n a^{m}

Lesson: Zero, negative, and fractional exponents

A negative exponent means take the reciprocal of the base and make the exponent positive.

a^{- n} = \frac{1}{a ^{n}}

Lesson: Zero, negative, and fractional exponents

A fraction raised to a negative power flips the fraction and makes the exponent positive.

(\frac{a}{b})^{- n} = (\frac{b}{a})^{n}

Lesson: Zero, negative, and fractional exponents

A fractional exponent converts to a root, where the denominator becomes the index of the root.

a^{\frac{m}{n}} = (n a)^{m}

Lesson: Zero, negative, and fractional exponents

For any base between 0 and 1, increasing the exponent decreases the value.

0 < a < 1

⟹ a^{3} < a^{2} < a

Lesson: Zero, negative, and fractional exponents

When two powers of the same base are equal, the exponents are equal.

a^{x} = a^{y}

⟹ x = y

(a \neq = 0, 1, - 1)

Lesson: Equations with exponents

Common base

When two sides use different bases that are powers of a common base, rewrite each side as a power of that common base before equating exponents.

p^{a} = p^{b}

⟹ a = b

Lesson: Equations with exponents

The square root of a product is the product of the square roots.

ab = a \cdot b

Lesson: Roots and radicals

The product and quotient of two square roots equal the square root of the product and quotient.

a \cdot b = ab

and

\frac{a}{b} = \frac{a}{b}

Lesson: Roots and radicals

Combine radicals that share the same radicand by adding or subtracting their coefficients.

a x + b x = (a + b) x

Lesson: Roots and radicals

To eliminate a radical from a denominator, multiply the fraction by a form of 1.

\frac{a}{b} = \frac{a}{b} \cdot \frac{b}{b} = \frac{a b}{b}

and

\frac{a}{b + c} = \frac{a}{b + c} \cdot \frac{b - c}{b - c} = \frac{a ( b - c )}{b - c ^{2}}

Lesson: Roots and radicals

The principal square root of a squared variable is its absolute value.

x^{2} = ∣ x ∣

Lesson: Roots and radicals

The cube root of a real number cubed

3 x^{3} = x

Lesson: Roots and radicals

Algebra

34 formulas

Course: Linear equations Course: Algebraic expressions and factoring Course: Quadratic equations Practice: Algebra

Quadratic formula

Solve any quadratic equation.

x = \frac{- b \pm b ^{2} - 4 a c}{2 a}

Difference of squares

Factor quickly without long multiplication.

a^{2} - b^{2} = (a + b) (a - b)

Linear equation

Solve for a single variable.

a x + b = 0

\Rightarrow x = - \frac{b}{a}

The solution to a linear equation in standard form

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

Lesson: Solving linear equations

Collecting terms for a linear equation with variables on both sides

a x + b = c x + d

⟹ x (a - c) = d - b

Lesson: Solving linear equations

Writing equations

Translate a sentence into an equation by assigning a variable to the unknown and replacing words with operators.

is

\to=,

of

\to \times,

more than

\to +,

less than

\to -

Lesson: Writing equations from word problems

A system of two independent linear equations in two unknowns has a unique solution

a_{1} x + b_{1} y = c_{1}

a_{2} x + b_{2} y = c_{2}

Lesson: Systems of linear equations

Substitute an isolated variable into the other equation

x = \frac{c - b y}{a}

⟹ a_{2} (\frac{c - b y}{a}) + b_{2} y = c_{2}

Lesson: Systems of linear equations

Scale the equations so one variable's coefficients are opposites, then add.

(a_{1} x + b_{1} y = c_{1}) \times k_{1}

and (a_{2} x + b_{2} y = c_{2}) \times k_{2}

⟹ a_{1} k_{1} = - a_{2} k_{2}

Lesson: Systems of linear equations

Two-type mixture from total count and total value

x + y = n,

p_{1} x + p_{2} y = V

Lesson: Systems of linear equations

Solving for an expression

When the target is a constant multiple of one given equation's left side, multiply the entire equation by that factor.

if a x + b y = c

and the target is k (a x + b y),

then k (a x + b y) = k c

Lesson: Solving for an expression

A target expression is determined when it is a linear combination of the given equations.

a \cdot E_{1} + b \cdot E_{2} = T

Lesson: Solving for an expression

Solve for the variable with the smaller coefficient, stepping the variable with the larger coefficient.

y = \frac{c - a x}{b}

(step x

if a > b)

Lesson: Integer solutions to linear equations

Divisibility shortcut

For $a x + b y = c$ , the second variable is an integer only when the remaining total is divisible by the other coefficient.

y = \frac{c - a x}{b}

Lesson: Integer solutions to linear equations

The distributive property

a (b + c) = ab + a c

Lesson: Simplifying algebraic expressions

A negative sign before parentheses reverses every term inside.

- (a - b + c) = - a + b - c

Lesson: Simplifying algebraic expressions

The product of two binomials

(a + b) (c + d) = a c + a d + b c + b d

Lesson: Simplifying algebraic expressions

Factoring a sum by pulling out the greatest common factor

ab + a c = a (b + c)

Lesson: Simplifying algebraic expressions

The three special products

(a + b)^{2} = a^{2} + 2 ab + b^{2},

(a - b)^{2} = a^{2} - 2 ab + b^{2},

(a + b) (a - b) = a^{2} - b^{2}

Lesson: Special products

The square of a binomial

(a \pm b)^{2} = a^{2} \pm 2 ab + b^{2}

Lesson: Special products

The sum of two squares equals the square of their sum minus twice their product.

a^{2} + b^{2} = (a + b)^{2} - 2 ab

Lesson: Special products

Simplifying an algebraic fraction by cancelling a shared factor

\frac{a x}{a y} = \frac{x}{y}

where a \neq = 0

and y \neq = 0

Lesson: Algebraic fractions

Cancel shared multiplicative factors, not added terms

\frac{a \cdot b}{a \cdot c} = \frac{b}{c}

but

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

Lesson: Algebraic fractions

The sum of two fractions with unlike denominators

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

Lesson: Algebraic fractions

Multiplying and dividing two algebraic fractions

\frac{a}{b} \cdot \frac{c}{d} = \frac{a c}{b d}

and

\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c} = \frac{a d}{b c}

Lesson: Algebraic fractions

The zero product property for a factored quadratic equation

If (x - a) (x - b) = 0,

then x - a = 0

or x - b = 0

Lesson: Solving quadratics by factoring

Standard form of a quadratic equation

a x^{2} + b x + c = 0

Lesson: Solving quadratics by factoring

A product relation with a dependency becomes a quadratic in one variable

x y = k

and y = a x + b

⟹ x (a x + b) = k

Lesson: Quadratics in word problems

The product of two consecutive integers

n (n + 1) = k

Lesson: Quadratics in word problems

Revenue as the product of price and quantity

R = P \times Q

Lesson: Quadratics in word problems

A quadratic with roots r and s

(x - r) (x - s) = 0

Lesson: Roots and coefficients of a quadratic

Sum and product of the roots of $a x^{2} + b x + c = 0$

r_{1} + r_{2} = - \frac{b}{a}

and

r_{1} \times r_{2} = \frac{c}{a}

Lesson: Roots and coefficients of a quadratic

Sum of the squares of the roots

r^{2} + s^{2} = (r + s)^{2} - 2 r s

Lesson: Roots and coefficients of a quadratic

Discriminant

Δ = b^{2} - 4 a c

Δ > 0 : two real roots

Δ = 0 : one real root

Δ < 0 : no real roots

Inequalities and absolute value

14 formulas

Course: Absolute value and the number line Practice: Inequalities

Flip on multiply

Flip the inequality when multiplying or dividing by a negative.

a < b,

c < 0

\Rightarrow a c > b c

Triangle inequality

The sum of absolute values is an upper bound.

∣ a + b ∣ \leq ∣ a ∣ + ∣ b ∣

The distance from x to zero

∣ x ∣ = x

if x \geq 0

and ∣ x ∣ = - x

if x < 0

Lesson: Absolute value as distance

The distance between two points on a number line

∣ a - b ∣ = ∣ b - a ∣

Lesson: Absolute value as distance

The sign of a variable revealed by its absolute value

∣ x ∣ = x

⟹ x \geq 0,

∣ x ∣ = - x

⟹ x \leq 0,

x^{2} = ∣ x ∣

Lesson: Absolute value as distance

The value of an expression with nested absolute value bars, evaluated from the inside out.

∣ a - ∣ b ∣∣

Lesson: Absolute value as distance

Bounds on a combined absolute value

∣ x + y ∣ \leq ∣ x ∣ + ∣ y ∣

∣ x - y ∣ \geq ∣ x ∣ - ∣ y ∣

Lesson: Absolute value as distance

An absolute value equation splits into two linear equations.

∣ c x - d ∣ = b

⟹ c x - d = b

or c x - d = - b

Lesson: Absolute value equations and inequalities

A value within b units of a

∣ x - a ∣ < b becomes a - b < x < a + b

Lesson: Absolute value equations and inequalities

The value of x that is equidistant from a and b

x = \frac{a + b}{2}

Lesson: Absolute value equations and inequalities

A range from p to q written as an absolute value inequality

∣ x - \frac{p + q}{2} ∣ \leq \frac{q - p}{2}

Lesson: Absolute value equations and inequalities

The ranges where an operation changes a number, found by factoring the inequality

x^{2} > x

⟹ x (x - 1) > 0

⟹ x > 1

or x < 0

Lesson: Number ranges on the number line

Distance between two points on the number line

∣ a - b ∣

Lesson: Number ranges on the number line

The four test ranges for number properties

x < - 1,

- 1 < x < 0,

0 < x < 1,

x > 1

Lesson: Number ranges on the number line

Percents and interest

22 formulas

Course: Percents Course: Interest and growth Practice: Percentages

Percent change

The old value is always the denominator.

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

Percent of

What portion one value is of another.

\frac{part}{whole} \times 100

Successive percent change

Each percent applies to the current value, not the original.

(1 + r_{1}) (1 + r_{2}) - 1

The part equals the whole times the percent over 100

part = \frac{percent}{100} \times whole

Lesson: Percent, part, and whole

Percent change from an original value to a new value

Percent Change = \frac{New - Original}{Original} \times 100

Lesson: Percent change

Original value before a percent change of p percent

O = \frac{N}{1 + \frac{p}{100}}

Lesson: Percent change

The overall multiplier for a value that changes by p% and then by q%

Overall multiplier = (1 + \frac{p}{100}) (1 + \frac{q}{100})

Lesson: Combined percent changes

Net multiplier for a rise and fall of the same percent

(1 + \frac{p}{100}) (1 - \frac{p}{100}) = 1

- \frac{p ^{2}}{10000}

Lesson: Combined percent changes

The percent r that undoes a p% change

(1 + \frac{p}{100}) (1 + \frac{r}{100}) = 1

Lesson: Combined percent changes

Overall multiplier for three successive percent changes

(1 + \frac{p}{100}) (1 + \frac{q}{100}) (1 + \frac{r}{100})

Lesson: Combined percent changes

Overall multiplier for a product when two factors change by percents

M = (1 + \frac{p}{100}) (1 + \frac{q}{100})

Lesson: Combined percent changes

A quantity p percent more than a base B

A = B (1 + \frac{p}{100})

Lesson: Percent comparisons

For chained comparisons, multiply the decimal multipliers.

A = m_{1} \times m_{2} \times C

Lesson: Percent comparisons

Two percent sentences with unknowns form a system of equations

A = B \times r_{1}

and A = C \times r_{2}

⟹ B \times r_{1} = C \times r_{2}

Lesson: Percent comparisons

Interest earned

I = P \times r \times t

Lesson: Simple interest

Total interest from two split deposits

I = P_{1} r_{1} t_{1} + P_{2} r_{2} t_{2}

Lesson: Simple interest

Final balance with compound interest

balance = P (1 + \frac{r}{n})^{n t}

Lesson: Compound interest

The value after a constant factor change over several periods

V = V_{0} \times r^{n}

Lesson: Growth and decay by a constant factor

Final value after decay

V = V_{0} \times (1 - \frac{r}{100})^{n}

Lesson: Growth and decay by a constant factor

Final value after an increase of p percent and a decrease of p percent

V_{final} = V_{0} \times (1 + \frac{p}{100}) \times (1 - \frac{p}{100})

Lesson: Growth and decay by a constant factor

Linear growth adds a constant amount, while exponential growth applies a constant factor.

V = V_{0} + d \cdot n

and

V = V_{0} \times (1 + r)^{n}

Lesson: Growth and decay by a constant factor

The growth ratio of end to start, independent of the principal

\frac{end}{start} = (1 + r)^{n}

Lesson: Interest and growth in Data Sufficiency

Ratios

10 formulas

Course: Ratios and proportions Practice: Ratios

Ratio to fraction

The fraction of the total that belongs to part a.

\frac{a}{a + b}

Cross multiplication

Solve for an unknown in a proportion.

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

\Rightarrow a d = b c

The actual amounts behind a ratio a:b

Amount_{1} = a x

and Amount_{2} = b x

Lesson: Understanding ratios

Combine two ratios that share a term

a : b

and b : c

\to a : b : c (shared

term

scaled

to

match)

Lesson: Combining ratios with a shared term

New amounts after a one-part change

New ratio = \frac{a x \pm change}{b x}

Lesson: Ratios that change

New ratio from known changes to both parts

\frac{a x + p}{b x + q} = \frac{c}{d}

Lesson: Ratios that change

Direct proportion keeps the quotient constant, and inverse proportion keeps the product constant.

Direct: \frac{y _{1}}{x _{1}} = \frac{y _{2}}{x _{2}}

Inverse: x_{1} y_{1} = x_{2} y_{2}

Lesson: Direct and inverse proportion

Matching units in matching positions

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

⟹ a \times d = b \times c

Lesson: Direct and inverse proportion

The product of two inversely proportional quantities stays constant across two states.

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

Lesson: Direct and inverse proportion

The total from a ratio with multiplier k

Total = k \times (sum of ratio parts)

Lesson: Ratio problems in Data Sufficiency

Rates and work

19 formulas

Course: Rates and speed Course: Work problems Practice: Rates

Rate

The base formula. Everything in rates follows from it.

rate = \frac{work}{time}

Combined work rate

Rates add. Times do not.

\frac{1}{t _{combined}} = \frac{1}{t _{1}}

+ \frac{1}{t _{2}}

Average speed (equal distances)

The harmonic mean. Never average the speeds directly.

\frac{2 \times s _{1} \times s _{2}}{s _{1} + s _{2}}

Distance equals rate multiplied by time

d = r t

Lesson: Distance, rate, and time

Distance equals rate multiplied by time, but only when their time units match.

d = r \times t

Lesson: Distance, rate, and time

Average speed for a multi-leg trip

Average Speed = \frac{Total Distance}{Total Time}

Lesson: Distance, rate, and time

Total quantity produced or consumed

Q = r t

Lesson: Distance, rate, and time

The ratio of times for the same distance is the inverse of the ratio of rates.

\frac{t _{1}}{t _{2}} = \frac{r _{2}}{r _{1}}

Lesson: Distance, rate, and time

The ratio of distances for the same time equals the ratio of rates.

\frac{d _{1}}{d _{2}} = \frac{r _{1}}{r _{2}}

Lesson: Distance, rate, and time

Average speed for two legs of equal distance

Average Speed = \frac{2 ab}{a + b}

Lesson: Average speed

Time to close the gap when moving toward each other

t = \frac{gap}{r _{1} + r _{2}}

Lesson: Two travelers and relative speed

Time for a single leg at speed r over distance d

t = \frac{d}{r}

Lesson: Currents, wind, and changing speeds

Average speed for two legs

Average Speed = \frac{d _{1} + d _{2}}{t _{1} + t _{2}}

Lesson: Rate problems in Data Sufficiency

Time to complete a remaining fraction of a job

time = \frac{f}{r}

Lesson: Understanding work rates

Total work across phases with different rates

W = r_{1} t_{1} + r_{2} t_{2} + \dots + r_{n} t_{n}

Lesson: Advanced work problems

One completed job from a group of workers

(r_{A} + r_{B} + \dots + r_{n}) \times t = 1

Lesson: Advanced work problems

Combined rate equals the sum of individual rates

\frac{1}{x} + \frac{1}{x + d} = \frac{1}{c}

Lesson: Advanced work problems

Relationship between rate ratio and time ratio

\frac{r _{A}}{r _{B}} = \frac{t _{B}}{t _{A}}

Lesson: Work problems in Data Sufficiency

The unknown rate is the combined rate minus the known rate.

r_{u nk n o w n} = r_{co mbin e d} - r_{k n o w n}

Lesson: Work problems in Data Sufficiency

Mixtures and weighted averages

7 formulas

Course: Mixtures and weighted averages Practice: Word problems

The combined average, given each group's average and size

combined average = \frac{a _{1} n _{1} + a _{2} n _{2} + \dots}{n _{1} + n _{2} + \dots}

Lesson: Weighted averages

The combined average is the total sum divided by the total count.

Combined Average = \frac{n _{A} \cdot Average _{A} + n _{B} \cdot Average _{B}}{n _{A} + n _{B}}

Lesson: Finding the mix from the averages

The ratio of group sizes is the inverse of the ratio of their distances from the combined average.

\frac{n _{A}}{n _{B}} = \frac{d _{2}}{d _{1}}

Lesson: Finding the mix from the averages

The size of a group from its ratio share and the total count

n_{A} = (\frac{r _{A}}{r _{A} + r _{B}}) \times N

Lesson: Finding the mix from the averages

The final concentration is the total amount of the ingredient divided by the total volume.

c = \frac{c _{1} v _{1} + c _{2} v _{2}}{v _{1} + v _{2}}

Lesson: Mixtures and concentrations

Combined concentration of two solutions

c_{1} v_{1} + c_{2} v_{2} = c (v_{1} + v_{2})

Lesson: Mixtures and concentrations

The ratio of the weights is determined by the distances from the combined average to the group averages.

\frac{w _{A}}{w _{B}} = \frac{∣ C - B ∣}{∣ C - A ∣}

Lesson: Weighted averages and mixtures in Data Sufficiency

Statistics

8 formulas

Course: Descriptive statistics Practice: Statistics

Mean

The average. Sum divided by count.

\frac{\sum x _{i}}{n}

Weighted average

When values have different weights or frequencies.

\frac{\sum w _{i} x _{i}}{\sum w _{i}}

Standard deviation

Measures spread. The GMAT rarely requires computing it by hand.

\frac{\sum ( x _{i} - x ˉ ) ^{2}}{n}

The mean equals the sum divided by the count.

mean = \frac{sum}{count}

Lesson: The mean as a total

The position of the median in an ordered list of $n$ values

median position = \frac{n + 1}{2}

Lesson: Median and mode

The range of a set of values

Range = Maximum - Minimum

Lesson: Range and standard deviation

Maximum of one value in a set with a fixed sum

Target_{m a x} = Sum

- \sum Other minimums

Lesson: Maximizing and minimizing one value

Mean and median of a symmetric set

Mean = Median

when

the

data

is

evenly

spaced

or

symmetric

about

its

center

Lesson: Statistics in Data Sufficiency

Overlapping sets

5 formulas

Course: Overlapping sets Practice: Word problems

Two overlapping groups

The total counts each member once by adding the two groups, subtracting the overlap they share, and adding the people in neither.

total = A

+ B

- both

+ neither

Lesson: Two overlapping groups

The parts of any row or column add up to the total.

cell_{1} + cell_{2} = row or column total

Lesson: The double-set matrix

Total members across three groups, given the exactly-two count, the all-three count, and the neither count

total = A

+ B

+ C

- (exactly two)

- 2 \times (all three)

+ neither

Lesson: Three overlapping groups

Maximum and minimum overlap when two groups share a fixed total with no neither count

max both = min (A, B)

min both = max (A + B - total, 0)

Lesson: Minimum and maximum overlap

The relationship between cells, rows, columns, and the total in a two-by-two table.

Row Total = a + b

and

Column Total = a + c

Lesson: Overlapping sets in Data Sufficiency

Counting, probability, functions and sequences

45 formulas

Course: Counting Course: Probability Course: Functions and sequences Practice: Combinatorics Practice: Probability Practice: Algebra

Basic probability

Favorable outcomes divided by total outcomes.

P (A) = \frac{favorable}{total}

Complement rule

When it is easier to count what does not happen.

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

Independent events

Multiply when events are independent.

P (A and B) = P (A) \times P (B)

A function evaluated at a given input

f (a) = the rule applied to a

Lesson: Understanding functions

Given a function rule and an output value, solve for the unknown input

f (x) = y

Lesson: Understanding functions

Evaluating a composition of two functions

f (g (x))

Lesson: Understanding functions

A custom operation applies a defined rule to its inputs.

a \oplus b = rule (a, b)

Lesson: Custom operations and symbols

Finding an identity value for a custom operation

a \otimes e = a

Lesson: Custom operations and symbols

The term at position n in a sequence that adds the same amount each step

a_{n} = a_{1} + (n - 1) d

Lesson: Arithmetic and geometric sequences

The term at position n in a sequence that multiplies by the same ratio each step

a_{n} = a_{1} r^{n - 1}

Lesson: Arithmetic and geometric sequences

The common difference of an arithmetic sequence from two known terms

d = \frac{a _{m} - a _{k}}{m - k}

Lesson: Arithmetic and geometric sequences

The sum of a set of evenly spaced terms

S_{n} = \frac{n ( a _{1} + a _{n} )}{2}

Lesson: Arithmetic and geometric sequences

The constant added in an arithmetic sequence; The constant multiplied in a geometric sequence

d = a_{n + 1} - a_{n}

r = \frac{a _{n + 1}}{a _{n}}

Lesson: Arithmetic and geometric sequences

The nth term as a function of earlier terms

a_{n} = f (a_{n - 1}, \dots)

Lesson: Recursive sequences

Each term is the rule applied to the previous term

a_{n} = f (a_{n - 1})

Lesson: Recursive sequences

The next term depends on the two previous terms.

a_{n} = f (a_{n - 1}, a_{n - 2})

Lesson: Recursive sequences

The inverted recursive rule to find the previous term

a_{n - 1} = f^{- 1} (a_{n})

Lesson: Recursive sequences

The position of a target term within a repeating cycle

r = n mod p

Lesson: Recursive sequences

Total outcomes when choosing one option from each independent stage

total = a \times b \times c \times \dots

Lesson: The fundamental counting principle

Total outcomes when repetition is allowed across k slots

n \times n \times \dots \times n = n^{k}

Lesson: The fundamental counting principle

Total arrangements of distinct items in sequence without repetition

n \times (n - 1) \times (n - 2) \times \dots

Lesson: The fundamental counting principle

Outcomes in either of two disjoint cases A or B

∣ A \cup B ∣ = ∣ A ∣ + ∣ B ∣

Lesson: The fundamental counting principle

Allowed outcomes by the complement method

∣ A ∣ = ∣ T ∣ - ∣ A^{c} ∣

Lesson: The fundamental counting principle

The number of ways to arrange r of n distinct objects in order

\frac{n !}{( n - r )!}

Lesson: Arrangements

The number of arrangements of n distinct objects where a specific group of k objects must be adjacent

(n - k + 1)! \times k!

Lesson: Arrangements

Non-adjacent items

The number of ways to arrange $n$ unrestricted items and $k$ restricted items so that no two restricted items are adjacent

n! \times (k n + 1) \times k!

Lesson: Arrangements

The number of distinct arrangements of n objects where some are identical

\frac{n !}{n _{1} ! \times n _{2} ! \times \dots \times n _{k} !}

Lesson: Arrangements

Distinct circular arrangements of n distinct objects

(n - 1)!

Lesson: Arrangements

The number of ways to choose r items from n items when order does not matter

(r n) = \frac{n !}{r ! ( n - r )!}

Lesson: Combinations

Total selections from distinct pools

Total = (r _{1} n _{1}) \times (r _{2} n _{2}) \times \dots

Lesson: Combinations

The number of unordered pairs among n items

(2 n) = \frac{n ( n - 1 )}{2}

Lesson: Combinations

Ordered selections of r items chosen from n

_{n} P_{r} = \frac{n !}{( n - r )!} = (r n) \times r!

Lesson: Choosing and then arranging

Total ways to select from multiple groups and then arrange

Total = (r _{1} n _{1}) \times (r _{2} n _{2}) \times \dots \times (r_{1} + r_{2} + \dots)!

Lesson: Choosing and then arranging

Total ways to form a group across multiple valid compositions

Total = ((r _{1} n _{1}) \times (r _{2} n _{2}))

+ ((s _{1} n _{1}) \times (s _{2} n _{2}))

+ \dots

Lesson: Choosing and then arranging

The fewest draws needed to guarantee an outcome

fewest draws = (longest unlucky sequence that avoids the outcome)

+ 1

Lesson: Guaranteeing an outcome

Fewest draws to guarantee a matching pair with $k$ distinct colors

k + 1

Lesson: Guaranteeing an outcome

Fewest draws to guarantee n matching items of any kind when k kinds have enough supply

k (n - 1) + 1

Lesson: Guaranteeing an outcome

Probability of an event, as favorable outcomes divided by total outcomes

P = \frac{favorable outcomes}{total outcomes}

Lesson: Basic probability

The probability of event A or event B occurring

P (A or B) = P (A)

+ P (B)

- P (A and B)

Lesson: Basic probability

The probability that all independent events occur

P (all) = P (A) \times P (B) \times P (C) \times \dots

Lesson: Independent events and at least one

Probability of at least one success among independent events

P (at least one) = 1 - P (none)

Lesson: Independent events and at least one

Probability of a sequence of draws without replacement

P (A and B) = P (A) \times P (B ∣ A)

Lesson: Selection without replacement

The probability of A given B

P (A ∣ B) = \frac{P ( A and B )}{P ( B )}

Lesson: Conditional probability

Two events A and B are independent when the conditional probability equals the unconditional probability

P (A ∣ B) = P (A)

Lesson: Conditional probability

Probability of an event from its complement

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

Lesson: Probability in Data Sufficiency

Data Insights

33 formulas

Course: Data Sufficiency Course: Table Analysis Course: Graphics Interpretation Course: Two-Part Analysis Course: Multi-Source Reasoning Practice: Data Sufficiency Practice: Table Analysis Practice: Graphics Interpretation Practice: Two-Part Analysis Practice: Multi-Source Reasoning

The AD/BCE branch after testing statement (1) alone

(1) sufficient

\to A or D; (1) insufficient

\to B, C, or E

Lesson: Data Sufficiency and the AD/BCE grid

A linear equation in one variable yields exactly one solution.

a x + b = c

\Rightarrow x = \frac{c - b}{a},

a \neq = 0

Lesson: Value questions: sufficient means exactly one value

The roots of a squared relation

x^{2} = k

⟹ x = \pm k

Lesson: Value questions: sufficient means exactly one value

A linear equation in two variables

a x + b y = c

Lesson: Value questions: sufficient means exactly one value

A yes/no statement is sufficient when every allowed value gives the same answer

sufficient

⟺ the

answer

is

constant

across

all

allowed

values

Lesson: Yes/no sufficiency

Target quantities

A value question is sufficient when the statement determines one value for the target expression, not necessarily one value for each variable.

Sufficient

⟺ target

expression

has

exactly

one

value

Lesson: Rephrasing the question: solving for what is asked

The number of independent linear equations required to solve for n unknowns

n independent linear equations

⟺ n unknowns are fixed

Lesson: When counting equations fails

A linear equation with a positive integer constraint

a x + b y = c

(x, y \in Z^{+})

Lesson: When counting equations fails

Actual value when the axis label gives a scale factor

actual value = scale reading \times scale factor

Lesson: Bar graph dropdowns

Value of a bar where $V$ is the value of the bar and $s$ is the scale mark at the top of the bar.

V = s

Lesson: Bar graph dropdowns

The change in a quantity over time

change = end value

- start value

Lesson: Line graphs and change over time

Percent change equals the difference between the end and start values divided by the start value.

percent change = \frac{end value - start value}{start value}

Lesson: Line graphs and change over time

Difference between two series at time t

Δ_{t} = V_{A} (t) - V_{B} (t)

Lesson: Line graphs and change over time

The trend line equation

y = m x + b

Lesson: Scatter plots, trend lines, and correlation

The actual amount for a category equals its share of the pie times the total amount of the pie.

amount = share \times total

Lesson: Pie charts: parts of a whole

Percent change from a graphic, where the base is the earlier or reference value

percent change = \frac{larger value - smaller value}{base value}

Lesson: Differences and percent change from a graphic

The change in the actual amount of a segment when the total also changes

Δ A = (S_{2} \times T_{2}) - (S_{1} \times T_{1})

Lesson: Differences and percent change from a graphic

The ratio or percent of a part to a whole

ratio or percent = \frac{part}{whole}

Lesson: Ratios and percents from graphics

A subgroup's share of the total equals its share within the category times the category's share of the total.

\frac{S}{T} = \frac{S}{C} \times \frac{C}{T}

Lesson: Ratios and percents from graphics

The legend defines the encoding for every series and region.

value = encoding (series, category)

Lesson: Compound and custom graphics

The procedure for a statement naming a column and a condition

sort by column

\to read rank or value

\to compare against condition

Lesson: Sorting and comparing rows

Count of rows passing every condition, where each $F_{i}$ is the set of rows that pass one filter

count = ∣ F_{1} \cap F_{2} \cap \dots \cap F_{n} ∣

Lesson: Filtering and conditional counts

Weighted average of values with corresponding weights

Weighted average = \frac{\sum ( value \times weight )}{\sum weights}

Lesson: Totals, averages, and medians from a column

Ratio of two columns for a given row

Row Ratio = \frac{Column A}{Column B}

Lesson: Ratios and percents from a table

Amount in a row from a percent column and its base column

A = \frac{p}{100} \times B

Lesson: Ratios and percents from a table

Test a strict association claim by sorting one column and checking the other for reversals.

Sort Column A ascending

\to Scan Column B for reversals

Lesson: Rank agreement and correlation

A statement is supported if and only if it holds for every listed row.

Supported

⟺ \forall row \in Table,

Statement (row) = True

Lesson: Inference from a table: must be true versus could be true

Linked equations

Write the given relation between two unknowns x and y as an equation; the correct pair is the one that satisfies it.

f (x, y) = 0

Lesson: Building the linked equation for quantitative two-part questions

Translating a word problem into a linked equation

Q_{1} \circ Q_{2} = R

Lesson: Building the linked equation for quantitative two-part questions

How to fill each column

selection = the

statement

that

fills

the

role

exactly

as

the

argument

states

it

Lesson: Verbal and logical two-part questions

The effective value of a rule is the value stated in the most recent tab that addresses it.

Effective Rule = Latest Tab Value

Lesson: Multi-source reasoning

The combined result from two sources

result = value from tab A \circ value from tab B

Lesson: Combining facts across sources

A conditional rule applies its outcome only when the condition is met.

If Condition, then Outcome

Lesson: Applying conditional rules to data

GMAT formula sheet PDF

171 formulas, two-sided, US Letter

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