GMAT Quantitative Reasoning Cheat Sheet: Topics, Formulas, and Examples

Essential formulas at a glance

These are entry points, not substitutes for the full conditions in the topic sections. Throughout the guide, percentage rates used in multiplication are decimals: 12% means $r=0.12$, not $r=12$.

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Integers, signs, parity, and divisibility

Recognize the number set before calculating. Integers include negative whole numbers, zero, and positive whole numbers; a prime is an integer greater than 1 with exactly two positive divisors. The number 1 is neither prime nor composite, 2 is the only even prime, and zero is even.

• Signs: An even number of negative factors produces a positive product; an odd number produces a negative product. A zero factor makes the product zero.
• Parity: Even + even and odd + odd are even; even + odd is odd. A product is odd only when every integer factor is odd.
• Consecutive integers: Write $n,n+1,n+2$. For consecutive even or odd integers, use $n,n+2,n+4$, with the parity of $n$ specified.
• Zero rules: $0/a=0$ for $a\ne0$, but division by zero is undefined. Do not divide an equation by a variable before checking whether that variable can be zero.

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For a positive integer, test possible prime divisors only up to its square root when checking primality. If a composite integer is $ab$ with both factors greater than its square root, their product would exceed the integer, so at least one factor must be no larger than that square root.

Trap: A prime number need not be odd because 2 is prime. A negative integer can be odd, and “positive” never includes zero.

Prime factors, GCF, LCM, and factorials

Prime factorization turns divisibility into bookkeeping. For positive integers, the greatest common factor (GCF, also called GCD or HCF) uses the minimum shared prime exponents; the least common multiple (LCM) uses the maximum exponents from either number.

$$\gcd(a,b)\operatorname{lcm}(a,b)=ab$$

This identity is for two positive integers. Do not extend it by simply multiplying the GCF and LCM of three numbers.

If $N=p_1^{e_1}p_2^{e_2}\cdots p_k^{e_k}$, with distinct primes and positive integer exponents, then:

$$\text{Number of positive divisors of }N=\prod_{i=1}^{k}(e_i+1)$$

A perfect square has even exponents in its prime factorization; a perfect cube has exponents divisible by 3. The exponent of a prime $p$ in $n!$, for a nonnegative integer $n$, is:

$$\left\lfloor\frac np\right\rfloor+ \left\lfloor\frac n{p^2}\right\rfloor+ \left\lfloor\frac n{p^3}\right\rfloor+\cdots$$

Stop once the terms become zero. Here $\lfloor x\rfloor$ is the greatest integer no greater than $x$; $n!=n(n-1)\cdots1$, and $0!=1$. Trailing zeros in $n!$ count factors of 10, so count factors of 5 because factors of 2 are more numerous.

Remainders, units digits, and inclusive counting

For an integer $N$ divided by a positive integer $d$:

$$N=dq+r,\qquad 0\le r<d$$

The remainder must be nonnegative and less than the divisor. For example, $-7=5(-2)+3$, so the nonnegative remainder on division by 5 is 3, not $-2$.

For sums and products, replace numbers by their remainders and reduce again. If $a$ leaves remainder $r$ and $b$ leaves remainder $s$ modulo $d$, then $a+b$ has the remainder of $r+s$, and $ab$ has the remainder of $rs$. Do not cancel or divide remainders as though every modular equation were an ordinary equation.

• Units digits: Work modulo 10 and look for a repeating cycle. Powers of 3 cycle through 3, 9, 7, 1, so $3^{14}$ ends in 9.
• Place value: A three-digit integer with digits $a,b,c$ is $100a+10b+c$, with $a\ne0$.
• Inclusive interval: There are $U-L+1$ integers from integer $L$ through integer $U$, if $U\ge L$.
• Multiples in an interval: For positive integer $k$ and integer endpoints $L\le U$, the count is:

$$\left\lfloor\frac Uk\right\rfloor- \left\lfloor\frac{L-1}{k}\right\rfloor$$

Trap: “Between” can be ambiguous in everyday language; obey the stated strict or inclusive boundaries. Between 10 and 30 inclusive, the multiples of 4 are 12, 16, 20, 24, and 28, so the count is $7-2=5$.

Fractions, decimals, and estimation

Fraction operations require nonzero denominators. Cancel common factors in a product, not terms in a sum.

$$\frac ab+\frac cd=\frac{ad+bc}{bd}, \qquad \frac ab\cdot\frac cd=\frac{ac}{bd}$$ $$\frac ab\div\frac cd=\frac{ad}{bc} \quad\text{with }b,c,d\ne0$$

For fractions with positive denominators, compare $a/b$ and $c/d$ by comparing $ad$ and $bc$. If signs are uncertain, resolve them before cross-multiplying an inequality.

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A reduced fraction has a terminating decimal exactly when its positive denominator has no prime factors except 2 and 5. Thus $7/40$ terminates, while $7/30$ does not; reduce first because $3/6=1/2$.

Scientific notation writes a positive number as $a\times10^k$, where $1\le a<10$. Keep powers of 10 separate from the leading factors to avoid losing decimal places. For example, $0.0048\times0.025=(4.8\times10^{-3})(2.5\times10^{-2})=1.2\times10^{-4}$.

Estimation rule: Use bounds that preserve the answer decision. If options are close together, coarse rounding may destroy the distinction; if you need a lower bound on a positive product, round both factors downward rather than mixing directions.

Percents, reverse percents, and successive changes

Use the base explicitly. “Percent of,” “percent greater than,” and “percentage-point increase” are different instructions.

$$\text{part}=\frac p{100}\times\text{whole}$$ $$\text{new}=\text{old}(1+r),\qquad \text{old}=\frac{\text{new}}{1+r}$$

Here $r$ is a signed decimal change and $1+r\ne0$ for the reverse formula. A decrease of 20% uses $r=-0.20$; an increase of 20% uses $r=0.20$.

$$\text{Successive-change multiplier}=(1+r_1)(1+r_2)\cdots(1+r_n)$$

The net decimal change is that product minus 1. Do not add percentages unless they apply to the same unchanged base.

• Percent change: For a positive original amount, $(\text{new}-\text{old})/\text{old}\times100\%$.
• Percentage points: A rate rising from 20% to 25% rises 5 percentage points, but its relative increase is $5/20=25\%$.
• Percent more versus less: If $A=(1+r)B$ for $r>0$, then $B$ is $r/(1+r)$ less than $A$, expressed as a decimal fraction.
• Percent error: When a question specifies the positive true value as the base, absolute relative error is $|\text{estimate}-\text{true}|/\text{true}\times100\%$. Follow any different definition given in the prompt.

Ratios, proportions, and variation

Represent a ratio with a shared multiplier. If $A:B=a:b$, write $A=ak$ and $B=bk$ rather than assuming the quantities equal the ratio numbers themselves.

$$A=\frac{a}{a+b}T,\qquad B=\frac{b}{a+b}T$$

These share formulas assume positive ratio parts and total $T=A+B$. A proportion $a/b=c/d$ gives $ad=bc$ when $b,d\ne0$.

• Linked ratios: If $A:B=2:3$ and $B:C=4:5$, match the $B$ terms: $A:B:C=8:12:15$.
• Direct variation: $y=kx$, so $y/x=k$ for $x\ne0$.
• Inverse variation: $y=k/x$, so $xy=k$, with $x\ne0$.
• Joint variation: $y=kxz$ when the prompt says $y$ varies directly with both $x$ and $z$.

Trap: A part-to-part ratio is not a part-to-whole fraction. If women to men is $3:2$, the fraction of the group who are women is $3/5$, not $3/2$. Check whether quantities must be integers before choosing a convenient value for $k$.

Exponents, roots, and magnitude

For integer exponents, use the rules below wherever both sides are defined. Nonzero bases are required for division, zero powers, and negative powers.

$$a^m a^n=a^{m+n},\qquad \frac{a^m}{a^n}=a^{m-n},\qquad (a^m)^n=a^{mn}$$ $$(ab)^n=a^n b^n,\qquad a^0=1,\qquad a^{-n}=\frac1{a^n}$$

For fractional powers, the simple real-number rules are safest with positive bases: $a^{m/n}=\sqrt[n]{a^m}$, where $m$ is an integer and $n$ is a positive integer. Even roots require nonnegative radicands over the reals; an even root symbol denotes the principal nonnegative root.

$$\sqrt{a^2}=|a|, \qquad \sqrt{ab}=\sqrt a\sqrt b\quad(a,b\ge0)$$

If $x^2=49$, then $x=\pm7$, but $\sqrt{49}=7$. The expression $\sqrt{a+b}$ is not generally $\sqrt a+\sqrt b$.

• Between 0 and 1: Squaring makes a positive number smaller; taking its square root makes it larger.
• Above 1: Squaring makes the number larger; taking its square root makes it smaller.
• Negative bases: Parentheses matter: $(-3)^2=9$, but $-3^2=-9$.
• Useful squares: Know $1^2$ through $15^2$, and recognize perfect-square factors such as $72=36\cdot2$.

Algebraic identities and rational expressions

Recognize a structure before expanding everything. These identities often reveal cancellation or an easier substitution.

$$(a+b)^2=a^2+2ab+b^2,\qquad (a-b)^2=a^2-2ab+b^2$$ $$a^2-b^2=(a-b)(a+b)$$ $$a^3-b^3=(a-b)(a^2+ab+b^2)$$ $$a^3+b^3=(a+b)(a^2-ab+b^2)$$

Keep domain restrictions when simplifying. The expression $(x^2-9)/(x-3)$ simplifies to $x+3$, but the original still excludes $x=3$. Cancellation does not make an undefined original expression defined.

Trap: From $x(x-4)=0$, the solutions are $x=0$ and $x=4$. Dividing by $x$ without considering zero loses a valid solution. Similarly, $(a+b)/a$ is $1+b/a$, not $b$.

Linear equations, systems, and quadratics

For $ax+b=c$, $x=(c-b)/a$ if $a\ne0$. When $a=0$, the equation may be an identity or a contradiction instead of having one solution. With a system, use substitution or elimination and check whether the equations are independent.

$$ax^2+bx+c=0 \quad\Longrightarrow\quad x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}, \qquad a\ne0$$

For real coefficients, the discriminant $b^2-4ac$ determines the number of real roots: positive gives two distinct roots, zero gives a repeated root, and negative gives no real root. Factoring is often simpler than using the quadratic formula.

If the roots are $r_1,r_2$, their sum is $-b/a$ and their product is $c/a$. This can answer a question about the roots without solving for each root individually.

For equations with variable denominators, list excluded values first. For radical equations, isolate the radical, square only as needed, and substitute every candidate into the original equation because squaring can introduce extraneous solutions.

Inequalities and absolute value

Adding or subtracting the same quantity preserves an inequality. Multiplying or dividing by a positive quantity preserves it; multiplying or dividing by a negative quantity reverses it. When the sign is unknown, separate cases rather than guessing.

$$-2x>6\quad\Longrightarrow\quad x<-3$$

For product or rational inequalities, find zeros and undefined points, divide the number line into intervals, and test a value in each interval. Exclude denominator zeros even when a non-strict inequality would otherwise include an endpoint.

Absolute value is distance from zero. For $c\ge0$:

$$|u|=c\iff u=c\text{ or }u=-c$$ $$|u|\le c\iff -c\le u\le c$$ $$|u|\ge c\iff u\le-c\text{ or }u\ge c$$

For $c<0$, $|u|=c$ and $|u|\le c$ have no real solutions. An inequality $|u|\ge c$ is then true for every real $u$. For strict versions, change the endpoints accordingly.

Functions and newly defined operations

Function notation tells you to substitute into a rule. If $f(x)=2x^2-3$, then $f(a+1)=2(a+1)^2-3$; substitute the whole input with parentheses before simplifying.

$$(f\circ g)(x)=f(g(x))$$

Composition is generally not commutative: $f(g(x))$ need not equal $g(f(x))$. A newly defined symbol is only as meaningful as its definition; if $a\star b=a^2-b$, apply that rule instead of treating $\star$ as multiplication.

• Domain: Denominators cannot be zero, and real even roots cannot have negative radicands.
• Piecewise rules: Choose the branch using the input's condition before calculating.
• Inverse versus reciprocal: $f^{-1}(x)$, when an inverse exists, is not the same notation as $1/f(x)$.

Trap: If $f(x)=x^2$, then $f(a+b)=(a+b)^2$, not $a^2+b^2$. Read every new operation as a fresh definition, even when its symbol resembles a familiar one.

Translating word problems and converting units

Define the unknown in the units the question requests. Then translate one relationship at a time, keeping totals, differences, and ratios separate.

On a narrow screen, scroll the table horizontally to see every column.

For unit conversion, multiply by a fraction equal to 1 in different units. For example, $90\text{ km/h}\times(1000\text{ m}/1\text{ km})\times(1\text{ h}/3600\text{ s})=25\text{ m/s}$.

In integer-allocation problems, rounding has a purpose. If a vehicle carries at most 8 people, transporting 35 people requires $\lceil35/8\rceil=5$ vehicles; $\lceil x\rceil$ means the smallest integer at least $x$. This is not ordinary rounding to the nearest integer.

Trap: Two people's age difference remains constant, but their age ratio changes over time. The phrase “three times as old” must apply at the stated date, not automatically at every date.

Distance, speed, and average rates

For constant speed, distance equals speed times time. When a trip has stages, calculate each stage's distance and time, then add.

$$D=rt,\qquad v_{\rm avg}=\frac{D_1+D_2+\cdots}{t_1+t_2+\cdots}$$

Two special cases are worth remembering:

$$\text{Equal times:}\quad v_{\rm avg}=\frac{u+v}{2}$$ $$\text{Equal distances:}\quad v_{\rm avg}=\frac{2uv}{u+v} \quad(u,v>0)$$

For constant speeds along the same line, objects moving toward each other close a gap at $u+v$; a faster object chasing a slower one closes it at $u-v$, assuming $u>v$. Meeting or catch-up time equals the initial gap divided by the appropriate closing speed.

Work rates, machines, and filling tanks

If a worker completes one job in $a$ time units at a constant rate, the rate is $1/a$ jobs per time unit. Add rates for independent workers contributing to the same job; subtract a drain or opposing process when appropriate.

$$\text{work}=\text{rate}\times\text{time},\qquad \frac1T=\frac1a+\frac1b$$ $$T=\frac{ab}{a+b}$$

The two-worker shortcut assumes both rates remain constant and they work together throughout. If they start or stop at different times, calculate work completed in each time interval instead.

For identical machines or workers with unchanged productivity, output is proportional to number × rate per machine × time. The phrase “identical” matters; different workers cannot automatically be counted as equal units of output.

Mixtures and concentration

Track the quantity of the substance, not just the percentages. With compatible units and additive quantities, concentration multiplied by amount gives the amount of the ingredient.

$$c_1V_1+c_2V_2=c_f(V_1+V_2)$$

Concentrations are decimals, and $V_1,V_2$ must use the same quantity basis. For two concentrations $c_L<c_H$ mixed to a target $c_L<c<c_H$:

$$\frac{V_L}{V_H}=\frac{c_H-c}{c-c_L}$$

This ratio method is a rearrangement of the weighted-average equation, not a separate rule to use without understanding the units. A target concentration must lie between the component concentrations when positive amounts are combined.

For a thoroughly mixed container, removing fraction $f$ removes that same fraction of every ingredient. After replacing the removed volume with pure solvent and repeating the same procedure $n$ times, concentration is $c_0(1-f)^n$, assuming the mixture is homogeneous each time and total volume is restored.

Revenue, profit, discounts, and interest

Separate the base of each percentage. Let $C$ be cost, $S$ selling price, and $P_f$ profit for an item.

$$P_f=S-C,\qquad \text{markup rate}=\frac{S-C}{C},\qquad \text{profit margin}=\frac{S-C}{S}$$

The markup formula requires $C>0$, and the margin formula requires $S>0$. For quantity $q$, revenue is unit price × $q$; a simple fixed-plus-variable cost model is $F+cq$. If unit selling price is $p>c$, break-even quantity is $F/(p-c)$; if only whole units can be sold, round up to reach or exceed break-even.

For simple interest, with principal $P$, decimal rate $r$ per time unit, and $t$ matching time units:

$$I=Prt,\qquad A=P(1+rt)$$

For compound interest with periodic rate $r$ over $n$ periods and no additional deposits or withdrawals:

$$A=P(1+r)^n$$

If a nominal annual rate $j$ is compounded $m$ times per year over $t$ years, use $A=P(1+j/m)^{mt}$ when $mt$ is the stated number of compounding periods. Interest earned is $A-P$, not $A$.

Trap: Apply tiered prices or taxes to the units or amounts within each tier unless the problem explicitly says the final rate applies to the whole purchase. Do not treat a marginal rate as an average rate.

Means, medians, range, and standard deviation

The arithmetic mean converts between a total and a count. For $n>0$ values:

$$\bar x=\frac{x_1+\cdots+x_n}{n},\qquad \text{sum}=n\bar x$$

For groups with positive total count:

$$\bar x_{\rm combined}=\frac{n_1\bar x_1+n_2\bar x_2}{n_1+n_2}$$

When one value changes from $a$ to $b$ in a fixed-size set, the mean changes by $(b-a)/n$. To add a new value $x$, use $\bar x_{\rm new}=(n\bar x+x)/(n+1)$; when removing $x$, use $(n\bar x-x)/(n-1)$ for $n>1$.

• Median: Sort first. For odd $n$, use position $(n+1)/2$; for even $n$, average positions $n/2$ and $n/2+1$.
• Mode: The most frequent value or values; a set need not have a unique mode.
• Range: Maximum minus minimum.
• Symmetric, equally spaced set: The mean and median equal $(\text{first}+\text{last})/2$.

For a population of $n>0$ values, population standard deviation is:

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

The sample standard-deviation convention uses $n-1$ instead of $n$, with $n>1$. Follow the definition in the question; this reference uses the population version above.

The transformation rules are often more useful than lengthy arithmetic. Adding a constant to every value changes the mean and median by that constant but leaves standard deviation and range unchanged. Multiplying every value by $k$ multiplies the mean by $k$ and multiplies standard deviation and range by $|k|$. Standard deviation is zero exactly when all values are equal.

Overlapping sets and two-way groups

Let $|A|$ mean the number of elements in set $A$. For two finite sets:

$$|A\cup B|=|A|+|B|-|A\cap B|$$

If $N$ is the whole population, neither equals $N-|A\cup B|$. The number in exactly one of the two sets is $|A|+|B|-2|A\cap B|$, and “A only” is $|A|-|A\cap B|$.

For three sets, inclusion-exclusion is:

$$|A\cup B\cup C| =|A|+|B|+|C| -|A\cap B|-|A\cap C|-|B\cap C| +|A\cap B\cap C|$$

Each pairwise intersection in this formula includes the triple intersection. If the prompt gives “A and B only,” that is a different region.

For two separate yes/no classifications, a two-way table is often clearer than a Venn diagram. Use row totals, column totals, and the grand total to reconstruct missing cells; do not assume statistical independence simply because there are two classifications.

Counting, permutations, and combinations

First decide whether order matters, whether repetition is allowed, and whether the objects are distinguishable. A formula cannot make those choices for you.

• Multiplication principle: Multiply the number of choices at successive stages when those counts describe the available choices on each path.
• Addition principle: Add counts for disjoint alternatives. If alternatives overlap, remove double-counting.
• Ordered choices with replacement: $n^r$ for $r$ positions, each allowing any of the same $n$ choices.
• Ordered choices without replacement: $nP_r=n!/(n-r)!$.
• Unordered choices without replacement: $\binom nr=n!/[r!(n-r)!]$.
• Arrange a multiset: $n!/(n_1!n_2!\cdots)$ when identical-item counts sum to $n$.
• Circular arrangements: $(n-1)!$ for $n\ge1$ distinct objects when rotations are equivalent but reflections remain distinct.

For permutations and combinations, $n,r$ are integers with $0\le r\le n$. Restriction problems may be easier through cases, a complement, or treating items that must stay together as a block; account for internal block arrangements when they differ.

Trap: A three-digit number cannot start with zero. Its first position can have a different choice count from later positions, so “ten choices everywhere” may be wrong.

Probability and conditional probability

Counting favorable outcomes divided by total outcomes works only when the elementary outcomes being counted are equally likely. Otherwise, add the actual probabilities.

$$P(A^c)=1-P(A)$$ $$P(A\cup B)=P(A)+P(B)-P(A\cap B)$$ $$P(A\cap B)=P(A)P(B\mid A)$$ $$P(B\mid A)=\frac{P(A\cap B)}{P(A)} \qquad(P(A)>0)$$

For independent events, $P(B\mid A)=P(B)$, so the intersection becomes $P(A)P(B)$. Mutually exclusive events cannot occur together, so their intersection is zero; mutual exclusivity and independence are not interchangeable.

For $n$ independent trials with the same success probability $p$, exactly $k$ successes has the probability below, where $n,k$ are integers, $0\le k\le n$, and $0<p<1$. The endpoint models $p=0$ and $p=1$ are certain failure and certain success, respectively.

$$\binom nk p^k(1-p)^{n-k}$$

For the same model, at least one success is $1-(1-p)^n$. Without replacement, update the counts after each draw unless a valid counting method already accounts for the dependence.

Sequences, series, and partial sums

An arithmetic sequence has a constant difference $d$. For positive integer $n$:

$$a_n=a_1+(n-1)d,\qquad S_n=\frac n2(a_1+a_n)$$

A geometric sequence has a constant ratio $r$:

$$a_n=a_1r^{n-1},\qquad S_n=a_1\frac{1-r^n}{1-r}\quad(r\ne1)$$

When $r=1$, the sum is $na_1$. When $r=0$, retain the given first term and set every later term to zero rather than evaluating $0^0$. If an infinite geometric series is explicitly involved, the usual convergence rule is $|r|<1$, giving $a_1/(1-r)$; a finite sum does not need that convergence condition.

Useful finite sums include:

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

For a recursive sequence, calculate a few terms and obey its initial condition. If $S_n$ is the sum of the first $n$ terms, then $a_n=S_n-S_{n-1}$ for $n\ge2$.

A final problem-solving checklist

Use this routine to decide whether a formula is doing the right job. It is a reasoning checklist, not a promise that every problem requires algebraic calculation.

• Name the target: Are you finding a value, difference, ratio, remainder, maximum, or number of possibilities?
• List restrictions: Positive, integer, distinct, nonzero, within an interval, or without replacement?
• Choose a model: Equation, rate table, weighted total, prime factorization, cases, or complementary probability?
• Check the denominator: Original price, cost, revenue, total time, subgroup size, or all possible outcomes?
• Choose an efficient method: Direct calculation, factoring, back-solving from choices, testing permissible numbers, or estimation?
• Verify the result: Does it satisfy the original conditions, use the requested units, and answer the actual question?

For full worked problems, use our GMAT Quant questions explained and Quant practice questions with hidden answers. Those pages provide practice; this page provides the reference to consult when a concept or condition is unclear.

Keep other preparation decisions separate. Use the timing strategy for pacing, the practice-test guide for choosing a mock, and the score hub for interpreting reported scores.