Study notes · Reference · Cheat sheet
The quant formula cheat sheet
Every formula the GMAT and GRE can test fits on one page. Memorize the ones you use weekly, and drill the rest until they are reflexes. Formulas do not solve questions by themselves, but not knowing them guarantees you will not.
The formulas are standard mathematics; what gets re-checked here is the scope, meaning which topics each publisher says its exam can test. The byline is a masthead name, explained in full on who writes this, and the sourcing rules are set out on the standards page.
Bookmark this sheet and pair it with the drills in theGMAT GRE Prep practice bank. Each section below links to a worked example where the formula earns its keep. For the verbal half of the test, the matching reference is the GRE vocabulary list.
Quick-reference table
Use this table as a one-screen index. Each row points to the section that explains the formula and shows it in a worked question.
| Topic | Formula at a glance | Worked question |
|---|---|---|
| Percent change | (new − old) / old × 100% | Successive discount |
| Simple interest | I = P × r × t | Simple interest |
| Ratio setup | a : b = ak : bk | Ratio of new arrivals |
| Exponent addition | a^m × a^n = a^(m+n) | Adding exponents |
| Absolute value | |x − c| < d means c − d < x < c + d | Absolute value integers |
| Linear equation | ax + b = cx + d | Linear equation 6x |
| Pythagorean theorem | a² + b² = c² | Right triangle area |
| Line on a graph | y = mx + b; m = (y₂ − y₁) / (x₂ − x₁) | Line x-intercept |
| Combined work | 1/a + 1/b = 1/together | Combined work rate |
| Mean (missing value) | mean = sum / count | Average missing number |
| Combinations | C(n,k) = n! / (k!(n−k)!) | Committee with women |
| Probability | favorable / total equally likely outcomes | Dice sum of seven |
| Remainders | cycles of units digits; n mod k | Remainder power cycle |
Arithmetic and percents
- Percent change
- (new − old) / old × 100%
- Successive changes
- Multiply factors: +20% then −20% is 1.2 × 0.8 = 0.96
- Simple interest
- I = P × r × t (see the worked example)
- Compound interest
- A = P(1 + r/n)^(nt)
- Ratio setup
- a : b = ak : bk; keep the multiplier k (see this question)
Exponents and roots
- Product rule
- a^m × a^n = a^(m+n)
- Quotient rule
- a^m / a^n = a^(m−n)
- Power of a power
- (a^m)^n = a^(mn)
- Negative exponent
- a^(−n) = 1 / a^n
- Fractional exponent
- a^(1/n) = the nth root of a
- Same terms added
- a^n + a^n = 2·a^n (see this question)
Algebra
- Difference of squares
- a² − b² = (a − b)(a + b)
- Perfect squares
- (a ± b)² = a² ± 2ab + b²
- Quadratic formula
- x = (−b ± √(b² − 4ac)) / 2a
- Absolute value inequality
- |x − c| < d means c − d < x < c + d
- Distance between points
- √((x₂ − x₁)² + (y₂ − y₁)²)
- Slope
- m = (y₂ − y₁) / (x₂ − x₁)
Geometry
- Triangle area
- (1/2) × base × height (legs in a right triangle)
- Pythagorean theorem
- a² + b² = c²; memorize 3-4-5, 5-12-13, 8-15-17
- Circle area and circumference
- A = πr², C = 2πr
- Rectangle and square
- A = lw; square diagonal = s√2
- Triangle angle sum
- 180°; exterior angle = sum of remote interiors
- Cube and cylinder volume
- V = s³; V = πr²h
Rates and work
- Distance
- d = r × t
- Combined work
- Add rates: 1/a + 1/b = 1/together (see this question)
- Average speed
- total distance / total time, never the average of speeds
- Consecutive integers
- n, n+1, n+2 …; sum of n terms = n × median
Statistics
- Mean
- sum / count; work with sums to undo it (example)
- Median
- Middle value of an ordered list; average the middle two if count is even
- Weighted average
- (w₁x₁ + w₂x₂) / (w₁ + w₂)
- Range
- max − min
- Standard deviation (concept)
- How spread out values are; you compare spreads, you never compute it
Counting and probability
- Combinations
- C(n,k) = n! / (k!(n−k)!) (see this question)
- Permutations
- P(n,k) = n! / (n−k)!
- At least one
- 1 − P(none); complement first
- Probability
- favorable / total equally likely outcomes (example)
- Independent events
- P(A and B) = P(A) × P(B)
- Mutually exclusive events
- P(A or B) = P(A) + P(B)
Number properties
- Even and odd parity
- odd + odd = even; even + odd = odd; odd × odd = odd; even × anything = even
- Divisibility by 3
- A number is divisible by 3 if the sum of its digits is divisible by 3 (same rule for 9, using the digit sum)
- Divisibility by 4
- Last two digits form a number divisible by 4
- Prime numbers under 30
- 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 (1 is not prime; 2 is the only even prime)
- GCF and LCM
- GCF shares the lowest exponents across prime factorizations; LCM shares the highest
- Factorial
- n! = n × (n−1) × … × 1; note 0! = 1
- Remainder notation
- x mod k = r means x = kq + r, with 0 ≤ r < k (see remainder cycle)
Sequences and coordinate geometry
- Arithmetic sequence
- a_n = a_1 + (n − 1)d; sum of n terms = n × (a_1 + a_n) / 2
- Geometric sequence
- a_n = a_1 × r^(n − 1); sum of n terms = a_1 × (1 − r^n) / (1 − r), r ≠ 1
- Midpoint of a segment
- ((x₁ + x₂)/2, (y₁ + y₂)/2)
- Distance between points
- √((x₂ − x₁)² + (y₂ − y₁)²)
- Slope
- m = (y₂ − y₁) / (x₂ − x₁)
- Line equation
- y = mx + b; b is the y-intercept (see x-intercept question)
- Parallel and perpendicular slopes
- parallel: m₁ = m₂; perpendicular: m₁ × m₂ = −1
Extra geometry
- Trapezoid area
- (1/2) × (base₁ + base₂) × height
- Parallelogram area
- base × height (not base × side)
- Regular polygon interior angle
- (n − 2) × 180° / n, where n is the number of sides
- Sphere surface and volume
- A = 4πr²; V = (4/3)πr³
- Cone volume
- V = (1/3)πr²h
- Inscribed angle (circle)
- half of the central angle that subtends the same arc
- Sum of interior angles
- (n − 2) × 180°, where n is the number of sides
How to actually memorize these
- Recall, do not reread. Cover the formula column and reproduce it from memory. Rereading feels fluent and is nearly useless.
- Attach each formula to one question. The links above exist for this: a formula tied to a solved problem sticks far better than a formula on a list.
- Spaced review. Ten minutes at the start of every study session, from week 2 of the 8-week plan onward.