Tip: print this page - it works as a one-page cheat sheet for practice sessions (no formula sheet is provided on the real test).
Percentages
Percent problems make up a large share of Arithmetic Reasoning questions. These three formulas cover finding a percent of a number, finding a percent change, and working backward to an original value.
| Formula | When to Use It |
|---|---|
| part = percent (as a decimal) × whole | Find a percent of a number, e.g. "What is 30% of 80?" |
| percent change = (new - old) / old × 100 | Find the percent increase or decrease between two values |
| original = new / (1 ± rate) | Work backward to the value before a percent change - use + if the value increased, - if it decreased |
Ratios and Proportions
When two ratios are set equal to each other, cross multiplication turns the proportion into a simple equation you can solve for the missing term.
| Formula | When to Use It |
|---|---|
| a/b = c/d, so ad = bc | Cross-multiply to solve a proportion for a missing term |
Distance-Rate-Time
Motion problems almost always come down to one relationship between distance, rate, and time, plus one rule for averaging speed over a multi-leg trip.
| Formula | When to Use It |
|---|---|
| d = r × t | Solve for distance, rate, or time when given the other two (r = d/t, t = d/r) |
| average speed = total distance / total time | Multi-leg trips at different speeds - don't just average the speeds, divide total distance by total time |
Work Problems
Combined-work problems ask how long two people or machines take to finish a job when working together.
| Formula | When to Use It |
|---|---|
| 1/t = 1/a + 1/b | Two workers with individual completion times a and b work together; t is their combined time |
Averages
Averages show up across both Arithmetic Reasoning and Mathematics Knowledge, from a simple mean to problems that ask for a missing value.
| Formula | When to Use It |
|---|---|
| mean = sum / count | Find the average of a list of values |
| missing value = (target mean × new count) - current sum | Find the value needed to bring the group to a target average |
| median = middle of a sorted list | Find the middle value once values are ordered least to greatest (average the two middle values if the count is even) |
| mode = most frequent value | Find the value that appears most often in a list |
Exponents
Mathematics Knowledge tests these five exponent rules directly and inside more complex expressions.
| Formula | When to Use It |
|---|---|
| xa · xb = xa+b | Multiplying same-base powers - add the exponents |
| xa / xb = xa-b | Dividing same-base powers - subtract the exponents |
| (xa)b = xab | Raising a power to a power - multiply the exponents |
| x0 = 1 | Any nonzero base raised to the zero power equals 1 |
| x-a = 1/xa | Rewrite a negative exponent as a positive-exponent fraction |
Algebra
These three identities cover most of the algebra questions on Mathematics Knowledge, from multiplying binomials to factoring.
| Formula | When to Use It |
|---|---|
| (a+b)(c+d) = ac + ad + bc + bd | Multiply two binomials (FOIL: First, Outer, Inner, Last) |
| a2 - b2 = (a+b)(a-b) | Factor a difference of squares |
| x2 + bx + c = (x+m)(x+n), where m+n=b and mn=c | Factor a quadratic by finding two numbers that add to b and multiply to c |
Geometry
Geometry questions test area, perimeter, circumference, and volume for a handful of shapes that come up again and again.
| Formula | When to Use It |
|---|---|
| A = lw | Area of a rectangle (length × width) |
| P = 2(l+w) | Perimeter of a rectangle |
| A = ½bh | Area of a triangle (base × height, divided by 2) |
| A = πr2 | Area of a circle |
| C = 2πr | Circumference of a circle |
| V = lwh | Volume of a rectangular box |
| V = πr2h | Volume of a cylinder |
Pythagorean Theorem
The Pythagorean theorem finds a missing side of a right triangle. Memorizing common triples saves time by skipping the calculation entirely.
| Formula | When to Use It |
|---|---|
| a2 + b2 = c2 | Find a missing side of a right triangle; c is the hypotenuse (the longest side) |
| Common triples: 3-4-5, 5-12-13, 8-15-17 | Recognize these side ratios instantly instead of calculating them |
Interest
Simple interest problems appear on Arithmetic Reasoning as a straightforward plug-and-solve formula.
| Formula | When to Use It |
|---|---|
| I = P × r × t | Simple interest: principal × rate (as a decimal) × time (in years) |