Add fractions
$$\frac{a}{b}+\frac{c}{d}=\frac{ad+bc}{bd}$$
Rewrite two fractions over a common denominator, then add their numerators.
Middle schoolArithmetic formulas, grouped by topic.
13 arithmetic formulas across 7 topics, free and without an account. Arithmetic formulas turn everyday quantities into something you can compute with: parts of a whole, rates over time, and money that grows. Almost every one of them is really a statement about proportion, so the reliable move is to write the relationship first and substitute second. Watch the units, because a rate formula with mismatched units gives a number that looks fine and is wrong.
Every arithmetic formula on one page
Each entry links to its own page, where the symbols are defined, the conditions are stated, and the formula is substituted through a worked example. Levels covered here: Middle school, High school.
Fractions
3 formulas in fractions.
Multiply fractions
$$\frac{a}{b}\cdot\frac{c}{d}=\frac{ac}{bd}$$
Multiply numerators together and denominators together.
Middle schoolDivide fractions
$$\frac{a}{b}\div\frac{c}{d}=\frac{a}{b}\cdot\frac{d}{c}$$
Multiply by the reciprocal of the divisor.
Middle schoolPercent
2 formulas in percent.
Percent of a quantity
$$\text{part}=\frac{p}{100}\cdot\text{whole}$$
Find a stated percentage of a whole quantity.
Middle schoolPercent change
$$\%\text{ change}=\frac{\text{new}-\text{original}}{|\text{original}|}\cdot100\%$$
Compare a change with the magnitude of its starting value.
Middle school · High schoolRates
2 formulas in rates.
Distance, rate, and time
$$d=rt$$
Connect distance traveled, constant rate, and elapsed time.
Middle school · High schoolUnit rate
$$\text{unit rate}=\frac{\text{quantity}}{\text{number of units}}$$
Express a comparison per one unit.
Middle schoolFinance
3 formulas in finance.
Simple interest
$$I=Prt,\qquad A=P(1+rt)$$
Calculate interest only on the original principal.
Middle school · High schoolCompound interest
$$A=P\left(1+\frac{r}{n}\right)^{nt}$$
Grow an initial amount when interest is added repeatedly.
High schoolContinuous compounding
$$A=Pe^{rt}$$
Model growth when interest is compounded continuously.
High schoolRatios
1 formula in ratios.
Equivalent proportions
$$\frac{a}{b}=\frac{c}{d}\iff ad=bc$$
Test or solve an equality between two ratios by cross multiplication.
Middle schoolAverages
1 formula in averages.
Arithmetic average
$$\bar{x}=\frac{x_1+x_2+\cdots+x_n}{n}$$
Share a total equally among all observations.
Middle school · High schoolMeasurement
1 formula in measurement.
Percent error
$$\%\text{ error}=\frac{|\text{measured}-\text{accepted}|}{|\text{accepted}|}\cdot100\%$$
Compare measurement error with an accepted reference value.
Middle school · High schoolKeep going with arithmetic
A formula sheet is a reference, not a method. Use the subject guide to see where these relationships come from, a calculator to check a substitution you are unsure about, and practice questions to find out whether you can actually retrieve them.
- Study arithmetic — the subject hub with lessons, worked methods, and common mistakes.
- Arithmetic calculator — enter your own values and compare the result with your working.
- Arithmetic Practice — retrieval practice across every topic listed above.
- Math formula library — search all 215 formulas by name, symbol, or condition.
Formulas in the other subjects
Mathematics does not stop at a subject boundary, and neither do its formulas. Each subject page below lists its own relationships, conditions, and worked examples.
Algebra
Review the core algebra relationships, their restrictions, and worked substitutions.
Geometry
Review the core geometry relationships, their restrictions, and worked substitutions.
Trigonometry
Review the core trigonometry relationships, their restrictions, and worked substitutions.
Precalculus
Review the core precalculus relationships, their restrictions, and worked substitutions.
Calculus
Review the core calculus relationships, their restrictions, and worked substitutions.
Statistics
Review the core statistics relationships, their restrictions, and worked substitutions.
Probability
Review the core probability relationships, their restrictions, and worked substitutions.
Linear Algebra
Review the core linear algebra relationships, their restrictions, and worked substitutions.
Differential Equations
Review the core differential equations relationships, their restrictions, and worked substitutions.
Discrete Math
Review the core discrete math relationships, their restrictions, and worked substitutions.