Fractions Calculator

What does this fractions calculator compute?

A fraction a/b is a numerator over a nonzero denominator. This calculator adds, subtracts, multiplies, and divides proper/improper fractions and mixed numbers; simplifies to lowest terms; and converts between fractions and decimals.

The published formula or method is Add/subtract: L = LCM(b,d); [a(L/b) ± c(L/d)] / L (equivalent (ad ± bc)/bd). Multiply ac/bd. Divide a/b × d/c.. The following references support the stated mathematics or method:

• Margaret Stevens — The Maths Workbook (University of Oxford): Algebraic addition/subtraction, multiplication/division, distributive rules, equivalent fractions, common denominators and reciprocals. Ratios are interpreted as quotients. Equality of ratios with nonzero denominators permits cross multiplication, and solving a proportion uses the chapter’s equation transformations. A zero divisor is excluded; integer quotient/remainder uses the separately cited Euclidean-division reference. Mathematical reference only; not implementation validation or an endorsement of CalculatorX.

Which inputs and limits should you check?

Follow the input domain, units, branch or solve direction, and rounding policy in the specification. Preserve exact outputs and any additional candidates or warnings.

The worked example uses 3/4 + 1/6 with LCD = LCM(4, 6) = 12; 9/12 + 2/12. Compare it with the Example calculations and the complete result for the same inputs.

How can you verify and cite this result?

CalculatorX is responsible for this page. The mathematical references above explain the model; implementation checks and their scope are documented separately in the math.fractions v1.0.8 Evidence and capability contract. Read the displayed verification and production state together with the specification's numerical limits.

Use the clean calculator URL to cite the tool. A shared input link identifies a particular calculation. Content updates, numerical verification, and domain reviews have separate dates and scopes.

What is a fraction?

A fraction a/b has:

• numerator a — how many equal parts you have
• denominator b — how many equal parts make one whole (b ≠ 0)

Examples: 3/8 of a pie is three of eight equal slices. Remaining after eating three of eight is 5/8.

Exact fractions and decimal conversion limits

Fraction fields are decimal safe integers with absolute value at most 2^53−1. Integer text may include a sign and surrounding whitespace, at most64 trimmed characters and128 raw characters. Required numerators and denominators never default from explicit empty or missing values; whole parts alone may be absent or empty and mean0. An unknown mode, operation, machine field, wrong type or conflicting alias is rejected. operator aliases op, decInput aliases decimal, n/d alias the first numerator/denominator and c aliases the second numerator. Known inactive fields are type-checked then omitted from normalized inputs.

A mixed input has magnitude |whole|×|denominator|+|numerator|. A negative whole or negative numerator selects a negative magnitude once; a negative denominator flips that sign. These are signed mixed-number parts, not an algebraic sum of independently signed terms. For example −2 and1/3 means −7/3. Fraction arithmetic and every LCD/product/intermediate must remain safe integers; even an eventually reducible result is rejected if an intermediate exceeds that envelope.

From decimal first normalizes a finite decimal value to binary64, then converts its shortest decimal serialization to a reduced base10 rational using integer arithmetic. Thus0.1 becomes1/10 and1e-7 becomes1/10000000; this is not the exact binary fraction underlying float0.1. JSON numbers and decimal strings permit ordinary finite binary64 rounding, but a nonzero value that converts to zero is rejected as NUMERIC_UNDERFLOW. A reduced numerator or denominator beyond the safe bound is rejected as VALUE_OUT_OF_RANGE:1e-16 cannot be represented in this envelope. There is no continued-fraction substitution or silent zero approximation.

conversion_source records the normalized decimal serialization for from-dec and is null in other modes. rational_exact:true describes the rational n/d, not its decimal preview. To decimal and secondary decimal displays are rounded binary64 previews:12 significant digits normally,9 in scientific form at |x|≥1e9 or0<|x|<1e-6. Use the exact fraction and complete steps for reproducible arithmetic. Copy JSON and CSV preserve the full structured fractions_output, all active raw inputs, diagnostic status and calculation version. An older share version may show the current engine with a notice; it does not replay a historical runtime.

Interactive arithmetic runs locally without an account. Shared URLs are sent for server rendering; REST/MCP send inputs for execution. The shared history helper can store up to20 recent calculations in browser localStorage. Clear resets the current calculation and does not promise to erase history. Analytics can load under site configuration; retention, deletion, masking and processing region are not established here.