Math Calculator

Fraction Calculator

Calculate exact fractions, mixed numbers and decimal forms.

Fraction inputs

Keep exact values and follow the working.

First numerator

Signed integer source text; decimals are supported through the decimal input format.

Operation and source formats

Calculation mode

Worked operation

All four results remain in the table; choose which method to expand.

First value format

First input format

Changing format loads a relevant example for this value.

First denominator

Any nonzero signed integer. Negative denominators are normalized explicitly.

Second value format

Second input format

Changing format loads a relevant example for this value.

Second numerator

Signed integer source text; decimals are supported through the decimal input format.

Second denominator

Any nonzero signed integer. Negative denominators are normalized explicitly.

Rounded decimal places

Only the decimal display is rounded; exact values are retained.

Decimal expansion digit limit

A long expansion is explicitly marked as truncated.

Integer source fields support sixty digits. Decimal sources support sixty significant digits and scientific exponents from negative one hundred through positive one hundred; repeating entries support sixty total digits. Rounded displays use half-away-from-zero rounding and compact scientific notation for very large or sub-resolution nonzero values. Very long exact expressions scroll on screen; print uses a compact notice while CSV retains all digits.

Sum

\frac{23}{20}

Mixed-number decomposition

\left(1+\frac{3}{20}\right)

Rounded decimal: $1.15$
Percentage: $115\%$

Results use exact integer arithmetic. Reduced fractions have positive denominators; the minus sign of a mixed number applies to the entire value. Rounded decimal and percentage views do not replace the exact answer.

Decimal expansion

1.15

Exact terminating decimal expansion.

Compare the two input values

\frac{3}{4}\gt\frac{2}{5}

This comparison uses exact integer products, even when the rounded decimal displays look identical.

Formula for the selected method

\frac{a}{b}+\frac{c}{d}=\frac{ad+bc}{bd}

Each source is normalized before the operation. Addition and subtraction use a least common denominator; multiplication can cancel cross factors; division requires a nonzero divisor. Reciprocals require a nonzero value.

Your fraction calculation step by step

Normalize the first value
$\frac{3}{4}=\frac{3}{4}$
Convert the source to an integer fraction, put any sign outside a positive denominator, then reduce by the greatest common divisor.
Greatest common divisor for the first value
$g=\gcd(\left|3\right|,\left|4\right|)=1$
Dividing both integer parts by this positive divisor preserves the exact value. A zero numerator reduces to zero with denominator one.
Normalize the second value
$\frac{2}{5}=\frac{2}{5}$
Convert the source to an integer fraction, put any sign outside a positive denominator, then reduce by the greatest common divisor.
Greatest common divisor for the second value
$g=\gcd(\left|2\right|,\left|5\right|)=1$
Dividing both integer parts by this positive divisor preserves the exact value. A zero numerator reduces to zero with denominator one.
Find the least common denominator
$L=\operatorname{lcm}(4,5)=20$
Use the least positive common multiple of the normalized denominators instead of multiplying them unnecessarily.
Rewrite and combine the numerators
$\frac{15}{20}+\frac{8}{20}=\frac{23}{20}=\frac{23}{20}$
Combine numerators over the shared denominator and reduce the result to lowest terms.

Inputs saved with this calculation

Calculation mode: all
Worked operation: add
First source value: format: fraction; numerator: 3; denominator: 4
Second source value: format: fraction; numerator: 2; denominator: 5
Rounded decimal places: 6
Maximum decimal expansion digits: 40

All result forms

Every input value and requested operation is retained in the table. Fractions and mixed decompositions are exact; approximate decimal or percentage displays are marked. Undefined quotient and reciprocal cases have an explanation rather than a false zero.

Fraction calculation details
Quantity	Reduced fraction	Mixed decomposition	Rounded decimal	Percentage
First value	$\frac{3}{4}$	$\frac{3}{4}$	$0.75$	$75\%$
Second value	$\frac{2}{5}$	$\frac{2}{5}$	$0.4$	$40\%$
Sum	$\frac{23}{20}$	$\left(1+\frac{3}{20}\right)$	$1.15$	$115\%$
Difference	$\frac{7}{20}$	$\frac{7}{20}$	$0.35$	$35\%$
Product	$\frac{3}{10}$	$\frac{3}{10}$	$0.3$	$30\%$
Quotient	$\frac{15}{8}$	$\left(1+\frac{7}{8}\right)$	$1.875$	$187.5\%$

What Can You Create?

Work with a fraction's value and its different representations

Exact arithmetic with two values

Add, subtract, multiply or divide fractions without converting the inputs into rounded decimals first. Choose all four arithmetic results together or focus on one method. Compare the same exact inputs with an integer-based relationship, or swap their order to see how subtraction and division change while addition and multiplication retain the same result.

Mixed-number and decimal conversion

Enter a signed mixed number or a decimal directly, then inspect the normalized integer fraction and its reduced form. Parenthesized repeating digits distinguish an infinite recurring decimal from a finite approximation. Mixed decompositions show how whole units and a proper remainder describe the same magnitude, with an explicit convention for a negative sign on the complete mixed number.

Working that explains each operation

Follow greatest-common-divisor reduction, least-common-denominator rewriting, cross-cancellation and reciprocal steps. Rounded decimal and percentage views sit beside exact values rather than replace them. A long-division view identifies a terminating or detected repeating expansion and states when its digit budget was reached, keeping a shortened display from becoming a false claim about the exact number.

Formulas and worked examples

Fraction formulas and step-by-step examples

For integer numerators $a,c$ and nonzero denominators $b,d$ , normalize denominator signs before working with the values. A reduced fraction removes the positive greatest common divisor from both integer parts. Division has the additional requirement $c\ne0$ , because a valid fraction with a zero numerator is zero and cannot be inverted. These conditions are different from rejecting a zero source denominator.

\frac{a}{b}=\frac{a/g}{b/g},\qquad g=\gcd(|a|,|b|)

\frac{a}{b}\pm\frac{c}{d}=\frac{ad\pm bc}{bd},\qquad \frac{a}{b}\times\frac{c}{d}=\frac{ac}{bd},\qquad \frac{a}{b}\div\frac{c}{d}=\frac{ad}{bc}

The product of the denominators is a valid common denominator for addition and subtraction, but the calculator uses their least common multiple after normalization when a smaller shared denominator is available. Once both denominators are positive, the scaled integer numerators can also be compared exactly. A mixed-number sign is interpreted on the full magnitude, so a negative mixed source becomes $-(w+n/d)$ for a nonnegative whole magnitude $w$ and a proper fractional part $n/d$ .

Normalize the two default source fractions
Start with $3/4$ and $2/5$ . Both denominators are already positive and both numerator-denominator pairs have greatest common divisor $1$ . Their exact values remain $3/4$ and $2/5$ . If the first source were $-3/(-4)$ , normalization would give the same positive value $3/4$ . A source such as $3/0$ would instead be invalid; it is never replaced by a more convenient fraction.
Add or subtract equal-sized fractional parts
The least common denominator is $\operatorname{lcm}(4,5)=20$ . Rewrite the values as $15/20$ and $8/20$ . Addition gives $(15+8)/20=23/20$ , with mixed decomposition $1+3/20$ and decimal $1.15$ . Subtraction gives $(15-8)/20=7/20$ . The numerators change when the denominators are scaled; changing only the denominator would change the source value and would not be an equivalent-fraction step.
Multiply directly or divide by a nonzero reciprocal
Multiplication gives $3/4\times2/5=6/20=3/10$ . Division replaces the second value with its reciprocal, giving $3/4\div2/5=3/4\times5/2=15/8$ . The default all-operations view therefore retains the four results $23/20$ , $7/20$ , $3/10$ and $15/8$ . If the second numerator becomes zero while its denominator remains valid, the first three operations still have valid results but the quotient is undefined.
Verify a repeating decimal and a signed mixed value
For $x=0.1\overline6$ , shifting the nonrepeating prefix and one repeating digit gives $100x-10x=16-1=15$ , hence $x=15/90=1/6$ . The parenthesized source syntax records the indefinitely repeating block explicitly. Separately, a mixed source with signed whole part $-2$ and fractional part $1/3$ represents $-(2+1/3)=-7/3$ . Both sources become exact integer fractions before any arithmetic or display rounding is applied.

Distinguish an exact representation from a shortened display

The OpenStax fraction arithmetic overview reviews equivalent fractions and the standard operations. Its decimals and fractions section discusses terminating and repeating representations. The worked calculations here are independently evaluated examples. This tool additionally states its input bounds, mixed-sign convention, display rounding and expansion digit limit so the computed output has a clear interpretation.

Why Users Love This Tool

Preserve the exact value while making each step easier to inspect

Reliable integer arithmetic

Integer source text is retained without silent truncation or floating-point conversion.
Negative fraction denominators are normalized visibly; a zero denominator remains an error.
Common denominators and arithmetic products use exact integers beyond ordinary safe-number precision.
Zero divisors are explained independently of invalid source fractions, preserving other valid results.

Representations with clear limits

Reduced fractions and mixed decompositions are exact, while rounded equivalents are marked when approximate.
Repeating decimal input uses a defined parenthesis grammar and converts exactly before arithmetic.
Decimal expansion limits stop the display explicitly instead of pretending a partial cycle is complete.
Active-source records, editable sharing and complete CSV retain enough information to reproduce a result.

Perfect For

Fraction support for classroom practice and proportional quantities

Students learning the reason behind a rule

Use the worked-operation selector to focus on one arithmetic method while preserving all four answers in the table. Check why denominators must match for addition, why factors may cancel in a product, and why a divisor cannot be zero. Compare the reduced answer with an exact mixed decomposition and a decimal expansion so the relationships between representations stay visible during practice.

People combining or scaling measurements

Work with fractional quantities before choosing a rounded display for practical use. Source values must refer to compatible quantities when they are added or compared; this calculator does not infer measurement units or convert between them. Mixed input can express a whole amount plus a fractional remainder, while exact arithmetic avoids accumulating display-rounding errors when several manual calculations are checked separately.

Teachers comparing examples and edge cases

Test a negative denominator, a zero numerator, an improper fraction or a recurring decimal without losing the entered source. Show why two nearby fractions may share a rounded decimal yet remain unequal. The all-results view and print record can support a lesson with a clear distinction between a valid zero value, an invalid zero denominator and an undefined division by zero.

How It Works

How it works in three quick steps.

Choose the operation and identify each source format

Start with all four arithmetic operations or focus on addition, subtraction, multiplication, division, comparison, simplification or a reciprocal. Each required value can use integer numerator and denominator fields, a mixed number, or a decimal source. Changing a source format loads an example for that value while changing the operation preserves existing sources.

Enter exact values and select readable display settings

Enter signed integers without decimal truncation. Fraction denominators may be negative but cannot be zero; mixed-number fractional parts use positive denominators and an overall sign on the whole part. Decimal sources accept scientific e notation or a parenthesized repeating block. Choose rounded decimal places and a long-division digit limit independently of the exact arithmetic.

Follow the method and save every result form

Review normalized inputs, the least common denominator or cross-cancellation steps, and the exact result. Compare mixed decompositions, rounded decimal and percentage forms, plus terminating, repeating or explicitly truncated decimal expansions. In the all-operations view, select the worked method while retaining all four results. Copy, share, download the full CSV or print and save a PDF with the source record.

Download & Print

Keep the source, exact values and worked method together

Download every requested result

The CSV contains the active operation, input formats and original source strings, followed by every requested result row. Exact numerator and denominator fields remain strings, preserving large integers for later inspection. Mixed forms, rounded decimals, percentages and expansion statuses stay attached to the same value. A comparison also retains its common denominator and scaled numerators instead of exporting an isolated ordering label.

Print the working or save a PDF

Review the browser's print preview before choosing a printer or Save as PDF. Editing controls and site navigation are removed while the result, mathematical steps, input record and complete native table remain. Landscape orientation gives the result-form columns more room. Very long exact expressions and expansions use an explicit compact notice in print; the CSV remains the complete digit-preserving record for those unusually large cases.

Share an editable fraction example

A shared URL fragment contains only active validated fields, including decimal source text and the chosen display settings. A recipient can inspect the same assumptions and edit the example. Copy summary provides a plain-text record for notes. Reset restores the original fractions and clears the fragment. Changing the arithmetic operation preserves current operands, while changing an input format deliberately loads a relevant example for that value.

FAQ

Frequently Asked Questions

What can this fraction calculator calculate?

The tool supports addition, subtraction, multiplication, division, comparison, simplification and reciprocals, with an all-operations view for the four arithmetic results. Each input can be an integer fraction, a signed mixed number or a decimal, including an explicitly repeating block. Results retain reduced fractions, mixed decompositions, rounded decimals and percentages. A separate decimal expansion distinguishes terminating, repeating and truncated output. The method and source record remain attached to the answer for learning or later verification.

How do I simplify a fraction or enter a negative denominator?

Enter the integer numerator and any nonzero integer denominator. The tool moves the sign to the normalized numerator and divides both parts by their greatest common divisor. For example,

12/(-18)=-2/3

. Negative denominators are supported directly rather than silently changed to positive inputs. A zero numerator with a valid denominator reduces to

0/1

, while a zero denominator is an invalid source fraction and receives a validation message instead of an invented replacement.

How should I type a negative mixed number?

Choose mixed-number format, enter the signed whole part and use a nonnegative proper fractional numerator over a positive denominator. The sign applies to the entire magnitude: a whole part of

-2

with a fractional part of

1/3

represents

-(2+1/3)=-7/3

, not

-2+1/3

. A whole part entered as negative zero applies the negative sign to a proper fraction. If the fractional numerator is improper, use the ordinary fraction format instead of silently carrying values between fields.

Can I convert terminating or repeating decimals into fractions?

Yes. Select decimal format and enter a finite decimal, scientific e notation or a repeating block in parentheses. The literal input 0.1(6) means

0.1\overline{6}=1/6

, while 0.(3) means

0.\overline{3}=1/3

. Parentheses describe infinitely repeating digits rather than an uncertain approximation. The parser converts these source digits to an exact integer fraction before reduction. A finite entry such as

0.333

is therefore

333/1000

, and should not be mistaken for

1/3

Why do addition and subtraction need a common denominator?

A denominator identifies the size of the fractional parts being counted. Addition and subtraction combine equal-sized parts, so the calculator rewrites normalized inputs over their least common denominator before combining numerators. For

3/4+2/5

, the shared denominator is

20

and the numerators become

15

and

8

, giving

23/20

. Using the least common multiple avoids unnecessary growth in intermediate integers. The final answer is reduced again so the exact result is shown in lowest terms.

How does cross-cancellation help multiplication?

Factors in a numerator and the opposite denominator can be cancelled before multiplying because the same common divisor is removed from the product's numerator and denominator. For

2/3\times9/10

, cancellation leaves

1/1\times3/5=3/5

. The calculator shows both cancellation divisors and the remaining integer fractions. It still uses exact integer arithmetic for the complete operation, so cancellation is a transparent explanation and a way to keep intermediate numbers smaller rather than a change in the mathematical value.

When is division by a fraction or a reciprocal undefined?

A valid source fraction requires a nonzero denominator. Separately, dividing by a valid fraction whose numerator is zero is undefined because that value is zero and has no reciprocal. The calculator keeps those two cases distinct. An invalid source denominator stops the calculation and highlights the input. A valid zero divisor leaves addition, subtraction and multiplication available in the all-operations view, while the quotient row explains why division is undefined. The reciprocal of zero is likewise unavailable.

How are rounded decimals different from repeating-decimal expansions?

The rounded decimal display uses a chosen number of decimal places with half-away-from-zero rounding. The long-division view instead traces actual decimal digits and detects a repeating remainder. A completed repeating cycle is marked with an overline, while an expansion that reaches the selected digit limit is explicitly labeled truncated and ends with an ellipsis. Truncation is not rounding and does not claim the shown digits form a complete cycle. Neither view changes the exact fraction stored by the calculation.

Can this compare very large fractions accurately?

Comparison uses exact integer arithmetic with positive normalized denominators rather than comparing rounded decimal displays. Integer source fields support up to sixty digits, so nearby large values can remain distinct even beyond the precision of ordinary binary floating-point arithmetic. Decimal sources also retain the entered digits under their stated limits. Two equal rounded decimals do not prove the fractions are equal. The comparison and common-denominator records show the exact relationship, and CSV retains full numerator and denominator strings.

Can I save or share the full fraction calculation?

Copy summary preserves active source settings, exact fractions, mixed forms and decimal-expansion statuses in plain text. The editable share link includes only active validated fields, preserving source strings rather than converting them to floating-point numbers. CSV includes every requested result row and all supporting source values. The print view retains mathematical working and the input record; choose Save as PDF in the browser to keep a document. Very long exact expressions use a compact print notice while the CSV retains every digit.

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About This Tool

About this exact fraction arithmetic and conversion tool

Toolarithm's Fraction Calculator treats an input as an exact rational value before presenting alternative forms. A numerator and denominator, a mixed number and a repeating decimal can describe the same value, but their written appearance can make different mathematical properties easier to see. The editor therefore keeps source formats distinct while normalizing their values for arithmetic. This allows a learner to compare representations without confusing a shortened decimal display with the fraction itself.

The arithmetic engine uses integer strings and rational operations rather than converting everything into binary floating-point numbers. Input fractions reduce by a greatest common divisor, normalized denominators stay positive, and addition or subtraction can use a least common denominator. Multiplication exposes cross-cancellation factors. Division and reciprocal calculations make their nonzero requirements explicit. Comparison uses exact integer relationships, preserving distinctions between large or closely spaced fractions that may appear identical after decimal rounding.

Display choices are deliberately separate from those exact calculations. Mixed decompositions show a whole magnitude and proper remainder, while a negative sign applies to the complete mixed value. Rounded decimal and percentage views use a stated rounding rule. Long division detects completed recurring patterns and marks a digit-limited expansion as truncated when necessary. Accessible LaTeX and MathML keep the authored guide and computed working available to assistive technology, including the server-rendered default results without JavaScript. Sharing, copying, CSV and print preserve the assumptions and result status so a saved answer remains mathematically reviewable rather than becoming an unexplained fraction in isolation.

Editorial Transparency

Who maintains this page

Ownership and review

Written and maintained by the Toolarithm editorial team. No review date is shown without a maintained editorial record. No independent professional review is claimed.

Methodology

Equations are checked by substituting worked examples into the stated relationship and comparing the result with the interactive implementation.

Read the editorial methodology

Dates and sources

Review dates change only after a substantive method or content check.

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Calculators hub

Fraction Calculator

Fraction inputs

Operation and source formats

Sum

Decimal expansion

Compare the two input values

Formula for the selected method

Your fraction calculation step by step

Normalize the first value

Greatest common divisor for the first value

Normalize the second value

Greatest common divisor for the second value

Find the least common denominator

Rewrite and combine the numerators

Inputs saved with this calculation

All result forms

Work with a fraction's value and its different representations

Exact arithmetic with two values

Mixed-number and decimal conversion

Working that explains each operation

Fraction formulas and step-by-step examples

Normalize the two default source fractions

Add or subtract equal-sized fractional parts

Multiply directly or divide by a nonzero reciprocal

Verify a repeating decimal and a signed mixed value

Distinguish an exact representation from a shortened display

Preserve the exact value while making each step easier to inspect

Reliable integer arithmetic

Representations with clear limits

Fraction support for classroom practice and proportional quantities

Students learning the reason behind a rule

People combining or scaling measurements

Teachers comparing examples and edge cases

How it works in three quick steps.

Choose the operation and identify each source format

Enter exact values and select readable display settings

Follow the method and save every result form

Keep the source, exact values and worked method together

Download every requested result

Print the working or save a PDF

Share an editable fraction example

Frequently Asked Questions

Related calculators

Percentage Calculator

Ratio Calculator

GCD and LCM Calculator

About this exact fraction arithmetic and conversion tool

Who maintains this page

Ownership and review

Methodology

Dates and sources

Explore more calculators