Fraction Calculator
Enter a fraction expression such as 2 3/4 ÷ 2/3. Get the exact reduced answer, mixed number, decimal, visual model, step-by-step explanation and a reproducible verification.
Fraction workspace
Interpreted as: (2 + 3/4) ÷ (2/3)
The fields below only build the expression — they feed the exact same parser and engine as the text box.
Answer
- Mixed number
- 4 1/8
- Decimal
- 4.125
- Percentage
- 412.5%
Result details
- Interpreted as: (2 + 3/4) ÷ (2/3).
- Dividing by a fraction is multiplying by its reciprocal — flip the divisor: 11/4 ÷ 2/3 = 11/4 × 3/2.
- Multiply the remaining numerators and denominators: (11 × 3)/(4 × 2) = 33/8.
A measurement model: how many lengths of 2/3 fit into 11/4? Exactly 33/8 of them.
✓ Verified All exact checks passed. Every check below is an integer identity — never a rounded decimal comparison.
- Denominator strictly positive — passed: d = 8
- Numerator and denominator coprime — passed: GCD(|33|, 8) = 1
- Zero normalized as 0/1 — passed: not applicable (non-zero result)
- Cross multiplication: unreduced form ≡ reduced result — passed: 33 × 8 = 33 × 8 = 264
- Inverse: quotient × divisor = dividend — passed: 33/8 × 2/3 = 11/4, expected 11/4
Full calculation receipt (JSON)
{
"ok": true,
"mode": "calculate",
"input": "2 3/4 ÷ 2/3",
"normalizedInput": "(2 + 3/4) ÷ (2/3)",
"parserVersion": "1.0.0",
"engineVersion": "1.0.0",
"receiptVersion": "1.0.0",
"method": "Division by multiplying with the reciprocal",
"formula": "a/b ÷ c/d = a/b × d/c",
"substitution": "11/4 ÷ 2/3 = 33/8",
"steps": [
{
"text": "Interpreted as: (2 + 3/4) ÷ (2/3).",
"level": "quick"
},
{
"text": "Dividing by a fraction is multiplying by its reciprocal — flip the divisor: 11/4 ÷ 2/3 = 11/4 × 3/2.",
"level": "quick"
},
{
"text": "Why the flip works: division asks \"how many times does 2/3 fit into 11/4?\", and multiplying by 3/2 answers exactly that question.",
"level": "detailed"
},
{
"text": "Multiply the remaining numerators and denominators: (11 × 3)/(4 × 2) = 33/8.",
"level": "quick"
}
],
"exactResult": {
"n": "33",
"d": "8",
"text": "33/8",
"isNegative": false,
"isInteger": false,
"isZero": false,
"latex": "\\frac{33}{8}"
},
"mixedResult": {
"sign": 1,
"whole": "4",
"n": "1",
"d": "8",
"text": "4 1/8",
"latex": "4\\tfrac{1}{8}"
},
"decimalResult": {
"kind": "terminating",
"text": "4.125",
"preview": "4.125",
"intPart": "4",
"nonRepeating": "125",
"cycle": null,
"cycleLength": 0,
"truncated": false,
"exact": true
},
"percentageResult": {
"text": "412.5%"
},
"statuses": [
"Lowest terms",
"Improper fraction",
"Mixed number available",
"Terminating decimal"
],
"visual": {
"type": "grouping",
"dividend": {
"n": 11,
"d": 4,
"text": "11/4"
},
"divisor": {
"n": 2,
"d": 3,
"text": "2/3"
},
"result": {
"n": 33,
"d": 8,
"text": "33/8"
},
"resultApprox": 4.125,
"fallback": false
},
"visualDescription": "A measurement model: how many lengths of 2/3 fit into 11/4? Exactly 33/8 of them.",
"verification": {
"passed": true,
"checks": [
{
"id": "denominator-positive",
"label": "Denominator strictly positive",
"ok": true,
"detail": "d = 8"
},
{
"id": "lowest-terms",
"label": "Numerator and denominator coprime",
"ok": true,
"detail": "GCD(|33|, 8) = 1"
},
{
"id": "zero-normal-form",
"label": "Zero normalized as 0/1",
"ok": true,
"detail": "not applicable (non-zero result)"
},
{
"id": "cross-multiplication",
"label": "Cross multiplication: unreduced form ≡ reduced result",
"ok": true,
"detail": "33 × 8 = 33 × 8 = 264"
},
{
"label": "Inverse: quotient × divisor = dividend",
"ok": true,
"detail": "33/8 × 2/3 = 11/4, expected 11/4",
"id": "inverse-operation"
}
]
},
"warnings": [],
"errorCode": null
}Working with fractions — the complete guide
How to add fractions
Two fractions can only be added when they count the same kind of part, so the first job is a common denominator. Reduce each operand, take the least common multiple of the denominators as the LCD, rewrite each fraction as an equivalent over that LCD, add the numerators, and reduce the result. For 1/4 + 1/3 the LCD is LCM(4, 3) = 12, the equivalents are 3/12 and 4/12, and the sum is 7/12 — already in lowest terms. When the denominators are equal from the start (2/9 + 5/9 = 7/9) no LCD step is needed and the calculator skips it rather than showing a redundant LCM of equal numbers.
How to subtract fractions
Subtraction follows the same LCD path with two extra care points. First, the order matters: 1/3 − 3/4 = 4/12 − 9/12 = −5/12, a negative result, which is perfectly valid. Second, mixed numbers are safest converted to improper fractions before subtracting: 2 1/3 − 2/3 = 7/3 − 2/3 = 5/3 = 1 2/3. If you subtract the parts column-style instead, you sometimes need to borrow a whole (regroup 2 1/3 as 1 4/3) — the Detailed explanation level walks through when and why that regrouping happens.
How to multiply fractions
No common denominator is needed: multiply numerator by numerator and denominator by denominator. Before multiplying, cross-cancel any factor shared between a numerator and an opposite denominator — it keeps every intermediate number small and the final reduction trivial. For 2/3 × 9/4, cancel 2 with 4 (÷2) and 9 with 3 (÷3) to get 1/1 × 3/2 = 3/2 = 1 1/2. The word “of” between two numbers means the same multiplication: 1/2 of 1/3 = 1/2 × 1/3 = 1/6.
How to divide fractions
Dividing by a fraction is multiplying by its reciprocal: flip the divisor and multiply, then cross-cancel as usual. So 2/3 ÷ 4/9 = 2/3 × 9/4 = 3/2. The flip works because division asks “how many times does the divisor fit?”, and the reciprocal answers exactly that. Division by a fraction equal to zero is undefined and the calculator refuses it with a structured error instead of guessing. Every quotient is verified by multiplying back: 3/2 × 4/9 = 2/3 ✓.
Mixed and improper fractions
An improper fraction like 9/4 and a mixed number like 2 1/4 are the same value in different clothes. To go improper → mixed, divide: 9 ÷ 4 = 2 remainder 1, so 9/4 = 2 1/4. To go mixed → improper, rebuild the numerator: 2 × 4 + 1 = 9, so 2 1/4 = 9/4. A negative mixed number applies its sign to the whole value: −2 3/4 = −11/4, not −2 + 3/4. The workspace accepts 2 3/4, “2 and 3/4” and the compact 2½ as the same input.
Simplifying fractions
A fraction is in lowest terms when its numerator and denominator share no factor but 1. Divide both by their greatest common divisor: for 18/24, GCD(18, 24) = 6, so 18/24 = 3/4. The Simplify mode shows the GCD, the division applied to both parts, and a lowest-terms proof (the new GCD is 1). It also standardizes signs — a negative denominator moves its sign up, so 6/−8 becomes −3/4.
Fraction and decimal conversions
A fraction a/b converts to a decimal by long division, and the denominator decides the shape: if b factors into only 2s and 5s the decimal terminates (3/8 = 0.375); any other prime factor forces a repeating cycle (1/3 = 0.(3), 1/6 = 0.1(6), 1/7 = 0.(142857)). Going the other way, a terminating decimal sits over a power of ten (0.6 = 6/10 = 3/5), and a repeating decimal uses the identity x = (all digits − non-repeating digits) / (10ᵏ · (10ᶜ − 1)) — which is why 0.(3) is exactly 1/3, not approximately. Percentages just divide by 100: 62.5% = 62.5/100 = 5/8.
Equivalent fractions and common denominators
Multiplying numerator and denominator by the same non-zero number changes the name of a fraction, never its value: 1/4 = 3/12 because 1 × 12 = 3 × 4 (cross products agree). The least common denominator of a set of fractions is the LCM of their denominators; each fraction scales by LCD ÷ its own denominator. For 1/4 and 1/6, LCD = LCM(4, 6) = 12 and the equivalents are 3/12 and 2/12 — the form used to add, subtract or compare them exactly.
Negative fractions and zero
The three forms −1/2, 1/−2 and −(1/2) are one value; the calculator normalizes every result so the sign lives on the numerator and the denominator is strictly positive. Zero is always stored as 0/1. A zero numerator is fine (0/5 = 0); a zero denominator is not a number at all, and dividing by a zero fraction is refused for the same reason — the reciprocal of 0 does not exist. Adding opposite values cancels exactly: 1/−2 + 1/2 = 0.
Common mistakes
| Mistake | Wrong working | Correct method |
|---|---|---|
| Adding denominators | 1/2 + 1/3 = 2/5 ✗ | 1/2 + 1/3 = 3/6 + 2/6 = 5/6 ✓ — rewrite over the LCD first; only the numerators add. |
| Forgetting the reciprocal | 2/3 ÷ 4/9 = 8/27 ✗ | 2/3 ÷ 4/9 = 2/3 × 9/4 = 3/2 ✓ — flip the divisor before multiplying. |
| Reading a mixed number as multiplication | 2 3/4 = 2 × 3/4 = 3/2 ✗ | 2 3/4 = 2 + 3/4 = 11/4 ✓ — the whole part is added, never multiplied. |
| Leaving the result unreduced | 2/3 × 9/4 = 18/12 (stopping here) ✗ | 18/12 = 3/2 = 1 1/2 ✓ — divide by GCD(18, 12) = 6, or cross-cancel first and skip the big numbers. |
| Using a zero denominator | 5/0 = 0 or “infinity” ✗ | 5/0 is undefined ✓ — no number times 0 gives 5; the calculator returns a ZERO_DENOMINATOR error. |
Examples
Every example below is computed by this page’s engine (load any of them with one click above): 1/2 + 1/3 = 5/6 · 3/4 − 1/6 = 7/12 · 2/3 × 4/9 = 8/27 · 2 3/4 ÷ 2/3 = 33/8 = 4 1/8 · 1/2 of 1/3 = 1/6 · 2/3 × 2/3 × 2/3 = 8/27 · 1/2 + 1/3 × 3/4 = 3/4 (multiplication binds first) while (1/2 + 1/3) × 3/4 = 5/8 · 0.125 + 1/8 = 1/4 · 0.6 = 3/5 · 0.(3) = 1/3 · 62.5% = 5/8 · compare: 5/8 < 2/3 because 5 × 3 = 15 < 16 = 2 × 8.
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Frequently asked questions
How do I add fractions with different denominators?
Rewrite both fractions over the least common denominator — the LCM of the two denominators — then add only the numerators and reduce. For 1/4 + 1/3 the LCD is 12, so 3/12 + 4/12 = 7/12. The Steps view shows the LCD, both equivalent fractions and the final reduction for every sum.
How do I subtract mixed numbers?
The safest route is to convert each mixed number to an improper fraction first: 2 1/3 − 2/3 = 7/3 − 2/3 = 5/3 = 1 2/3. The column method also works but sometimes needs borrowing (regrouping 2 1/3 as 1 4/3); the Detailed explanation level shows when that regrouping happens and why.
How do I multiply fractions by whole numbers?
Write the whole number over 1 and multiply straight across: 4 × 2/3 = 4/1 × 2/3 = 8/3 = 2 2/3. Cross-cancel first when the whole number shares a factor with the denominator: 6 × 5/9 cancels 6 and 9 by 3 to give 2/1 × 5/3 = 10/3.
How do I divide one fraction by another?
Multiply the first fraction by the reciprocal of the second — flip the divisor and multiply: 2/3 ÷ 4/9 = 2/3 × 9/4 = 3/2. The calculator cross-cancels before multiplying and then verifies the quotient by multiplying it back by the divisor.
Why do you flip the second fraction when dividing?
Division asks how many times the divisor fits into the dividend. Multiplying by the reciprocal answers exactly that question: since b/c × c/b = 1, dividing by b/c and multiplying by c/b are the same operation. That is also why dividing by a fraction smaller than 1 makes the result bigger.
How do I simplify a fraction to lowest terms?
Divide numerator and denominator by their greatest common divisor. For 18/24, GCD(18, 24) = 6, so 18/24 = 3/4. Simplify mode shows the GCD, the division applied to both parts and a lowest-terms proof — the reduced pair has GCD 1, so no further factor exists.
How do I convert a mixed number to an improper fraction?
Multiply the whole part by the denominator and add the numerator, keeping the denominator: 2 1/4 = (2 × 4 + 1)/4 = 9/4. A negative mixed number applies its sign to the whole value: −2 3/4 = −11/4. The conversion is verified by dividing back to the mixed form.
How do I convert a decimal to a fraction?
Place the digits over the matching power of ten and reduce: 0.6 = 6/10 = 3/5, and 0.125 = 125/1000 = 1/8. Repeating decimals use the exact identity x = (all digits − non-repeating digits) ÷ (10^k · (10^c − 1)), which is how 0.(3) becomes exactly 1/3 and 0.1(6) becomes 1/6.
What does a repeating decimal mean?
A repeating decimal has a block of digits that recurs forever, written here in cycle notation: 1/3 = 0.(3), 1/6 = 0.1(6), 1/7 = 0.(142857). It appears whenever the reduced denominator has a prime factor other than 2 or 5. The repeating value is never shown as a finite exact decimal — the fraction is the exact form and any rounded preview is labeled display-only.
Can a fraction have a negative denominator?
As input, yes — 1/−2 is accepted. As a normalized value, no: the sign is moved to the numerator so the denominator is always positive, and 1/−2 is stored and shown as −1/2. The value is unchanged; the form is standardized so every equal value has exactly one representation.
What happens if the denominator is zero?
A zero denominator is not a number, so the calculator refuses it with the structured error ZERO_DENOMINATOR instead of returning infinity or zero. Dividing by a fraction equal to zero is refused the same way (DIVISION_BY_ZERO), because the reciprocal of 0 does not exist.
How does CalcDomain verify the answer?
Every calculation carries a receipt whose checks are exact integer identities: the denominator is positive, the result is in lowest terms (GCD = 1), the unreduced and reduced forms agree by cross multiplication, and the inverse operation recovers the first operand (for a ÷ b = c, it checks c × b = a). Only when every applicable check passes does the result show Verified.
Does the calculator use rounded decimals?
Never for the answer or its verification. All arithmetic runs on exact arbitrary-precision integers (BigInt), so 1/2 + 1/3 is exactly 5/6, not 0.8333…. Decimals appear only as clearly-labeled secondary forms, with repeating cycles marked exactly — like 5/6 = 0.8(3) — and rounded previews flagged as display-only.
Can I enter more than two fractions?
Yes — chains like 2/3 × 2/3 × 2/3 or 1/2 + 1/3 − 1/6 evaluate with standard precedence (parentheses first, then × and ÷, then + and −, left to right at equal level), up to 32 operands and 16 nesting levels. Compare and Common denominator modes also accept two or more comma-separated values.
Can I embed this calculator?
Yes. The embed at /embed/fraction-calculator runs the identical engine with no cookies, no tracking and visible CalcDomain attribution, and supports light/dark/system themes plus an optional initial expression and mode via URL parameters. The iframe snippet is on this page, free to copy.
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Methodology
Every value is stored as a pair of arbitrary-precision integers (BigInt numerator and denominator) normalized after each operation: the denominator is strictly positive, zero is always 0/1, the sign lives on the numerator, and numerator and denominator are coprime. Addition and subtraction rewrite the operands over the least common denominator (LCM of the denominators); multiplication cross-cancels shared factors before multiplying; division multiplies by the reciprocal and refuses a zero divisor. Decimal forms come from exact long division that separates the terminating case, the non-repeating prefix and the repeating cycle — a repeating decimal is never displayed as a finite exact value. A dedicated parser (no eval) reads fractions, mixed numbers, terminating and repeating decimals, Unicode vulgar fractions, symbolic and word operators and parentheses with standard precedence, and refuses ambiguous input instead of guessing. Each result carries a calculation receipt whose verification re-checks the invariants, the unreduced cross-multiplication identity and the inverse operation — all as exact integer identities, never rounded decimal comparisons.
Engine: fraction-engine v1.0.0 · Author: Ugo Candido · Last updated: · Every calculation runs locally in this browser — no expression, result or receipt is ever transmitted. Independent human mathematical review is a release gate tracked outside this page; the software verification above runs on every single calculation.