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Fraction Calculator

Add, subtract, multiply, divide and simplify fractions. Convert to decimal or mixed number with worked steps.

Result
5/6
1/2 + 1/3 = 5/6
Simplified
5/6
1/2 + 1/3 = 5/6
Decimal
0.833333

1/2 + 1/3 = 5/6 → simplified 5/6 (decimal 0.833333).

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  • Formula-verified

    Each calculator is unit-tested against authoritative sources.

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    Static-rendered pages. Sub-second loads on any device.

  • Works offline

    Visit once and it keeps working without an internet connection.

How to use the Fraction Calculator

  1. 1

    Enter the first fraction

    Type the numerator and denominator of the first fraction. Use whole numbers — for mixed numbers like 2 1/3, convert to improper first (multiply whole by denominator, add numerator: 7/3).

  2. 2

    Choose the operation

    Pick add, subtract, multiply or divide. Each uses a different rule: + and − need a common denominator, × multiplies straight across, ÷ flips the second fraction and multiplies (Keep, Change, Flip).

  3. 3

    Enter the second fraction

    Type the numerator and denominator of the second fraction. Watch out for zero denominators — division by zero is undefined and the calculator will return an error rather than a numeric result.

  4. 4

    Read the simplified result

    The result shows the answer reduced to lowest terms via Euclidean GCD, plus its decimal equivalent and (where applicable) mixed-number form. The worked steps panel reveals the full calculation procedure.

  5. 5

    Convert or share the answer

    Toggle between improper, mixed and decimal forms depending on context — improper for further calculations, mixed for cooking and construction, decimal for engineering. Copy the share link to send the exact problem.

What this calculator does

A fraction is a number written as numerator/denominator, representing num — den parts of a whole. 1/2 means "1 part out of 2", or 0.5 in decimal. The four operations follow distinct rules: addition and subtraction require a common denominator (multiply each side by what is missing); multiplication is straight across (a/b — c/d = ac/bd); division is "invert and multiply" (a/b — c/d = a/b × d/c = ad/bc). After any operation you simplify by dividing top and bottom by their GCD (greatest common divisor). An "improper" fraction (numerator − denominator) can also be written as a mixed number — a whole part plus a proper fraction.

Formula

a/b + c/d = (ad + bc)/(bd). a/b — c/d = ac/bd. a/b — c/d = ad/bc. simplify by dividing top and bottom by gcd.
a, c
Numerators (top of each fraction)
b, d
Denominators (bottom of each fraction); must be non-zero
gcd
Greatest common divisor — the largest integer dividing both numbers exactly

For addition and subtraction, b × d is a guaranteed common denominator but not always the LEAST common one — the result still simplifies correctly. For multiplication, multiply top — top and bottom — bottom; the result reduces if any factor of a shares a divisor with d or b shares a divisor with c. Division uses the rule "multiply by the reciprocal": flip the second fraction upside down, then multiply. Always simplify the final answer by Euclidean GCD reduction.

Worked examples

Example: 1/2 + 1/3

Common denominator is 6 (= 2 — 3). 1/2 = 3/6 and 1/3 = 2/6. 3/6 + 2/6 = 5/6.

gcd(5, 6) = 1, so 5/6 is already in lowest terms. Decimal: 0.8333—

Example: 3/4 × 8/9

3/4 × 8/9 = (3 — 8) / (4 × 9) = 24/36.

Simplify: gcd(24, 36) = 12. 24 ÷ 12 = 2; 36 ÷ 12 = 3. Result = 2/3. Decimal: 0.6667. Tip: you can also cross-cancel before multiplying — 3/4 × 8/9 → 1/1 × 2/3 = 2/3 (3 cancels with 9 leaving 1 and 3; 4 cancels with 8 leaving 1 and 2).

Example: 7/3 as a mixed number

7 ÷ 3 = 2 remainder 1.

Mixed number: 2 1/3 (two and one-third). Decimal: 2.3333…

Going the other way, 5 3/4 as an improper fraction = (5 × 4 + 3)/4 = 23/4.

Common use cases

  • Cooking and recipe scaling — halving 3/4 cup or doubling 1 1/3 tsp
  • Construction and woodworking — fractional inch measurements (3/8", 5/16")
  • Sewing and fabric — yardage and seam allowances
  • Music — note durations are fractions of a whole note (1/2, 1/4, 1/8)
  • Engineering drawings — imperial dimensions on technical drawings
  • Probability and statistics — combining fractional probabilities
  • Tutoring and homework — checking math worksheet answers
  • Chemistry — stoichiometric ratios as fractions
  • Finance — converting fractional shares, bond fractions (1/32 of a point in Treasuries)
  • Pharmacy — fractional doses for paediatric medication

What affects the result

  • Sign — negative numerators are common; negative denominators should be normalized to negative numerator
  • Zero denominators — division by zero is undefined; calculator returns an error or NaN
  • GCD algorithm — Euclidean GCD is the standard, fast even for very large integers
  • Mixed vs improper form — both are correct; choice is contextual (improper for further calculation, mixed for everyday reading)
  • Decimal precision — some fractions are exact in decimal (1/4 = 0.25), others repeat (1/3 = 0.333…)
  • Order of operations — for chained fraction expressions, follow standard PEMDAS / BODMAS rules

Tips

  • For addition/subtraction, the smallest common denominator (LCM) gives the cleanest result, but b × d always works
  • Cross-cancel before multiplying to keep numbers small and simplify in one step
  • For division, remember "Keep, Change, Flip" — keep the first fraction, change ÷ to ×, flip the second
  • To convert decimal to fraction: count decimal places (n), write as digits / 10ⁿ, then simplify. 0.375 = 375/1000 = 3/8
  • To convert repeating decimal to fraction: 0.3… = 1/3, 0.6… = 2/3, 0.16… = 1/6 — memorise the common ones
  • For mixed numbers in computation, convert to improper first (3 1/2 − 7/2), then convert back at the end

Mistakes to avoid

  • Adding numerators and denominators separately: 1/2 + 1/3 ≠ 2/5 (this is one of the most common math errors)
  • Forgetting to flip the second fraction when dividing — division is NOT "subtract straight across"
  • Not simplifying the final answer — 6/8 is correct but should be reduced to 3/4
  • Using the largest common denominator (b × d) and then forgetting to simplify the result
  • Sign errors with negative fractions — -1/2 × -1/3 = +1/6, not −1/6
  • Treating mixed numbers as multiplied — 2 1/3 means 2 + 1/3, NOT 2 × 1/3

Frequently asked questions

How do I add fractions with different denominators?

Find a common denominator. The simplest is to multiply: b × d is always a common denominator. Then rewrite each fraction: a/b = (a — d)/(b × d) and c/d = (c — b)/(b × d). Add the numerators. Simplify if possible.

How do I multiply fractions?

Multiply numerators together and denominators together: a/b — c/d = (a — c)/(b × d). You can cross-cancel before multiplying to keep numbers small — any common factor between any numerator and any denominator can be divided out.

How do I divide fractions?

Multiply by the reciprocal: a/b — c/d = a/b × d/c = (a — d)/(b — c). "Keep the first, Change the sign to —, Flip the second." Then simplify.

How do I simplify a fraction?

Find the greatest common divisor (GCD) of numerator and denominator, then divide both by it. 24/36: GCD = 12. Result = 2/3. The Euclidean algorithm finds GCD quickly: GCD(36, 24) = GCD(24, 12) = GCD(12, 0) = 12.

How do I convert an improper fraction to a mixed number?

Divide numerator by denominator. Quotient is the whole part; remainder over denominator is the fractional part. 17/5: 17 ÷ 5 = 3 remainder 2. Mixed: 3 2/5.

How do I convert a fraction to a decimal?

Divide the numerator by the denominator. 3/8 = 3 ÷ 8 = 0.375. Some fractions give terminating decimals (denominator is a product of 2s and 5s), others give repeating decimals (1/3 = 0.333…, 1/7 = 0.142857142857…).

Why is 1/2 + 1/3 not 2/5?

You cannot add numerators and denominators separately because the denominators represent different-sized pieces. 1/2 (a half) and 1/3 (a third) are different units. You must find a common denominator first: 1/2 = 3/6, 1/3 = 2/6, total = 5/6.

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Sources & references

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