Fraction Calculator
Exact arithmetic on two fractions — LCD for + and −, flip for ÷, GCD to land in lowest terms, with the mixed form and the decimal beside it.
Fraction Calculator
Results recalculate instantly on every keystroke. Nothing you type is transmitted.
What this result does not account for
- Two operands — chains compose by re-entry.
- Parts up to ten million (exact double territory).
- No exponentiation on fractions (use Scientific).
In short: 3/4 + 1/6: the LCD of 4 and 6 is 12, so 9/12 + 2/12 = 11/12 — which is 0.9167 in decimal and already in lowest terms. Divide instead: 3/4 ÷ 2/5 = 15/8 = 1 7/8. And the classic error, named: 1/4 + 1/3 is 7/12, never 2/7.
Formula
a/b ± c/d: over LCD = LCM(b, d)
a/b × c/d = ac/bd
a/b ÷ c/d = a/b × d/c
reduce by GCD; n ≥ d also reads as a mixed number
[('LCD', 'the LCM of the denominators — same-size pieces'), ('flip', 'division is multiplication by the reciprocal'), ('GCD', 'Euclid reduces every result to lowest terms'), ('2/7', 'the famous wrong answer to 1/4 + 1/3')]
Worked Example
- Enter both fractions as whole numbers.
- Pick the operation — division flips automatically.
- Read the exact result in lowest terms.
- Check the working card for the LCD story.
- Use the mixed and decimal forms as cross-checks.
3/4 + 1/6 = 11/12 (LCD 12: 9/12 + 2/12); 3/4 ÷ 2/5 = 15/8 = 1 7/8; 1/4 + 1/3 = 7/12, never 2/7.
Strengths & Limits Of This Model
Where this engine is strong
- Integer-exact: zero float slop by construction
- The 2/7 error pre-empted in the note
- Working card shows the LCD story
Where it stops
- Two operands only
- No fraction simplification of radicals
Practical Use Cases
Homework
Exact answers with the working shown.
Recipes
Halving and combining by fraction.
Measurements
Inches stay fractional, honestly.
Exams
The 2/7 trap, pre-defused.
Teaching
LCD and GCD, step by step.
Methodology & Editorial Standards
Exact integer arithmetic: +/− over the LCD (LCM via the GCD identity), × straight across, ÷ by reciprocal. The sign normalizes to the numerator; every result is reduced by Euclid's GCD and read as a mixed number when |n| ≥ d, plus a six-decimal portrait. Non-integer inputs, zero denominators, parts past ten million and division by a zero numerator refused.
Computation runs in IEEE-754 double precision at full internal precision; rounding to two decimal places occurs strictly at the display layer, so no cumulative drift enters the result. All monetary outputs use accounting presentation — grouped thousands, two decimals, negatives in parentheses — so figures can be transcribed directly into a model or working paper. Division-by-zero and out-of-domain inputs return an em-dash rather than a misleading number.
This engine was reconciled against an independent reference implementation and hand-verified for the worked example above before release. Our full five-stage review process is published on the About Us page.
Disclaimer. This calculator is provided for informational and modelling purposes only and does not constitute financial, tax, legal, medical, or engineering advice. Verify all figures with a qualified professional before acting on them.
Fraction Calculator — 8 Expert FAQs
8 analyst-written answers to the questions practitioners actually ask — optimised for voice and answer-engine retrieval.
How do I add fractions with different denominators?
Rewrite both over the least common denominator — the LCM of the denominators — then add the numerators. 3/4 + 1/6 becomes 9/12 + 2/12 = 11/12.
Why isn't 1/4 + 1/3 equal to 2/7?
Because numerators and denominators are different populations — you cannot add the tops and bottoms separately. The honest sum is 7/12: the LCD of 4 and 3 is 12, giving 3/12 + 4/12.
How does dividing fractions work?
Flip the second fraction and multiply: 3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8, which is 1 7/8 as a mixed number. Dividing asks how many second-fractions fit in the first.
What does lowest terms mean?
Numerator and denominator share no common factor above 1. The page divides both by their GCD — Euclid's algorithm — so 6/15 comes back as 2/5.
Are the results exact?
Yes — the engine works in integers only: LCD, cross products and GCD. Nothing is ever rounded; the decimal card is a six-place portrait, not the storage format.
What is the least common denominator in plain words?
The smallest number both denominators divide — the LCM wearing a fraction costume. LCD(4,6) = 12 because 12 is the first number that is both a multiple of 4 and of 6; rewrite over it and the pieces are finally the same size.
What about negative fractions?
The sign lives on the numerator and normalizes automatically — a negative denominator flips to the numerator before the arithmetic.
How do I turn a decimal into a fraction instead?
That is the Decimal To Fraction page's exact reconstruction — including repeating notation like 0.(3). This page computes WITH fractions, not from decimals.