Fraction Calculator

Every operation reduces to two steps: convert mixed numbers to improper fractions, then apply the rule for the chosen operator.

Addition and subtraction need a common denominator. The straightforward approach is to multiply the two denominators (giving a×d ± b×c over b×d), then simplify. Using the least common denominator (LCD) gives smaller intermediate numbers but the final simplified result is identical.

Multiplication is straight-across: (a/b) × (c/d) = ac/bd. No common denominator needed.

Division is multiplication by the reciprocal: (a/b) ÷ (c/d) = (a/b) × (d/c) = ad/bc. The 'flip and multiply' rule.

Every result is then simplified by dividing both numerator and denominator by their greatest common divisor (GCD), found here using the Euclidean algorithm.

1½ + ⅔: convert to improper → 3/2 + 2/3. Common denominator 6: 9/6 + 4/6 = 13/6. Simplify (already): 13/6 = 2 1/6 = 2.1667.

3¾ × 2⅖: convert → 15/4 × 12/5 = 180/20. Simplify: 180/20 = 9/1 = 9.0 exactly.

5⅙ ÷ ¾: convert → 31/6 ÷ 3/4 = 31/6 × 4/3 = 124/18. Simplify by GCD 2: 62/9 = 6 8/9 ≈ 6.889.

−¾ + ½: keep the sign on the improper form (−3/4 + 1/2 = −3/4 + 2/4 = −1/4). The calculator preserves sign through all four operations.

Recipes: doubling 2¾ cups + ⅔ cup of flour. Construction: stacking ⅝-in plywood with ¾-in subfloor for total thickness. Sewing patterns: combining ⅛-in seam allowances. Music: dotted notes (3/8 = quarter + eighth). Stock prices used to be quoted in eighths and sixteenths until decimalization in 2001.

In US construction, fractional inches dominate (¹⁄₁₆ to 1 1/2 inch). A fluent fraction calculator avoids the rounding errors that creep in when you convert to decimal, do the math, and convert back.

Imperial cooking measurements (1/3 cup, ¾ teaspoon) are still standard in US recipes — fraction arithmetic is the natural language of scaling them up or down.

Denominator can never be 0. Division by zero is undefined — the calculator returns '—' and skips the simplification step.

Mixed-number sign is carried by the whole part. −1¾ means −(1 + 3/4) = −7/4, not (−1) + 3/4 = −1/4. The calculator follows the standard convention (sign on the whole).

Don't simplify mid-step before the operator is applied — small errors compound. Convert to improper, do the math, then simplify the final result.

When mixing very large whole numbers with fractions, the improper-form numerator can grow into 8+ digits. JavaScript handles this safely up to 2^53 (≈9 × 10^15), well above any practical use.