SamCalculator

Fraction Calculator

Perform addition, subtraction, multiplication, and division of fractions with detailed step-by-step solutions, visual fraction models, decimal conversions, and simplified answers.

Fractions

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What Is a Fraction?

A fraction represents a part of a whole — written as a numerator over a denominator with a bar between them. The denominator tells you how many equal parts the whole is split into; the numerator tells you how many of those parts you have. The fraction 3/4 means three of four equal pieces. Every rational number — every value that terminates or repeats as a decimal — can be written as a fraction, which is why fractions sit at the foundation of arithmetic, algebra, and beyond.

This calculator pulls six tools into one workspace: arithmetic on two fractions, mixed-number arithmetic, single-fraction simplification, decimal-to-fraction conversion, fraction-to-decimal conversion with repeating-cycle detection, and a BigInt-powered big-number tab for fractions whose numerator or denominator exceeds JavaScript's 15-digit floating-point precision. Pair it with our percentage calculator, ratio calculator, common factor calculator, and prime factorization calculator for related work.

How Fraction Operations Work

Common denominators for + and −

Add or subtract fractions by rewriting both with the least common multiple of their denominators as a common bottom number, then add or subtract the numerators. The LCM is the smallest denominator that both inputs divide evenly.

Straight-across × and reciprocal ÷

Multiply two fractions by multiplying the numerators and denominators directly. Divide a fraction by another by multiplying by its reciprocal — flip the second fraction upside down, then multiply.

Reduce to lowest terms

Every answer is divided by the greatest common factor of its numerator and denominator so the result is in simplest form — both display-friendly and the canonical mathematical form.

Convert between forms

Every result is shown as a simplified fraction, mixed number, decimal, and percentage where they apply, so you can read whichever form fits your context — recipe (mixed), classroom (fraction), spreadsheet (decimal), or analytics (percent).

6 Ways to Use This Calculator

1

Homework checking

Verify a fraction arithmetic answer in seconds — the step-by-step explanation shows whether you missed a common denominator or a reduction.

2

Recipe scaling

Triple a 2 1/2 cup recipe to 7 1/2 with the mixed-numbers tab — exact arithmetic without converting to decimals.

3

Measurement work

Carpentry and machine work both use inch fractions. Add 3/8 + 9/16 to get 15/16 directly — no decimal rounding error.

4

Probability & odds

Combine probabilities like 1/6 × 1/6 = 1/36 for the chance of rolling double sixes. The simplify tab keeps every fraction in lowest terms.

5

Decimal ↔ fraction conversion

Convert 0.625 to 5/8 for an engineering drawing, or convert 5/16 to 0.3125 for a spreadsheet cell — both directions instantly.

6

Exact big-number arithmetic

Use the BigInt tab for cryptography homework, continued-fraction expansions, or any calculation where 15 digits of precision is not enough.

Best Practices

Always reduce when finishing. An answer like 35/180 is technically correct but communicates poorly; the simplified 7/36 is the same value in canonical form. This calculator reduces every result automatically, but doing it by hand is worth practising.

Convert mixed numbers to improper form before multiplying or dividing. A common student mistake is multiplying whole and fractional parts separately — 2 1/2 × 1 1/2 ≠ 2 1/4. Convert first to 5/2 × 3/2 = 15/4 = 3 3/4.

Prefer fractions over decimals when the value is exactly rational. 1/3 in decimal is 0.333… repeating forever; keep it as 1/3 through a long chain of arithmetic and convert to decimal only at the very end to avoid compounding rounding error.

Why Fractions Matter

School & exams

Fractions are foundational from elementary through high-school algebra. Confident fraction arithmetic unlocks every later topic — equations, functions, calculus, and beyond.

Engineering & trades

Inch fractions are the lingua franca of carpentry, plumbing, and machining in the US. Even in metric workshops, fractional inches show up on drawings and stock sizes.

Pure mathematics

Rational numbers — fractions of integers — are the cleanest closed system under +, −, ×, and ÷. Number theory, algebraic geometry, and probability theory all rest on the fraction.

Daily life

Recipes, money, sport scores, and music notation all use fractions because they communicate exact parts of a whole more naturally than long decimals.

Tricky Cases & Edge Behaviour

Negative fractions

The sign always lives on the numerator after simplification — −3/4 and 3/−4 represent the same value, and the calculator outputs the canonical form with a positive denominator.

Zero in the denominator

Division by zero is undefined; the calculator rejects any input with a zero denominator before computing. Likewise, dividing by 0/1 returns an error rather than infinity.

Repeating decimal cycles

1/3 = 0.333…, 1/7 = 0.142857… repeating in a six-digit cycle, 1/97 has a 96-digit cycle. The fraction-to-decimal tab detects and renders the cycle with an overline.

Very large operands

JavaScript's regular Number only holds 15 significant digits. The big-number tab uses BigInt arithmetic and is exact for numerators and denominators of any length.

Core Fraction Formulas

Every result this calculator produces comes from one of these identities. Let a/b and c/d be the operands and g = gcd(numerator, denominator) the greatest common factor of the result.

Addition

a/b + c/d = (a·d + b·c) / (b·d)

Subtraction

a/b − c/d = (a·d − b·c) / (b·d)

Multiplication

a/b × c/d = (a·c) / (b·d)

Division

a/b ÷ c/d = (a·d) / (b·c)

Simplification

(a/b) / g = (a/g) / (b/g)

Decimal

a/b = a ÷ b (long division)

Percentage

a/b × 100% = (a × 100)/b %

Mixed number

n/d = ⌊n/d⌋ remainder/d

Common Fraction Mistakes

Adding numerators and denominators

1/2 + 1/3 is not 2/5. Add only the numerators after rewriting with a common denominator: 3/6 + 2/6 = 5/6.

Forgetting to reduce

Leaving 6/8 as the final answer when 3/4 is simpler is technically correct but loses style points and obscures the underlying value.

Reciprocal on the wrong fraction

When dividing a/b ÷ c/d, flip the SECOND fraction (c/d → d/c), not the first. Flipping both leaves the value unchanged.

Multiplying mixed numbers as-is

2 1/2 × 1 1/2 = 5/2 × 3/2 = 15/4 = 3 3/4, NOT 2 1/4. Always convert to improper form first.

Mismatched signs

A negative sign on the whole part of a mixed number applies to the WHOLE number, not just the integer part. −2 3/4 = −(2 + 3/4) = −11/4.

Calculator rounding traps

Standard calculators reduce 1/3 to 0.333 and then 0.333 + 0.333 + 0.333 = 0.999, missing the exact 1. Use fraction arithmetic to preserve exactness.

Built for students, teachers, engineers, machinists, and anyone who needs exact rational arithmetic in the browser.

Every step uses exact BigInt arithmetic — no floating-point rounding anywhere in the result pipeline. See our methodology and editorial policy for the principles behind every calculator on the site.

Frequently Asked Questions

A fraction represents a part of a whole. It is written as one integer (the numerator) over another integer (the denominator), separated by a horizontal bar — for example 3/4 means three of four equal parts. Fractions with a numerator smaller than the denominator are called proper fractions; ones where the numerator is larger are improper fractions; and a whole number combined with a proper fraction is a mixed number.

Find the least common multiple (LCM) of the denominators, multiply each fraction so both share that common denominator, then add the numerators while keeping the denominator unchanged. Finally, reduce the result to lowest terms using the greatest common factor. For example, 2/7 + 3/8: the LCM of 7 and 8 is 56, so 2/7 = 16/56 and 3/8 = 21/56, giving 37/56.

Subtraction follows the same recipe as addition: convert both fractions to the LCM of the denominators, subtract the numerators, then reduce. The sign of the result follows the larger absolute value. The calculator handles negative inputs automatically.

Multiply the two numerators together to get the new numerator, multiply the two denominators together to get the new denominator, then reduce. For 5/12 × 7/15 you get 35/180, which reduces to 7/36 by dividing both top and bottom by the GCD of 5.

Dividing by a fraction is the same as multiplying by its reciprocal. To compute a/b ÷ c/d, flip the second fraction to d/c and multiply: (a × d)/(b × c). Then reduce. The calculator validates that the divisor is non-zero before computing.

A proper fraction has a numerator smaller than its denominator (e.g. 3/4). An improper fraction has a numerator equal to or larger than its denominator (e.g. 7/4). A mixed number combines a whole part with a proper fraction (e.g. 1 3/4). Every improper fraction can be rewritten as a mixed number and vice versa.

Find the greatest common factor (GCF) of the numerator and denominator, then divide both by it. For 84/126, the GCF is 42, so the simplified form is 2/3. The calculator displays the GCF and the division steps.

Count the digits after the decimal point, write the number without the decimal as the numerator, and write 10 raised to that digit count as the denominator. Then reduce. For example, 1.375 = 1375/1000 = 11/8 = 1 3/8 in mixed form.

Divide the numerator by the denominator using long division. The calculator shows both the terminating decimal (when the denominator's prime factors are only 2 and 5) and any repeating cycle for other fractions — for example 2/7 = 0.285714285714… repeating every six digits.

The big-number tab uses JavaScript BigInt for the underlying arithmetic, so it is exact regardless of input size. There is no precision loss when adding, subtracting, multiplying, or dividing fractions whose numerator and denominator are hundreds of digits long.

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