Fraction Calculator

About This Calculator

A fraction represents a part of a whole. The number above the line is the numerator (how many parts you have), and the number below is the denominator (how many equal parts the whole is divided into). In $\frac{3}{4}$, you have 3 out of 4 equal parts.

This Fraction Calculator handles eight operations in a single tool: addition, subtraction, multiplication, division, simplification, comparison, decimal-to-fraction conversion, and fraction-to-decimal conversion. Every calculation produces a step-by-step solution panel showing the LCD calculation, equivalent fractions, GCD simplification, and mixed number conversion. A visual pie chart representation accompanies each result so you can see the fraction, not just compute it.

Toggle mixed number mode to work with numbers like $2\frac{3}{4}$ directly. Purpose-built variants for cooking and construction come with contextual presets, and the built-in Quick Start scenarios let you explore realistic problems instantly.

How to Use

1. Select an operation from the dropdown: Add, Subtract, Multiply, Divide, Simplify, Compare, Decimal to Fraction, or Fraction to Decimal.
2. Toggle mixed numbers if you need whole-number parts (e.g., $1\frac{3}{8}$). The whole number field appears automatically.
3. Enter the first fraction — numerator and denominator. For Simplify, Fraction to Decimal, and Recipe Scaling, only the first fraction is needed.
4. Enter the second fraction for operations that require two inputs (Add, Subtract, Multiply, Divide, Compare).
5. For Decimal to Fraction, enter a decimal value instead. The calculator handles both terminating decimals (0.625) and repeating decimals (0.333...).
6. Click Calculate to see the result displayed as a simplified fraction, decimal, percentage, and mixed number (when applicable), along with a step-by-step solution and pie chart visualization.

All Eight Operations

• Add — Combines two fractions using the least common denominator.
• Subtract — Finds the difference, which may be negative.
• Multiply — Multiplies numerators and denominators directly, then simplifies.
• Divide — Multiplies by the reciprocal of the second fraction.
• Simplify — Reduces a single fraction to lowest terms using the GCD.
• Compare — Determines which of two fractions is greater, less, or equal.
• Decimal to Fraction — Converts a decimal (terminating or repeating) to a simplified fraction.
• Fraction to Decimal — Converts a fraction to its decimal equivalent and identifies whether it terminates or repeats.

Formulas

Addition (LCD Method)

When adding $\frac{a}{b} + \frac{c}{d}$, find the least common denominator $L = \text{lcm}(b, d)$:

Subtraction

Same approach, subtracting the adjusted numerators:

Multiplication

No common denominator needed. Multiply straight across:

Division

Flip the second fraction (take its reciprocal), then multiply. Requires $c \neq 0$:

Simplification (GCD)

Divide both numerator and denominator by their greatest common divisor. The GCD is computed using the Euclidean algorithm: $\gcd(a, b) = \gcd(b,\; a \bmod b)$, with base case $\gcd(a, 0) = a$.

LCM (Least Common Multiple)

Used to find the LCD for addition and subtraction:

Comparison (Cross Multiplication)

To compare $\frac{a}{b}$ and $\frac{c}{d}$, compute $a \times d$ and $b \times c$. If $ad > bc$, the first fraction is larger. If $ad = bc$, they are equal.

Mixed Number Conversion

To convert a mixed number $a\frac{b}{c}$ to an improper fraction:

To convert back, divide the numerator by the denominator:

Decimal to Fraction

For a terminating decimal with $n$ digits after the point, write it over $10^n$ and simplify. For a repeating decimal $0.\overline{r}$ with $k$ repeating digits:

For mixed repeating decimals with $m$ non-repeating digits followed by $k$ repeating digits:

Worked Examples

Example 1: Student Homework — Adding Unlike Fractions

A 5th grader needs to solve $\frac{1}{6} + \frac{3}{8}$ for homework. The denominators are different, so we need the LCD.

1. Find the LCD: $\text{lcm}(6, 8) = 24$.
2. Convert each fraction: $\frac{1 \times 4}{24} + \frac{3 \times 3}{24} = \frac{4}{24} + \frac{9}{24}$.
3. Add the numerators: $\frac{4 + 9}{24} = \frac{13}{24}$.
4. Check simplification: $\gcd(13, 24) = 1$ — already in lowest terms.

Example 2: Construction — Subtracting Mixed Numbers

A carpenter has a board that is $5\frac{3}{8}$ inches wide and needs to remove $1\frac{7}{16}$ inches for a dado joint. How wide is the remaining piece?

1. Convert to improper fractions: $5\frac{3}{8} = \frac{43}{8}$ and $1\frac{7}{16} = \frac{23}{16}$.
2. Find the LCD: $\text{lcm}(8, 16) = 16$.
3. Convert and subtract: $\frac{86}{16} - \frac{23}{16} = \frac{63}{16}$.
4. Simplify: $\gcd(63, 16) = 1$ — already in lowest terms.
5. Convert to mixed number: $\frac{63}{16} = 3\frac{15}{16}$.

Example 3: Cooking — Scaling a Recipe

A recipe calls for $\frac{3}{4}$ cup of flour. You want to double it. Doubling means multiplying by 2:

1. Set up the multiplication: $\frac{3}{4} \times \frac{2}{1}$.
2. Multiply numerators and denominators: $\frac{3 \times 2}{4 \times 1} = \frac{6}{4}$.
3. Simplify: $\gcd(6, 4) = 2$, so $\frac{6 \div 2}{4 \div 2} = \frac{3}{2}$.
4. Convert to mixed number: $\frac{3}{2} = 1\frac{1}{2}$ cups.

Mixed Numbers

A mixed number combines a whole number with a proper fraction: $2\frac{3}{4}$ means "two and three-quarters." Toggle the Use Mixed Numbers checkbox to enter whole-number parts directly.

Internally, the calculator converts every mixed number to an improper fraction before performing any operation. For example, $2\frac{3}{4} = \frac{2 \times 4 + 3}{4} = \frac{11}{4}$. After the calculation, if the result is an improper fraction, it is automatically converted back to a mixed number for display.

Negative mixed numbers: If you enter a negative whole part (say, $-3$ with fraction $\frac{1}{4}$), the calculator treats this as $-3\frac{1}{4} = -\frac{13}{4}$. The sign applies to the entire quantity, not just the whole part.

Decimal Conversions

Terminating vs. Repeating Decimals

A fraction produces a terminating decimal when its denominator (in lowest terms) has only 2 and 5 as prime factors. For instance, $\frac{3}{8}$ terminates because $8 = 2^3$: the result is 0.375 exactly.

If the denominator contains any other prime factor, the decimal repeats. The classic example is $\frac{1}{3} = 0.333\ldots = 0.\overline{3}$.

The Repeating Decimal Formula

To convert a purely repeating decimal $0.\overline{r}$ where $r$ has $k$ digits, use the algebraic trick: multiply by $10^k$, subtract the original, and solve.

For a mixed repeating decimal like $0.1\overline{6}$ (which equals $\frac{1}{6}$), the general formula accounts for both the non-repeating and repeating portions.

Reading the Step-by-Step Panel

After each calculation, the Step-by-Step Solution panel breaks the work into numbered steps. Here is what you will typically see:

1. Original problem — the fractions and operation as you entered them.
2. Mixed number conversion — if applicable, the conversion from mixed numbers to improper fractions.
3. LCD calculation — the least common denominator and how each fraction is adjusted (for addition and subtraction).
4. Numerator arithmetic — the actual addition, subtraction, or multiplication of numerators.
5. GCD simplification — finding the GCD and dividing both parts of the result.
6. Final result — the simplified fraction, mixed number, decimal, and percentage.

Each step includes the LaTeX formula so you can follow the exact math. Use this panel for homework verification or to learn the method yourself.

A Brief History of Fraction Notation

Fractions are ancient. The Rhind Papyrus (c. 1650 BC) from Egypt records tables of unit fractions — fractions with numerator 1 — written with a mouth-shaped hieroglyph above the denominator. The Egyptians worked almost exclusively with unit fractions, decomposing values like $\frac{2}{5}$ into sums such as $\frac{1}{3} + \frac{1}{15}$.

Indian mathematicians (Brahmagupta, c. 628 AD) wrote one number above another without a bar. Arab scholars later added the horizontal line (the vinculum), and al-Khwarizmi transmitted this notation westward. Fibonacci brought Hindu-Arabic numerals and fraction notation to Europe in 1202 through his Liber Abaci. The slash (/) as an inline separator emerged in the 18th century.

Common Misconceptions

Frequently Asked Questions

How do I add fractions with different denominators?

Find the least common denominator (LCD) of both denominators, convert each fraction to an equivalent fraction with that LCD, then add the numerators. For example, to add $\frac{1}{3} + \frac{1}{4}$, the LCD is 12: $\frac{4}{12} + \frac{3}{12} = \frac{7}{12}$. The calculator finds the LCD automatically and shows each conversion step.

What is a mixed number?

A mixed number has a whole-number part and a fraction part, like $3\frac{1}{2}$ (three and a half). To compute with mixed numbers, convert them to improper fractions first: $3\frac{1}{2} = \frac{7}{2}$. Toggle "Use Mixed Numbers" in the calculator to enter them directly.

When does a fraction produce a repeating decimal?

A fraction in lowest terms produces a repeating decimal when its denominator has a prime factor other than 2 or 5. So $\frac{3}{8}$ terminates (since $8 = 2^3$), but $\frac{1}{3}$ repeats (since 3 is not 2 or 5). The length of the repeating block depends on the denominator. For $\frac{1}{7}$, the block is 6 digits long: $0.\overline{142857}$.

What if the denominator is zero?

Division by zero is undefined in mathematics. The calculator validates inputs and shows a clear error message if any denominator is zero. Similarly, dividing by a fraction whose numerator is zero (e.g., $\frac{3}{4} \div \frac{0}{5}$) is caught before computation.

Can I enter negative fractions?

Yes. Enter a negative numerator or negative denominator — the calculator normalizes the sign. For instance, $\frac{3}{-4}$ is treated as $-\frac{3}{4}$. Negative results (such as subtracting a larger fraction from a smaller one) are displayed correctly.

Is this calculator free?

Yes, completely free with no sign-up required. The step-by-step solutions, pie chart visualizations, and all eight operations are available without restriction. Competitors like Mathway and Symbolab paywall their step-by-step explanations — this calculator does not.

References

• Weisstein, Eric W. “Fraction.” MathWorld — A Wolfram Web Resource. https://mathworld.wolfram.com/Fraction.html
• Encyclopaedia Britannica. “Euclidean algorithm.” https://www.britannica.com/science/Euclidean-algorithm
• Wikipedia. “Euclidean algorithm.” https://en.wikipedia.org/wiki/Euclidean_algorithm
• MacTutor History of Mathematics. “Babylonian mathematics.” University of St Andrews. https://mathshistory.st-andrews.ac.uk/HistTopics/Babylonian_mathematics/
• Weisstein, Eric W. “Least Common Multiple.” MathWorld — A Wolfram Web Resource. https://mathworld.wolfram.com/LeastCommonMultiple.html

Disclaimer

This calculator is provided for educational and informational purposes only. While the underlying algorithms (Euclidean GCD, LCD via LCM, cross-multiplication comparison) are mathematically precise for integer inputs, floating-point limitations may affect decimal representations of very long repeating decimals. Always verify critical calculations — particularly in professional contexts such as construction measurements, medication dosages, or financial computations — through independent means.