Add, subtract, multiply, or divide two fractions (including mixed numbers), or evaluate a full expression with order of operations—exact simplified results with steps.
Fraction arithmetic is exact when you keep numerators and denominators as integers. Addition and subtraction rewrite both fractions over a common denominator—the least common multiple (LCM) of the two denominators—then combine numerators and simplify by the greatest common divisor (GCD). Multiplication multiplies across; division inverts the second fraction and multiplies (invert-and-multiply). Mixed numbers such as 2 1/2 are converted to improper fractions first (2 1/2 = 5/2), so 2 1/2 × 1 1/3 becomes (5/2)×(4/3)=20/6=10/3=3 1/3. This calculator uses BigInt integer arithmetic so large intermediates stay precise, then reports the simplified fraction, an optional mixed number, and a decimal form. Expression mode parses strings like 1/2 + 3/4 × 2/3, applies parentheses when present, and follows standard order of operations so multiplication and division happen before addition and subtraction—yielding 1 for that example, not left-to-right 5/6. Zero denominators and dividing by zero are rejected with clear errors instead of NaN or Infinity.
Homework checks for adding unlike denominators, converting mixed-number word problems, verifying multi-step expressions before algebra, and comparing decimal approximations after exact simplification. Related tools can simplify a single fraction, convert decimals, or compare two fractions without performing an operation.
(a/b) + (c/d) = (ad + bc) / bd [then simplify] (a/b) − (c/d) = (ad − bc) / bd (a/b) × (c/d) = (ac)/(bd) (a/b) ÷ (c/d) = (a/b) × (d/c) = (ad)/(bc)
Numerators (integers)
Denominators (non-zero integers)
Least common multiple used for +/−
Greatest common divisor used to simplify
LCM of 2 and 3 is 6 → 3/6 + 2/6 = 5/6.
Invert and multiply: (3/4)×(5/2)=15/8 = 1 7/8 = 1.875.
First 3/4 × 2/3 = 1/2, then 1/2 + 1/2 = 1.
The calculator finds a common denominator (the LCM), rewrites each fraction, adds the numerators, then simplifies by dividing by the GCD. Example: 1/2 + 1/3 = 3/6 + 2/6 = 5/6.
Yes. Use the optional whole-number field with numerator and denominator (for example 2, 1, 2 for 2 1/2). In expression mode you can also type mixed numbers like 2 1/2.
Expression mode evaluates a full fraction expression such as 1/2 + 3/4 × 2/3, respecting parentheses and the usual order of operations (multiply and divide before add and subtract). That example equals 1.
Every result is simplified to lowest terms, then also shown as a mixed number when useful and as a decimal. Expand Show steps for LCM, invert-and-multiply, and GCD details.
A zero denominator is rejected, and dividing by a fraction equal to zero (numerator 0) returns a divide-by-zero error. Both keep the page from showing NaN or Infinity.
Yes. Numerators and denominators are handled with integer/BigInt arithmetic so results stay exact before any decimal display is derived.
Fractions are fundamental to mathematics, but performing operations on them can sometimes be tricky. Whether you're adding, subtracting, multiplying, or dividing, our Fraction Operations Calculator simplifies the process, providing accurate, step-by-step results. This guide will walk you through the essential rules for each operation and show you how to use our tool effectively.
Understanding how to manipulate fractions is crucial for various mathematical concepts, from basic algebra to advanced calculus. Each operation—addition, subtraction, multiplication, and division—has its own set of rules that, once mastered, make working with fractions much easier. Our calculator applies these rules automatically, helping you solve problems efficiently and verify your manual calculations. We'll cover the necessity of finding common denominators for addition and subtraction, the straightforward process of multiplication, and the 'invert and multiply' rule for division.
Here are the core formulas for fraction operations:
**Addition:** \( \frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd} \)
**Subtraction:** \( \frac{a}{b} - \frac{c}{d} = \frac{ad - bc}{bd} \)
**Multiplication:** \( \frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd} \)
**Division:** \( \frac{a}{b} \div \frac{c}{d} = \frac{ad}{bc} \)Let's say you need to subtract \( \frac{2}{3} \) from \( \frac{5}{6} \). Using the subtraction formula, we first find a common denominator, which is 6. So, \( \frac{5}{6} - \frac{2 \times 2}{3 \times 2} = \frac{5}{6} - \frac{4}{6} \). Now, subtract the numerators: \( \frac{5 - 4}{6} = \frac{1}{6} \). Our calculator performs these steps automatically, ensuring you get the correct, simplified answer every time, even converting to mixed numbers when applicable.
To use the Fraction Operations Calculator, simply enter the numerator and denominator for your first fraction in the respective input fields. Next, select the desired operation (add, subtract, multiply, or divide) from the dropdown menu. Finally, enter the numerator and denominator for your second fraction. Click 'Calculate' to see the result, which will include the unsimplified fraction, the simplified fraction, and any mixed number equivalent. The tool handles validation for zero denominators and division by zero.
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