How do you add or subtract fractions with unlike denominators?

Find the Least Common Denominator (LCD) of both fractions using the LCM of their denominators. Multiply each fraction's numerator and denominator by the factor needed to match the LCD. Then add or subtract the numerators while keeping the common denominator, and reduce the resulting fraction using the GCD.

How do you divide two fractions?

To divide two fractions, keep the first fraction unchanged, invert the second fraction (find its reciprocal by swapping its numerator and denominator), and multiply: (a / b) ÷ (c / d) = (a / b) × (d / c) = (a × d) / (b × c).

What is an improper fraction versus a mixed number?

An improper fraction has a numerator greater than or equal to its denominator (e.g. 7 / 4). A mixed number expresses the same quantity as a whole integer combined with a proper fraction (e.g. 1 3/4).

Fraction Calculator: Add, Subtract, Multiply & Divide Fractions

Rules of Fraction Arithmetic

A fraction represents a quotient of integers, written as $a/b$, where $a$ is the numerator (the dividend) and $b$ is the denominator (the divisor, where $b \neq 0$).

1. Addition and Subtraction

Fractions can only be combined by addition or subtraction once they share a common denominator. The optimal denominator is the Least Common Multiple (LCM) of the denominators:

a/b + c/d = (a·d + b·c) / (b·d)

2. Multiplication

Multiplying fractions requires no common denominator. Simply multiply numerators together and denominators together:

(a/b) × (c/d) = (a × c) / (b × d)

3. Division (The Reciprocal Rule)

Dividing by a fraction is mathematically identical to multiplying by its reciprocal (inverting numerator and denominator):

(a/b) ÷ (c/d) = (a/b) × (d/c) = (a × d) / (b × c)

Frequently Asked Questions

Why can a fraction denominator never be zero?

Division by zero is undefined in mathematics. A fraction asking for a denominator of zero is attempting to divide a quantity into zero parts, which has no consistent numerical meaning.

How do you simplify a fraction to lowest terms?

Find the Greatest Common Divisor (GCD) of the numerator and denominator using the Euclidean algorithm, and divide both numbers by that GCD. If the GCD is 1, the fraction is already in its simplest irreducible form.