How do you add fractions with different denominators?

To add fractions with unlike denominators, first find the Least Common Denominator (LCD), which is the Least Common Multiple (LCM) of the denominators. Multiply the numerator and denominator of each fraction by the factor needed to produce the LCD, add the numerators while keeping the common denominator, and reduce to lowest terms using the Greatest Common Divisor (GCD).

How do you multiply and divide mixed numbers?

First convert each mixed number into an improper fraction: (Whole × Denominator + Numerator) / Denominator. For multiplication, multiply the numerators straight across and the denominators straight across. For division, keep the first fraction, invert the second fraction into its reciprocal, and multiply.

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 combines an integer whole number and a proper fraction (e.g. 1 3/4). They represent identical values on the number line.

Fraction Calculator with Steps | Mixed Numbers & Simplifier

Comprehensive Rules for Fraction Arithmetic

Quick Answer & Fraction Rules

To add/subtract fractions, find the Least Common Denominator (LCD), convert each fraction, combine numerators, and simplify via GCD. To multiply, multiply numerators straight across and denominators straight across: (a/b) × (c/d) = (a×c)/(b×d). To divide, invert the second fraction (reciprocal) and multiply: (a/b) ÷ (c/d) = (a×d)/(b×c).

Fractions represent ratios and divisions of real numbers ($a/b$ where $b \neq 0$). While modern calculators immediately convert fractions into decimals, performing exact rational arithmetic preserves precision without rounding errors. Our tool executes all four core arithmetic operations with step-by-step proofs.

Addition & Subtraction (Common Denominators)

Fractions cannot be added or subtracted directly unless they share a common denominator. The algebraic formula requires finding the Least Common Multiple:

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

Multiplication & Division (Reciprocal Rule)

Multiplication multiplies straight across: (a / b) × (c / d) = (a × c) / (b × d). Division multiplies by the reciprocal of the divisor:

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

Step-by-Step Worked Example: Adding Mixed Fractions

Consider the addition problem 1 1/2 + 2 1/4:

Step 1 (Convert to Improper):

1 1/2 = (1 × 2 + 1) / 2 = 3/2, and 2 1/4 = (2 × 4 + 1) / 4 = 9/4

Step 2 (Find LCD):

The Least Common Denominator (LCD) of 2 and 4 is 4.

Step 3 (Scale Numerators):

3/2 = (3 × 2) / (2 × 2) = 6/4

Step 4 (Add Across):

6/4 + 9/4 = (6 + 9) / 4 = 15/4

Step 5 (Convert to Mixed Number):

15 ÷ 4 = 3 with remainder 3 → 3 3/4 (Decimal: 3.75)

Frequently Asked Questions

How does the simplifier find the lowest terms?

The simplifier executes the Euclidean Greatest Common Divisor (GCD) algorithm on the numerator and denominator. Dividing both terms by this common divisor guarantees the fraction cannot be reduced further.

What happens if the denominator is zero?

Division by zero is undefined in mathematics. Our calculator automatically enforces a safe minimum denominator of 1, preventing application crashes and invalid operations.

Can I share exact fraction calculations with others?

Yes. Every change to your fractions and operators dynamically updates the browser URL query string. Click the Share button to copy a permanent deep link to your exact calculation.