½ Fraction Calculator

How to Calculate with Fractions

A fraction represents a part of a whole, written as two integers separated by a line: the numerator (top number, indicating how many parts you have) and the denominator (bottom number, indicating how many equal parts the whole is divided into). For example, 3/4 means "3 out of 4 equal parts." Fractions are foundational in arithmetic, algebra, cooking, carpentry, music, and virtually every technical field.

Adding and Subtracting Fractions

To add or subtract fractions, they must share a common denominator. The easiest approach is to use the product of the two denominators as the common denominator, then adjust the numerators accordingly:

• Formula: a/b + c/d = (a×d + c×b) / (b×d)
• Example: 3/4 + 1/6 = (3×6 + 1×4) / (4×6) = (18 + 4) / 24 = 22/24 = 11/12
• Subtraction: Same process with minus: 3/4 − 1/6 = (18 − 4) / 24 = 14/24 = 7/12

Using the product of denominators always works but may produce a larger-than-necessary denominator. Using the Least Common Multiple (LCM) as the denominator gives a smaller number to work with. The result should always be simplified by dividing both parts by their Greatest Common Divisor (GCD).

Multiplying Fractions

Multiplication is the simplest fraction operation: multiply numerators together and denominators together. No common denominator is needed.

• Formula: a/b × c/d = (a×c) / (b×d)
• Example: 2/3 × 3/5 = 6/15 = 2/5 (after simplifying by GCD of 3)
• Practical example: A recipe calls for 3/4 cup of sugar, and you're making half the recipe: 3/4 × 1/2 = 3/8 cup.

A useful shortcut called "cross-cancellation" simplifies before multiplying: divide any numerator and any denominator by their common factors first, reducing the numbers you're working with.

Dividing Fractions

Dividing by a fraction is the same as multiplying by its reciprocal. Flip the second fraction (invert numerator and denominator) and multiply.

• Formula: a/b ÷ c/d = a/b × d/c = (a×d) / (b×c)
• Example: 3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8 = 1 7/8
• Practical example: How many 1/4-cup servings are in 3/4 of a cup? 3/4 ÷ 1/4 = 3/4 × 4/1 = 12/4 = 3 servings.

Simplifying Fractions

A fraction is in simplest form (lowest terms) when the GCD of the numerator and denominator is 1. To simplify, divide both the numerator and denominator by their GCD. Example: 24/36 → GCD(24, 36) = 12 → 24/12 = 2, 36/12 = 3 → simplified: 2/3.

Mixed Numbers and Improper Fractions

• Proper fraction: Numerator < denominator (e.g., 3/4). Represents less than one whole.
• Improper fraction: Numerator ≥ denominator (e.g., 7/4). Represents one or more wholes.
• Mixed number: A whole number plus a fraction (e.g., 1 3/4). Same as 7/4 as an improper fraction.
• Converting mixed to improper: Multiply whole number by denominator, add numerator: 2 3/5 = (2×5 + 3)/5 = 13/5
• Converting improper to mixed: Divide numerator by denominator: 13 ÷ 5 = 2 remainder 3 → 2 3/5

Frequently Asked Questions

Why can't I add fractions with different denominators directly? Fractions represent parts of different-sized "wholes" unless they share a denominator. Adding 1/2 + 1/3 directly would be like combining one half-apple slice with one third-orange slice — the units aren't comparable. Converting to a common denominator (sixths: 3/6 + 2/6 = 5/6) makes the parts equivalent and addable.

What is a unit fraction? A unit fraction has 1 as its numerator (1/2, 1/3, 1/7, etc.). Ancient Egyptian mathematics expressed nearly all fractions as sums of distinct unit fractions. Unit fractions appear in modern contexts like harmonic series in mathematics and music theory.

How do I work with negative fractions? Treat negative fractions exactly like positive ones, following the same rules for operations, then apply standard sign rules: negative × negative = positive; positive × negative = negative. The negative sign can be in the numerator, denominator, or in front of the fraction — all three are equivalent: −3/4 = 3/(−4) = −(3/4).