Fractions Guide: Types, Arithmetic & Conversions

Fractions are one of the foundational concepts in mathematics — and one of the most persistently misunderstood. Operations that seem counterintuitive (dividing by a fraction makes a number larger) trip up students and adults alike.

This guide covers the full range of fraction operations: what different types of fractions are, how to simplify them, how to add, subtract, multiply, and divide them correctly, how to work with mixed numbers, and how to convert between fractions, decimals, and percentages. Each operation is shown with the formula and a worked example.

A fraction has a numerator (top number) and a denominator (bottom number). The denominator indicates how many equal parts the whole is divided into; the numerator says how many of those parts are taken.

Proper fractions — numerator is less than the denominator. The value is between 0 and 1. Examples: 1/2, 3/4, 7/8.

Improper fractions — numerator is greater than or equal to the denominator. The value is 1 or more. Examples: 5/3, 8/4, 11/7.

Mixed numbers — a whole number combined with a proper fraction. Examples: 1 2/3, 3 1/4. Mixed numbers and improper fractions represent the same values (1 2/3 = 5/3) in different forms.

Equivalent fractions — different fractions representing the same value. 1/2 = 2/4 = 3/6 = 50/100. Multiply or divide both numerator and denominator by the same non-zero number to produce equivalent fractions.

Unit fractions — fractions with numerator 1, such as 1/2, 1/3, 1/7. They are the building blocks used in ancient Egyptian fraction notation and appear frequently in proportional reasoning.

A fraction is in its simplest form (also called lowest terms) when the numerator and denominator share no common factors other than 1.

Method: divide both by the GCD

The greatest common divisor (GCD) is the largest number that divides evenly into both the numerator and denominator.

• Simplify 36/48: GCD(36, 48) = 12. Divide both: 36÷12 = 3, 48÷12 = 4. Result: 3/4
• Simplify 15/25: GCD(15, 25) = 5. 15÷5 = 3, 25÷5 = 5. Result: 3/5
• Simplify 7/13: GCD(7, 13) = 1 (both are prime). Already in simplest form: 7/13

Euclid's algorithm for GCD (efficient for large numbers): repeatedly divide the larger number by the smaller and take the remainder, until the remainder is 0. The last non-zero remainder is the GCD.

GCD(36, 48): 48 ÷ 36 = 1 remainder 12 → 36 ÷ 12 = 3 remainder 0. GCD = 12.

To add or subtract fractions, the denominators must be the same. If they already are, add or subtract the numerators and keep the denominator.

Same denominator:

• 3/8 + 1/8 = (3+1)/8 = 4/8 = 1/2
• 5/7 − 2/7 = (5−2)/7 = 3/7

Different denominators — find the LCD first:

The least common denominator (LCD) is the LCM of the two denominators. Convert each fraction to an equivalent fraction with the LCD, then add or subtract.

• 1/3 + 1/4: LCD = 12. Convert: 4/12 + 3/12 = 7/12
• 5/6 − 1/4: LCD = 12. Convert: 10/12 − 3/12 = 7/12. Simplify: 7/12
• 3/5 + 2/3: LCD = 15. Convert: 9/15 + 10/15 = 19/15 = 1 4/15

Always simplify the result. If the result is an improper fraction, convert to a mixed number if context calls for it.

Multiplying and dividing fractions do not require a common denominator.

Multiplication: multiply numerators, multiply denominators, simplify.

• 2/3 × 3/5 = (2×3)/(3×5) = 6/15 = 2/5
• 4/7 × 7/8 = 28/56 = 1/2
• 3/4 × 8/9 = 24/36 = 2/3

Cross-cancellation shortcut: before multiplying, cancel any factor in a numerator with any matching factor in a denominator (including across the two fractions). This keeps numbers smaller.

Example: 4/9 × 3/8. Cancel 4 and 8 (÷4): 1/9 × 3/2. Cancel 3 and 9 (÷3): 1/3 × 1/2 = 1/6.

Division: multiply by the reciprocal of the second fraction. The reciprocal of a/b is b/a (flip numerator and denominator).

• 2/3 ÷ 4/5 = 2/3 × 5/4 = 10/12 = 5/6
• 3/4 ÷ 3/8 = 3/4 × 8/3 = 24/12 = 2
• 5/6 ÷ 1/3 = 5/6 × 3/1 = 15/6 = 5/2 = 2 1/2

Why dividing by a fraction less than 1 produces a larger result: dividing by 1/2 asks "how many halves fit in X?" — more than X whole units do, so the result is larger.

Converting a mixed number to an improper fraction:

Multiply the whole number by the denominator, add the numerator, and place the result over the original denominator.

• 2 3/4 = (2×4 + 3)/4 = 11/4
• 5 1/3 = (5×3 + 1)/3 = 16/3
• 3 2/5 = (3×5 + 2)/5 = 17/5

Converting an improper fraction to a mixed number:

Divide numerator by denominator. The quotient is the whole number; the remainder becomes the new numerator over the original denominator.

• 13/4: 13 ÷ 4 = 3 remainder 1 → 3 1/4
• 22/7: 22 ÷ 7 = 3 remainder 1 → 3 1/7
• 19/6: 19 ÷ 6 = 3 remainder 1 → 3 1/6

Arithmetic with mixed numbers: convert to improper fractions first, perform the operation, then convert back.

Example: 2 1/2 + 1 3/4 = 5/2 + 7/4 = 10/4 + 7/4 = 17/4 = 4 1/4

The Fraction Calculator handles all mixed number operations with exact results and step-by-step solutions.

Fraction to decimal: divide numerator by denominator.

• 1/4 = 1 ÷ 4 = 0.25
• 3/8 = 3 ÷ 8 = 0.375
• 2/3 = 2 ÷ 3 = 0.6̄ (repeating)
• 7/11 = 0.6̄3̄ (repeating)

Fractions whose denominators have only factors of 2 and 5 produce terminating decimals. All other fractions produce repeating decimals.

Fraction to percentage: convert to decimal, then multiply by 100.

• 3/4 = 0.75 = 75%
• 5/8 = 0.625 = 62.5%
• 1/3 ≈ 0.333 = 33.3%

Decimal to fraction: write the decimal over the appropriate power of 10, then simplify.

• 0.6 = 6/10 = 3/5
• 0.125 = 125/1000 = 1/8
• 0.36 = 36/100 = 9/25

Common fraction-decimal-percent equivalents worth memorizing:

• 1/2 = 0.5 = 50%
• 1/4 = 0.25 = 25%
• 3/4 = 0.75 = 75%
• 1/3 ≈ 0.333 ≈ 33.3%
• 2/3 ≈ 0.667 ≈ 66.7%
• 1/5 = 0.2 = 20%
• 1/8 = 0.125 = 12.5%
• 1/10 = 0.1 = 10%

These tools handle fraction arithmetic and conversions with exact results:

• Fraction Calculator — add, subtract, multiply, divide, and simplify fractions and mixed numbers with step-by-step solutions
• Proportion Calculator — solve for a missing value in a proportion (equivalent fraction equation)
• Percentage Calculator — convert fractions to percentages and solve percentage problems

Browse all Math Calculators for the full toolkit including slope, Pythagorean theorem, and statistics.