Fraction Calculator

Working with fractions is a fundamental math skill that many students find challenging—but it doesn't have to be. Our Fraction Calculator handles all four arithmetic operations (addition, subtraction, multiplication, division) with proper fractions, improper fractions, and mixed numbers. Results are automatically simplified to lowest terms and can be shown as improper fractions or mixed numbers. For learners, we provide optional step-by-step solutions that explain each step of the calculation.

What Are Fractions?

A fraction represents a part of a whole. It consists of:

• Numerator: The top number (how many parts you have)
• Denominator: The bottom number (how many equal parts make the whole)

Types of Fractions:

• Proper Fraction: Numerator < Denominator (e.g., 3/4, 1/2). Value less than 1.
• Improper Fraction: Numerator ≥ Denominator (e.g., 7/4, 5/2). Value greater than or equal to 1.
• Mixed Number: Whole number + proper fraction (e.g., 1¾, 2½). Easier for human reading.
• Equivalent Fractions: Different fractions representing the same value (e.g., 1/2 = 2/4 = 3/6).

How to Add and Subtract Fractions

Rule 1: Same Denominator (Common Denominator)

When denominators are equal, simply add/subtract the numerators and keep the denominator.

Example: 2/7 + 3/7 = (2+3)/7 = 5/7

Rule 2: Different Denominators (Most Common Case)

You MUST find a common denominator (preferably the Least Common Denominator, LCD).

1. Find the LCD of both denominators (smallest number both divide into evenly)

2. Convert each fraction to equivalent form with the LCD

3. Add/subtract the numerators

4. Simplify the result

1. Denominators 3 and 4 → LCD = 12

2. 1/3 = 4/12 (multiply numerator and denominator by 4)

1/4 = 3/12 (multiply numerator and denominator by 3)

3. 4/12 + 3/12 = 7/12 (already simplified)

1. LCD = 12

2. 5/6 = 10/12 (×2), 1/4 = 3/12 (×3)

3. 10/12 - 3/12 = 7/12

How to Multiply Fractions

Simply multiply the numerators together and denominators together. No common denominator needed.

Examples:

• 2/3 × 4/5 = (2×4)/(3×5) = 8/15
• 1/2 × 2/3 = 2/6 = 1/3 (simplify after multiplication)

Multiplying a Fraction by a Whole Number:

Write the whole number as a fraction over 1.

Example: 3 × 2/5 = 3/1 × 2/5 = 6/5 = 1⅕

How to Divide Fractions

To divide by a fraction, multiply by its reciprocal (flip the second fraction).

Dividing a Fraction by a Whole Number:

Write the whole number as a fraction, then multiply by its reciprocal.

Example: 2/3 ÷ 4 = 2/3 ÷ 4/1 = 2/3 × 1/4 = 2/12 = 1/6

Working with Mixed Numbers

Mixed numbers (e.g., 1¾) are easier for humans but harder for calculations. Convert them to improper fractions first:

Example: 2⅓ = (2×3 + 1)/3 = (6+1)/3 = 7/3

Example: 1¾ = (1×4 + 3)/4 = (4+3)/4 = 7/4

Example: 7/4 = 7 ÷ 4 = 1 remainder 3 → 1¾

Example: 11/3 = 11 ÷ 3 = 3 remainder 2 → 3⅔

How to Simplify Fractions (Reducing to Lowest Terms)

Find the Greatest Common Divisor (GCD) of numerator and denominator, then divide both by the GCD.

Method 1 (List Factors):

24/36. Factors of 24: 1,2,3,4,6,8,12,24. Factors of 36: 1,2,3,4,6,9,12,18,36. GCD = 12. 24÷12=2, 36÷12=3 → 2/3.

Method 2 (Prime Factorization):

24 = 2³ × 3, 36 = 2² × 3². Cancel common factors: 2² × 3 cancels, leaving 2/3.

Method 3 (Euclidean Algorithm for large numbers):

48/180. 180÷48=3 remainder 36, 48÷36=1 remainder 12, 36÷12=3 remainder 0. GCD = 12. 48/180 = 4/15.

Step-by-Step Solutions Mode

Toggle "Show Steps" to see the complete solution process:

1. Identify operation (+, -, ×, ÷)

2. If + or -, find LCD, convert, add/subtract numerators

3. If × or ÷, multiply numerators/denominators or multiply by reciprocal

4. Simplify result using GCD

5. Convert to mixed number if requested

Real-World Applications of Fractions

Cooking & Recipes:

A recipe calls for 3/4 cup of flour, but you want to make half the recipe: 3/4 × 1/2 = 3/8 cup. Or you have 2/3 cup of sugar but need 1/4 cup less: 2/3 - 1/4 = 8/12 - 3/12 = 5/12 cup.

Construction & Woodworking:

A board is 7/8 inch thick. You need to cut off 1/4 inch: 7/8 - 1/4 = 7/8 - 2/8 = 5/8 inch remaining.

Music:

Fractions represent note durations. A quarter note (1/4) plus half note (1/2) = 3/4 of a measure in 4/4 time.

Banking & Finance:

Interest rates are often fractions: 5¾% = 5.75% = 23/4%.

Sewing & Fabric:

Pattern requires 2⅓ yards of fabric, and you have 5⅚ yards. Remaining: 5⅚ - 2⅓ = 35/6 - 14/6 = 21/6 = 3½ yards.

Teaching & Learning:

Parents helping children with homework can verify answers and see step-by-step methods.

FAQ: Fraction Calculator

Fraction Calculator is optimized for fast browser-based use, so you can test multiple scenarios in seconds.