Fraction Calculator — Add, Subtract, Multiply, and Divide Fractions

How to Use the Fraction Calculator

1. Open the fraction calculator.
2. Enter the first fraction: type the numerator (top) and denominator (bottom) into the respective fields.
3. Select the operation: +, −, ×, or ÷.
4. Enter the second fraction.
5. For mixed numbers (e.g., 2 3/4), enter the whole number, numerator, and denominator separately.
6. Click Calculate. The result appears in simplified form with step-by-step working.

Fraction Operations

Simplification Techniques

Adding and Subtracting Fractions: Step by Step

To add or subtract fractions, they must share the same denominator. Follow these steps:

1. Find the Lowest Common Denominator (LCD) — the smallest multiple that both denominators divide into evenly.
2. Convert each fraction to an equivalent fraction with the LCD as the new denominator: multiply numerator and denominator by the same factor.
3. Add or subtract the numerators — keep the denominator the same.
4. Simplify the result by dividing numerator and denominator by their Greatest Common Divisor (GCD).

Worked example: 2/3 + 5/8

1. LCD(3, 8) = 24 (since 3 × 8 = 24 and they share no common factors)
2. 2/3 = 16/24 (multiply top and bottom by 8); 5/8 = 15/24 (multiply by 3)
3. Sum = (16 + 15)/24 = 31/24
4. 31 and 24 share no common factor → already simplified. Convert to mixed number: 1 7/24

Multiplying Fractions

Multiplication of fractions is the simplest of the four operations — no common denominator needed:

(a/b) × (c/d) = (a × c) / (b × d)

Cross-cancellation shortcut: before multiplying, simplify diagonally by cancelling common factors between any numerator and any denominator. This keeps numbers smaller.

Example: 4/9 × 3/8 — without cross-cancellation: 12/72 = 1/6. With cross-cancellation: cancel 4 and 8 to get 1 and 2; cancel 9 and 3 to get 3 and 1. Remaining: (1×1)/(3×2) = 1/6. Easier arithmetic, same answer.

Dividing Fractions

Division by a fraction is the same as multiplication by its reciprocal:

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

The phrase "keep, change, flip" (KCF) is a popular memory aid:

• Keep the first fraction as it is
• Change ÷ to ×
• Flip the second fraction (reciprocal)

Mixed Numbers

A mixed number has a whole part and a fractional part: 2 3/4 means "two and three-quarters."

To add, subtract, multiply, or divide mixed numbers, convert them to improper fractions first:

Whole a b/c = (a × c + b) / c

Example: 2 3/4 = (2 × 4 + 3) / 4 = 11/4

After calculating, convert the result back to a mixed number if the numerator exceeds the denominator (improper fraction).

Finding the GCD and LCD

Euclidean algorithm for GCD

The Euclidean algorithm efficiently finds the Greatest Common Divisor:

1. Divide the larger number by the smaller, noting the remainder
2. Replace the larger number with the smaller, and the smaller with the remainder
3. Repeat until the remainder is 0
4. The last non-zero remainder is the GCD

GCD(48, 18): 48 = 2 × 18 + 12; 18 = 1 × 12 + 6; 12 = 2 × 6 + 0. GCD = 6.

LCD using prime factorisation

Factor each denominator into primes. Take the highest power of each prime that appears in either factorisation, and multiply these together. LCD(12, 18): 12 = 2² × 3; 18 = 2 × 3². LCD = 2² × 3² = 4 × 9 = 36.

Fractions on a Number Line

Comparing fractions requires a common denominator or decimal conversion:

• 3/4 vs. 5/7: convert to same denominator 28: 21/28 vs. 20/28. Therefore 3/4 > 5/7.
• Alternatively: 3/4 = 0.75; 5/7 ≈ 0.714. Therefore 3/4 > 5/7.

To convert a fraction to a decimal: divide numerator by denominator. 3/4 = 3 ÷ 4 = 0.75. Some fractions produce repeating decimals: 1/3 = 0.333... = 0.3̄.

Practical Applications

Cooking and recipe scaling

Recipes often use fractional measurements (1/4 cup, 1/3 teaspoon, 3/4 lb). Scaling up or down requires fraction multiplication. To halve 2/3 cup: 2/3 × 1/2 = 2/6 = 1/3 cup. To triple 1/4 teaspoon: 1/4 × 3 = 3/4 teaspoon.

Construction and DIY measurements

Imperial measurements use fractions: 3/8 inch, 5/16 inch, 7/32 inch. Adding lengths: 3/8 + 5/16 = 6/16 + 5/16 = 11/16 inches. Subtracting: if a board is 2 1/4 inches wide and needs to be 1 5/8 inches wide, remove: 2 1/4 − 1 5/8 = 9/4 − 13/8 = 18/8 − 13/8 = 5/8 inches.

Common Questions

Why can't you add fractions by adding numerators and denominators separately?

Adding numerators and denominators separately (1/2 + 1/3 ≠ 2/5) gives the wrong answer because it ignores what the denominator represents — the size of each piece. 1/2 means one piece of size 1/2; 1/3 means one piece of size 1/3. To combine them you need to express both in terms of the same-sized pieces (common denominator) before adding. 1/2 + 1/3 = 3/6 + 2/6 = 5/6.

What does it mean for a fraction to be in lowest terms?

A fraction is in lowest terms (simplest form) when the numerator and denominator share no common factors other than 1 — their GCD is 1. 3/4 is in lowest terms (GCD(3,4) = 1); 6/8 is not (GCD(6,8) = 2 → simplifies to 3/4). Calculators always simplify results to lowest terms automatically.

What is a unit fraction?

A unit fraction has a numerator of 1: 1/2, 1/3, 1/4, 1/7, etc. Ancient Egyptian mathematics used only unit fractions (plus 2/3) — they wrote all fractions as sums of distinct unit fractions. For example, 3/4 = 1/2 + 1/4. Modern number theory still studies Egyptian fraction representations.