How do I add fractions with different denominators?
Last updated March 26, 2026
Find the least common denominator (LCD), convert both fractions to equivalent fractions with that denominator, then add the numerators. For 1/3 + 1/4: the LCD is 12, so 4/12 + 3/12 = 7/12.
How to Calculate
- 1
Find the least common denominator (LCD) of both fractions — the smallest number both denominators divide into evenly
- 2
Convert each fraction to an equivalent fraction with the LCD as its denominator by multiplying the numerator and denominator by the same factor
- 3
Add (or subtract) the numerators and keep the common denominator
- 4
Simplify the result by dividing the numerator and denominator by their greatest common factor (GCF)
The Formula
a/b + c/d = (a×d + c×b) / (b×d)The cross-multiplication method works for any two fractions: multiply each numerator by the other fraction’s denominator, then add (or subtract) the results over the product of the two denominators. This always produces a correct answer, though it may need simplifying. For multiplication, simply multiply numerators together and denominators together: (a/b) × (c/d) = (a×c)/(b×d). For division, flip the second fraction and multiply: (a/b) ÷ (c/d) = (a×d)/(b×c).
| Variable | Meaning |
|---|---|
| a, c | Numerators — the top numbers of each fraction |
| b, d | Denominators — the bottom numbers of each fraction |
| LCD | Least Common Denominator — smallest shared multiple of both denominators |
| GCF | Greatest Common Factor — largest number that divides both numerator and denominator evenly |
Common Examples
1/3 + 1/4
7/12
2/3 - 1/6
3/6 = 1/2
3/4 × 2/5
6/20 = 3/10
3/4 ÷ 2/3
9/8 = 1 1/8
1/2 + 2/3
7/6 = 1 1/6
5/8 - 1/4
3/8
1/3 as a decimal
0.333… (repeating)
3/8 as a decimal
0.375
The Four Fraction Operations
Fractions represent parts of a whole. The top number (numerator) tells you how many parts you have, and the bottom number (denominator) tells you how many equal parts the whole is divided into. There are four basic operations you can perform with fractions: addition, subtraction, multiplication, and division. Each follows its own set of rules.
Adding and Subtracting Fractions
Adding and subtracting fractions requires a common denominator — both fractions must have the same bottom number before you can combine them. If the denominators are already the same, simply add or subtract the numerators: 3/8 + 2/8 = 5/8.
When denominators differ, you need to find the Least Common Denominator (LCD), which is the smallest number that both denominators divide into evenly. For example, to add 1/3 + 1/4, the LCD of 3 and 4 is 12. Convert each fraction: 1/3 = 4/12 and 1/4 = 3/12. Now add the numerators: 4/12 + 3/12 = 7/12.
The cross-multiplication shortcut works for any two fractions: a/b + c/d = (a×d + c×b) / (b×d). This always produces a correct answer, though the result may need simplifying. For subtraction, use the same formula but subtract instead of add: a/b − c/d = (a×d − c×b) / (b×d).
Multiplying Fractions
Multiplication is the simplest fraction operation — no common denominator needed. Multiply the numerators together and the denominators together: (a/b) × (c/d) = (a×c) / (b×d).
For example, 3/4 × 2/5 = (3×2) / (4×5) = 6/20. Simplify by dividing both by their greatest common factor (GCF of 6 and 20 is 2): 6/20 = 3/10.
A helpful shortcut is cross-canceling before you multiply. Look for common factors between any numerator and any denominator. In 3/4 × 2/5, there are no shared factors to cancel, but in 2/3 × 3/4, you can cancel the 3s first: 2/1 × 1/4 = 2/4 = 1/2.
Dividing Fractions
To divide fractions, use the “flip and multiply” method: keep the first fraction, flip the second fraction (swap its numerator and denominator), then multiply. Mathematically: (a/b) ÷ (c/d) = (a/b) × (d/c) = (a×d) / (b×c).
For example, 3/4 ÷ 2/3 = 3/4 × 3/2 = 9/8. Since 9/8 is an improper fraction (the numerator is larger than the denominator), convert it to a mixed number: 9/8 = 1 1/8.
Why does this work? Dividing by a fraction is the same as asking “how many groups of that fraction fit into this one?” Flipping and multiplying is a mathematical shortcut for finding that answer.
Finding the Least Common Denominator (LCD)
The LCD is the smallest number that both denominators divide into evenly. There are two common methods:
- List multiples: Write out multiples of each denominator until you find a match. For 3 and 4: multiples of 3 are 3, 6, 9, 12, 15… and multiples of 4 are 4, 8, 12, 16… The first common multiple is 12.
- Prime factorization: Break each denominator into prime factors, then take the highest power of each prime. For 6 (2×3) and 8 (2³): LCD = 2³ × 3 = 24.
Simplifying Fractions
A fraction is simplified (or reduced) when the numerator and denominator share no common factors other than 1. To simplify, find the Greatest Common Factor (GCF) of the numerator and denominator, then divide both by it.
For example, to simplify 6/20: the factors of 6 are 1, 2, 3, 6 and the factors of 20 are 1, 2, 4, 5, 10, 20. The GCF is 2. Divide both by 2: 6/20 = 3/10.
Converting Fractions to Decimals
To convert a fraction to a decimal, divide the numerator by the denominator. Some fractions produce terminating decimals (they end), while others produce repeating decimals (a pattern that repeats forever).
Common fraction-to-decimal equivalents worth memorizing:
- 1/2 = 0.5
- 1/3 = 0.333… (repeating)
- 1/4 = 0.25
- 1/5 = 0.2
- 1/6 = 0.1666… (repeating)
- 1/8 = 0.125
- 2/3 = 0.666… (repeating)
- 3/4 = 0.75
- 3/8 = 0.375
- 5/8 = 0.625
- 7/8 = 0.875
Mixed Numbers and Improper Fractions
A mixed number combines a whole number and a fraction, like 1 1/8. An improper fraction has a numerator larger than its denominator, like 9/8. They represent the same value.
To convert an improper fraction to a mixed number, divide the numerator by the denominator. The quotient is the whole number and the remainder becomes the new numerator: 9 ÷ 8 = 1 remainder 1, so 9/8 = 1 1/8.
To convert a mixed number to an improper fraction, multiply the whole number by the denominator and add the numerator: 1 1/8 = (1×8 + 1)/8 = 9/8.
When performing operations with mixed numbers, it is usually easiest to convert them to improper fractions first, perform the operation, then convert the result back to a mixed number if needed.