Fractions appear constantly in everyday life — cooking, carpentry, splitting a bill, or converting measurements — yet many people freeze when they have to add 3/4 and 2/3 in their head. The good news is that each of the four fraction operations follows a simple, repeatable rule. Learn the rule once, and any fraction problem becomes mechanical.
This guide walks through addition, subtraction, multiplication, and division of fractions with worked examples for each, plus a section on mixed numbers. Skip to the operation you need, or use the fraction calculator to check your work instantly.
The Four Rules at a Glance
| Operation | Rule | Example |
|---|---|---|
| Addition (a/b + c/d) | Find a common denominator, add numerators | 1/2 + 1/3 = 3/6 + 2/6 = 5/6 |
| Subtraction (a/b − c/d) | Find a common denominator, subtract numerators | 3/4 − 1/6 = 9/12 − 2/12 = 7/12 |
| Multiplication (a/b × c/d) | Multiply across: (a×c)/(b×d) | 2/3 × 3/4 = 6/12 = 1/2 |
| Division (a/b ÷ c/d) | Multiply by reciprocal: (a/b) × (d/c) | 2/3 ÷ 4/5 = 2/3 × 5/4 = 5/6 |
How to Add Fractions
Adding fractions requires a common denominator — both fractions must share the same bottom number before you can add the tops.
Step 1: Check the denominators. If they’re the same, skip to step 3. If not, find a common denominator.
Step 2: Find the common denominator. The easiest method: multiply the two denominators together. You can also find the least common multiple (LCM) for cleaner numbers, but multiplying always works.
Step 3: Convert each fraction. Multiply the numerator and denominator of each fraction so both denominators become your common denominator.
Step 4: Add the numerators. Keep the common denominator.
Step 5: Simplify. Divide numerator and denominator by their Greatest Common Factor (GCF).
Example: 1/2 + 1/3
- Different denominators: 2 and 3
- Common denominator: 2 × 3 = 6
- Convert: 1/2 → 3/6 and 1/3 → 2/6
- Add: 3/6 + 2/6 = 5/6
- Already simplified (GCF of 5 and 6 is 1)
Example: 3/8 + 1/4
- Common denominator: 8 (since 4 goes into 8 evenly)
- Convert: 1/4 → 2/8
- Add: 3/8 + 2/8 = 5/8 ✓
How to Subtract Fractions
Subtraction follows the exact same steps as addition — the only difference is that you subtract the numerators in step 4.
Example: 5/6 − 1/4
- Different denominators: 6 and 4
- Common denominator: 6 × 4 = 24 (or LCM = 12, which keeps numbers smaller)
- Using 12: 5/6 → 10/12, 1/4 → 3/12
- Subtract: 10/12 − 3/12 = 7/12
- GCF(7, 12) = 1, already simplified ✓
Example: 7/10 − 3/10
Same denominator: 7/10 − 3/10 = 4/10 = 2/5 (simplified by dividing by 2)
Use the fraction calculator to verify any subtraction and see every step shown automatically.
How to Multiply Fractions
Multiplication is the easiest operation: no common denominator needed. Multiply straight across.
Rule: (a/b) × (c/d) = (a×c) / (b×d), then simplify.
Example: 2/3 × 3/4
- Multiply numerators: 2 × 3 = 6
- Multiply denominators: 3 × 4 = 12
- Result: 6/12 = 1/2 (simplified by GCF = 6)
Example: 5/6 × 4/9
- 5 × 4 = 20, 6 × 9 = 54
- 20/54 → GCF(20, 54) = 2 → 10/27
Cross-cancellation tip: Before multiplying, check if any numerator and diagonal denominator share a common factor. Cancel them first to keep numbers small. For 2/3 × 3/4: the 3 in the numerator of the second fraction and the 3 in the denominator of the first cancel to 1, giving 2/1 × 1/4 = 2/4 = 1/2. Same answer, smaller arithmetic.
How to Divide Fractions
Division uses the “keep, change, flip” method:
- Keep the first fraction as-is
- Change ÷ to ×
- Flip the second fraction (take its reciprocal)
- Multiply normally and simplify
Example: 3/4 ÷ 1/2
- Keep 3/4, change to ×, flip 1/2 to 2/1
- 3/4 × 2/1 = 6/4 = 3/2 = 1 1/2
Example: 5/8 ÷ 5/4
- Flip 5/4 → 4/5
- 5/8 × 4/5 = 20/40 = 1/2 ✓
Working with Mixed Numbers
A mixed number combines a whole number and a fraction (like 2 3/4). Before you can do arithmetic, convert it to an improper fraction:
Formula: Whole × Denominator + Numerator, over the original denominator
| Mixed Number | Conversion | Improper Fraction |
|---|---|---|
| 1 1/2 | (1×2)+1 / 2 | 3/2 |
| 2 3/4 | (2×4)+3 / 4 | 11/4 |
| 3 2/3 | (3×3)+2 / 3 | 11/3 |
| 4 1/8 | (4×8)+1 / 8 | 33/8 |
Example: 1 1/2 + 2/3
- Convert 1 1/2 → 3/2
- Add: 3/2 + 2/3 — find common denominator 6
- 9/6 + 4/6 = 13/6
- Convert back to mixed number: 13 ÷ 6 = 2 with remainder 1, so 2 1/6
Example: 2 3/4 × 1 1/3
- Convert: 2 3/4 → 11/4 and 1 1/3 → 4/3
- Multiply: 11/4 × 4/3 = 44/12 = 11/3 = 3 2/3
The fraction calculator handles mixed number input directly — type the whole number into the “Whole Number” field and let the calculator do the conversion.
Simplifying Fractions (Reducing to Lowest Terms)
A fraction is fully simplified when the numerator and denominator share no common factor other than 1.
Method: Find the GCF of the numerator and denominator, then divide both by it.
| Fraction | GCF | Simplified |
|---|---|---|
| 6/12 | 6 | 1/2 |
| 15/25 | 5 | 3/5 |
| 18/24 | 6 | 3/4 |
| 14/21 | 7 | 2/3 |
Finding the GCF: List the factors of each number and find the largest one they share, or use the Euclidean algorithm (divide the larger by the smaller, take the remainder, repeat until remainder is 0).
Fractions also relate directly to percentages — for quick conversions between the two, the percentage calculator is useful. When working out discounts (e.g., a 1/4-off sale = 25% off), the discount calculator gives you the final price instantly.
Frequently Asked Questions
What is a common denominator and why do I need one to add or subtract fractions? A common denominator is a shared bottom number that lets you compare or combine fractions. You can only add or subtract fractions when they represent equal-sized parts. 1/4 and 1/3 are different-sized pieces, so you can’t directly add them — you first convert both to twelfths (3/12 and 4/12) and then add to get 7/12.
Can the denominator be negative? By convention, fractions are written with a positive denominator. If you end up with a negative denominator, multiply both numerator and denominator by −1. For example, 3/(−4) = (−3)/4.
What is an improper fraction? An improper fraction has a numerator larger than (or equal to) its denominator, such as 7/4. It represents a value greater than 1. It can also be written as the mixed number 1 3/4. Both forms are equally correct; improper fractions are easier to multiply and divide.
How do I add three or more fractions? Add two fractions at a time. Find a common denominator for the first two, add them, then add the result to the third fraction. Or find a common denominator for all fractions at once (the LCM of all denominators) and convert each.
Do I always need to simplify my answer? In most school and everyday contexts, yes — a simplified fraction is the expected final form. The fraction calculator automatically simplifies every result for you.
The Bottom Line
Every fraction operation follows a simple pattern:
- Add/Subtract: common denominator first, then combine numerators
- Multiply: straight across, then simplify
- Divide: flip the second fraction, then multiply
For mixed numbers, convert to improper fractions before operating. And whenever you need to check your work or handle complex calculations, the fraction calculator shows the full step-by-step solution so you can see exactly where each number comes from.