“Work out 1/2 + 1/3” - did you pause for a moment? Adding and subtracting fractions needs an extra step called finding a common denominator, which makes it one of the trickier early math topics.
This article explains how to add and subtract fractions with worked examples: common denominators, reducing answers, how it differs from multiplication and division, and the mistakes to avoid.
Fraction basics
First, the vocabulary:
| Term | Meaning | Example |
|---|---|---|
| Numerator | The top number | the 2 in 2/3 |
| Denominator | The bottom number | the 3 in 2/3 |
| Proper fraction | Numerator smaller than denominator | 2/3, 5/8 |
| Improper fraction | Numerator equal to or larger than denominator | 5/3, 7/7 |
| Mixed number | A whole number plus a proper fraction | 1 2/3 (= 5/3) |
The denominator is how many equal parts the whole is cut into; the numerator is how many you have. 2/3 means “2 parts out of 3”.
What is a common denominator?
A common denominator means rewriting fractions so they share the same denominator. You cannot add or subtract numerators until the denominators match.
To add 1/2 and 1/3, use the least common multiple of the denominators (6):
1/2 = 3/6
1/3 = 2/6Whatever you multiply the denominator by, multiply the numerator by the same amount: 2 → 6 by ×3, so 1 → 3; 3 → 6 by ×2, so 1 → 2.
The steps for finding a common denominator
- Find the least common multiple of the two denominators
- Work out what each denominator must be multiplied by
- Multiply each numerator by the same number
If the least common multiple is not obvious, multiply the denominators together and reduce at the end - the result is the same.
For 3/4 + 5/6, the least common multiple of 4 and 6 is 12: 3/4 = 9/12 and 5/6 = 10/12, so 9 + 10 = 19. The answer is 19/12, or 1 7/12 as a mixed number.
How to add fractions
Take 1/2 + 1/3:
- Common denominator: 1/2 = 3/6, 1/3 = 2/6
- Add the numerators: 3 + 2 = 5
- Keep the denominator: 5/6
- The GCD of 5 and 6 is 1, so no reduction is needed
How to subtract fractions
Subtraction follows the same flow: 1/2 − 1/3 becomes 3/6 − 2/6, so 3 − 2 = 1/6.
A common mistake: Writing 1/2 + 1/3 = 2/5 is wrong. When adding or subtracting, never add or subtract the denominators. The denominator shows how the whole is divided, so once the fractions share it, only the numerators change.
How multiplication and division differ
Addition and subtraction need a common denominator; multiplication and division do not.
| Operation | Method | Example |
|---|---|---|
| Addition / subtraction | Common denominator, then combine numerators | 1/2 + 1/3 = 5/6 |
| Multiplication | Multiply across: numerators, then denominators | 2/3 × 3/4 = 6/12 = 1/2 |
| Division | Multiply by the reciprocal | 1/2 ÷ 1/3 = 3/2 = 1 1/2 |
Multiplication is simplest: top times top, bottom times bottom. Division becomes multiplication by flipping the second fraction (its reciprocal): 1/2 ÷ 1/3 = 1/2 × 3/1 = 3/2.
Converting mixed numbers
Convert mixed numbers to improper fractions first - it removes a class of mistakes. 1 2/3 = 5/3 (1 × 3 + 2). To go back, divide: 5 ÷ 3 = 1 remainder 2.
For example, 1 1/2 + 2 1/3 = 3/2 + 7/3 = 9/6 + 14/6 = 23/6 = 3 5/6.
Reducing fractions
Reducing means dividing the numerator and denominator by their greatest common divisor. For 12/18 the GCD is 6, so (12 ÷ 6)/(18 ÷ 6) = 2/3.
Always reduce your final answer. If the GCD is not obvious, try small primes in order (2, 3, 5, 7…): 24/36 → 12/18 → 6/9 → 2/3.
Checking your work: Turn the answer into a decimal to verify it. 5/6 = 0.8333…, and 1/2 + 1/3 = 0.5 + 0.3333… = 0.8333…, so the answer checks out.
A routine that prevents mistakes
Fraction problems are error-prone because of the number of steps. Repeating the same order keeps you on track:
- Convert mixed numbers to improper fractions
- For addition or subtraction, find the common denominator
- Combine the numerators
- Reduce the answer
- Convert back to a mixed number if needed
Step 4 is the most commonly forgotten - check the numerator and denominator before writing the final answer.
How to calculate fractions
The Tools Hub fraction calculator takes two fractions and an operator, and shows the reduced answer with the working steps.
Enter the fractions and operator
Type the numerators and denominators, then choose +, −, × or ÷.
Read the answer and working
The reduced fraction, mixed number and decimal appear along with the calculation steps.
Or simplify a single fraction
Switch to the simplify mode to reduce one fraction and see its decimal form.
The tool introduced in this article
Fraction Calculator
Exact results as a reduced fraction, mixed number and decimal, with the working shown. All in your browser.
Summary
- Adding and subtracting fractions is a three-step routine: common denominator, combine numerators, reduce
- Find a common denominator by using the least common multiple, multiplying numerators to match
- Never add or subtract the denominators - a classic mistake
- Multiplication and division need no common denominator (division flips the second fraction)
- Convert mixed numbers to improper fractions before calculating
- Reduce the answer by dividing by the greatest common divisor
- The Tools Hub calculator shows the answer and the working steps
Next time you need to work with fractions, use the routine above or the Tools Hub fraction calculator.
FAQ
How do I find a common denominator?
Use the least common multiple of the denominators, then multiply each numerator by the same factor. For 1/2 and 1/3, use 6: 1/2 = 3/6 and 1/3 = 2/6.
What is 1/2 + 1/3?
With a common denominator, 3/6 + 2/6 = 5/6. Only the numerators are added - the denominator stays the same.
When should I reduce a fraction?
At the end of the calculation: check the greatest common divisor of the numerator and denominator, and divide if it is greater than 1. 12/18 reduces to 2/3.
How can I check a fraction answer with decimals?
Divide the numerator by the denominator: 5/6 = 0.8333…. The fraction calculator shows the reduced form and the decimal together, so you can verify at a glance.
