Key takeaways
- a/b + c/d = (a×d + b×c) / (b×d).
- You can't add numerators and denominators straight across.
- Cross-multiplying gives both fractions a matching denominator automatically.
- Always simplify the final answer using the HCF, if possible.
In this guide
What is fraction addition?
A fraction like 2/5 represents 2 parts out of 5 equal pieces. Adding two fractions combines those parts into one total, even if the fractions have different denominators (bottom numbers).
Why does adding fractions matter?
Real amounts are rarely whole numbers. Recipes, measurements, and project progress are often described in fractions, so combining them correctly matters whenever you're mixing ingredients or tracking partial progress.
1/2 + 1/3 is not 2/5. The pieces are different sizes, so you can't add tops and bottoms straight across — you first need to compare equal-sized pieces.
The formula
| Symbol | What it means | Example |
|---|---|---|
| a | First fraction's numerator (top) | 2 |
| b | First fraction's denominator (bottom) | 5 |
| c | Second fraction's numerator | 1 |
| d | Second fraction's denominator | 4 |
How to add fractions step by step
- 1Cross-multiply the first pair. Multiply the first numerator (a) by the second denominator (d):
2 × 4 = 8. - 2Cross-multiply the second pair. Multiply the second numerator (c) by the first denominator (b):
1 × 5 = 5. - 3Add those two results together to get your new numerator:
8 + 5 = 13. - 4Multiply the two denominators (b × d) for your new denominator:
5 × 4 = 20. Simplify if possible using the HCF.
Enter your own two fractions in the calculator below to see every step.
Try it yourself
Pre-filled with the example — change any value.
Common mistakes to avoid
- Forgetting to simplify the final answer. A fraction like 6/8 should be reduced to 3/4.
- Mixing up addition with [fraction multiplication](/calculator/fraction-multiply). Adding fractions needs matching denominators; multiplying does not.
- Cross-multiplying the wrong pair. Always pair each numerator with the other fraction's denominator.
Tips and tricks
- Use the smallest common denominator (the LCM of b and d) instead of b×d for smaller numbers to work with, if you spot it easily.
- Estimate first. Two fractions each close to 1/2 should add up to something close to 1.
- Once you have a final answer, run it through the fraction simplifier to double-check it's in lowest terms.
Where you'll use it in real life
- Cooking: combining ingredient amounts measured in fractions, like cups or teaspoons.
- Projects: adding up parts of a task that are already partly finished.
- Measuring: working with lengths given in fractions of an inch or foot.
- School: a core skill for every later topic in algebra and fractions.
Quick summary
To add two fractions, cross-multiply to get a matching denominator, add the resulting numerators, and simplify. You never add numerators and denominators straight across. Try the calculator above with your own two fractions.
Worked example
You hiked 2/5 of a trail before lunch, then hiked another 1/4 of the trail after lunch. What fraction of the trail have you covered in total?
- 01a/b + c/d = (ad + bc) / bd
- 022/5 + 1/4 = (2×4 + 5×1) / (5×4)
- 03= (8 + 5) / 20
- 04= 13/20
- 05Simplified = 13/20
- 06As a decimal = 0.65
These numbers are pre-filled in the calculator above.
Frequently asked questions
How do you add fractions with different denominators?
Cross-multiply: multiply each numerator by the other fraction's denominator, add the results, and put that over the product of both denominators. Then simplify if possible.
How do you add fractions that already have the same denominator?
Just add the numerators and keep the denominator the same. For example, 1/6 + 2/6 = 3/6, which simplifies to 1/2.
Why do you need a common denominator to add fractions?
Fractions can only be combined directly when the pieces are the same size. A common denominator makes both fractions use equal-sized pieces before adding.
Can the sum of two fractions be more than 1?
Yes. If the fractions together represent more than a whole, like 3/4 + 1/2, the sum will be greater than 1 (in this case, 5/4).
