A fraction represents part of a whole. The numerator is above the fraction bar and tells you how many parts are being considered. The denominator below it tells you how many equal parts make up the whole. When calculating, the key question is whether the fractions have the same denominator.
Rules and formulas
To add or subtract fractions, use a common denominator: a/b + c/d = (a×d + c×b)/(b×d). To multiply, multiply the numerators and denominators. To divide, multiply by the reciprocal of the second fraction. You can then simplify by dividing the numerator and denominator by their greatest common divisor.
Two examples
For 1/4 + 1/6, a common denominator is 12. Therefore 1/4 = 3/12 and 1/6 = 2/12. The sum is 5/12.
For 3/5 − 1/10, convert 3/5 to 6/10. Then 6/10 − 1/10 = 5/10. Dividing numerator and denominator by 5 gives 1/2. Use the fraction calculator to check your working.
When are fractions useful?
Fractions appear in cooking, measurements, geometry, probability and school mathematics. They are particularly useful when you want an exact result and a decimal representation would repeat indefinitely.
Common mistakes
- Adding numerators and denominators directly.
- Dividing only the numerator or only the denominator when simplifying.
- Forgetting to take the reciprocal when dividing.
- Dividing by zero or using a zero denominator.
Check your answer by converting the fraction to a decimal or by reversing the calculation.
Additional practical tip
A useful check is to convert the result to a common denominator or compare it with a decimal representation. With negative or mixed numbers, first convert everything to a consistent form. This makes sign errors and incorrect scaling factors easier to spot.
Read next: Related guide and calculator.
For worked explanations and practice, see Serlo’s guide to adding and subtracting fractions (German-language resource).
Find a common denominator
When denominators differ, find a common multiple, preferably the least common multiple. For 1/4 and 1/6, that number is 12. Multiply 1/4 by 3 to get 3/12, and multiply 1/6 by 2 to get 2/12. Only then add the numerators: 3/12 + 2/12 = 5/12. Subtraction works in the same way, except that you subtract the numerators. To simplify a fraction, divide the numerator and denominator by the same number. If you are unsure, check the result as a decimal or reverse the calculation. This helps you identify whether a mistake happened during conversion, arithmetic or simplification.
Check fractions in everyday situations
When cooking or sharing something, fractions can often be compared by imagining equal-sized parts. Make sure the parts are the same size. One half is not automatically the same as two arbitrary pieces of a different whole.