Adding and Subtracting Fractions Calculator
Enter two fractions (or mixed numbers), choose to add or subtract, and get the answer reduced to lowest terms. The step-by-step panel shows exactly how to find the least common denominator, rewrite each fraction, combine the numerators, and simplify the result. Switch between proper-fraction and mixed-number modes to match your homework or coursework.
Formula
Worked example
1/3 + 1/4: the LCD of 3 and 4 is 12. Rewrite: 4/12 + 3/12 = 7/12. The GCD of 7 and 12 is 1, so 7/12 is already simplified.
How to add fractions
To add two fractions, you need a common denominator. If the denominators are already the same (like fractions), simply add the numerators and keep the denominator. If the denominators differ (unlike fractions), find the Least Common Denominator (LCD), which is the smallest multiple that both denominators divide into evenly. Rewrite each fraction as an equivalent fraction over the LCD, then add the numerators. Finally, reduce the result by dividing the numerator and denominator by their Greatest Common Divisor (GCD). For example, to add 1/3 and 1/4: the LCD of 3 and 4 is 12, giving 4/12 + 3/12 = 7/12.
How to subtract fractions
Subtracting fractions follows exactly the same process as adding, with one change: instead of adding the numerators once you have a common denominator, you subtract them. Find the LCD of the two denominators, convert each fraction to an equivalent form with that LCD, then subtract the second numerator from the first. Reduce the result to lowest terms. For example, 3/4 - 1/3: LCD is 12, so 9/12 - 4/12 = 5/12. Remember that subtracting a negative fraction is the same as adding its positive equivalent.
Adding and subtracting mixed numbers
A mixed number combines a whole number with a fraction, such as 2 1/3. To add or subtract mixed numbers, the most reliable method is to first convert each mixed number to an improper fraction. Multiply the whole number by the denominator and add the numerator: 2 1/3 becomes (2 x 3 + 1) / 3 = 7/3. Then add or subtract the improper fractions as you would any fractions. Convert the improper-fraction answer back to a mixed number if needed by dividing the numerator by the denominator; the quotient is the whole part and the remainder over the denominator is the fractional part.
Finding the Least Common Denominator
The Least Common Denominator (LCD) is the same as the Least Common Multiple (LCM) of the two denominators. One reliable method is to use prime factorization: factor each denominator, then take the highest power of every prime that appears in either factorization. Their product is the LCM. A faster shortcut is to use the formula LCD = (d1 x d2) / GCD(d1, d2), where GCD is the Greatest Common Divisor. For example, LCD of 6 and 9: GCD(6,9) = 3, so LCD = (6 x 9) / 3 = 18. This calculator computes the LCD automatically so you never have to do this by hand.
Common fraction addition and subtraction identities
| Rule | Formula | Example |
|---|---|---|
| Like denominators (add) | a/c + b/c = (a+b)/c | 1/5 + 2/5 = 3/5 |
| Like denominators (subtract) | a/c - b/c = (a-b)/c | 4/7 - 1/7 = 3/7 |
| Unlike denominators | a/b + c/d = (ad+bc) / bd | 1/3 + 1/4 = 7/12 |
| Subtracting with unlike denominators | a/b - c/d = (ad-bc) / bd | 3/4 - 1/3 = 5/12 |
| Mixed number to improper | w a/b = (w*b + a) / b | 2 1/3 = 7/3 |
| Result always simplified | Divide numerator and denominator by GCD | 6/9 = 2/3 |
Quick-reference rules that apply regardless of the specific values.
Frequently asked questions
Why do fractions need a common denominator to be added or subtracted?
Fractions represent parts of a whole, and the denominator tells you how many equal parts that whole is divided into. You can only directly count up or take away parts when they are the same size. Adding 1/3 and 1/4 is like adding one slice of a pie cut into 3 equal pieces to one slice of a pie cut into 4 equal pieces; the slices are different sizes, so you cannot simply add 1+1. Converting both fractions to twelfths (4/12 and 3/12) makes the parts equal-sized, so addition gives a meaningful result.
What is the difference between the LCD and the LCM?
When working with fractions, the Least Common Denominator (LCD) is the Least Common Multiple (LCM) of the denominators. The two terms describe the same number from different perspectives: LCM is a property of two integers, while LCD refers to how you use that LCM to create a shared denominator. So LCD(3, 4) = LCM(3, 4) = 12.
Do I always have to use the LCD, or can I use any common denominator?
Any common multiple of the denominators will work for adding or subtracting fractions; you do not have to use the smallest one. Using a larger common denominator just means you have a bigger fraction to simplify at the end. For example, for 1/3 + 1/4 you could use 24 as the common denominator (8/24 + 6/24 = 14/24), then reduce 14/24 to 7/12. Using the LCD (12) saves a simplification step but gives the same answer.
How do I add a whole number and a fraction?
A whole number n is equivalent to the fraction n/1. To add it to a fraction a/b, simply place n over 1 and proceed normally: n/1 + a/b = (n*b + a) / b. For example, 3 + 2/5 = 15/5 + 2/5 = 17/5, which equals the mixed number 3 2/5. This calculator handles this automatically when you enter 0 in the fractional numerator field in mixed-number mode.
How do I reduce a fraction to its simplest form?
Find the Greatest Common Divisor (GCD) of the numerator and denominator, then divide both by it. If the GCD is 1, the fraction is already in its simplest (lowest) form. For example, 6/9: GCD(6,9) = 3, so 6/9 = 2/3. The calculator applies this step automatically and always returns the result in lowest terms.
What happens when the result is negative?
If you subtract a larger fraction from a smaller one, the numerator of the result will be negative, giving a negative fraction. For example, 1/4 - 3/4 = -2/4 = -1/2. The calculator handles negative values correctly and displays the sign in the numerator of the simplified result.