Adding & Subtracting Fractions Calculator

Why you need a common denominator: You can only add or subtract fractions when they share the same denominator. Think of it like units — you can’t add apples and oranges directly. 1/3 means “1 out of 3 equal parts”; 1/4 means “1 out of 4 equal parts.” To combine them you need a common unit size.

Finding the LCD (Least Common Denominator): The LCD is the Least Common Multiple (LCM) of the two denominators.

LCD(a, b) = |a × b| / GCD(a, b)

Conversion step: Multiply each fraction’s numerator by (LCD ÷ its own denominator).

The formula: a/b ± c/d = (a × LCD/b ± c × LCD/d) / LCD

Simplification: Divide both numerator and denominator by their GCD to reach lowest terms.

Mixed number form: If the result is an improper fraction (numerator > denominator), it can also be written as a whole number + fraction, e.g. 7/4 = 1 and 3/4.

Worked example: 1/3 + 1/4

Step 1: Find LCD: LCM(3, 4) = 12 Step 2: Convert: 1/3 → 4/12, 1/4 → 3/12 Step 3: Add numerators: 4 + 3 = 7 Step 4: Result: 7/12 Step 5: Simplify: GCD(7, 12) = 1 → already in lowest terms → 7/12

The denominators never get added. This is the single most common mistake, and it is worth saying plainly: 1/3 + 1/4 is not 2/7. Check it against something you can picture. 2/7 is smaller than either 1/3 or 1/4, and adding two positive amounts cannot leave you with less than you started with. The right answer, 7/12, sits just above a half, which is about what a third plus a quarter should feel like.

When the LCD is not the product. Multiplying the two denominators always gives a common denominator, but often not the smallest one. For 1/6 + 1/4, the product is 24 and the answer comes out 10/24, which then has to be simplified to 5/12. Using the LCD of 12 gets you to 5/12 in one step with no simplifying at all. That is the whole reason the GCD appears in the formula above.

Subtracting can go negative, and that is fine. 1/4 − 3/4 is −1/2. The calculator keeps the sign on the numerator and always leaves the denominator positive, which is the standard convention: you write −1/2, not 1/−2.

A whole-number answer is written as a whole number. 5/6 + 1/6 makes 6/6, which simplifies to 1, not to 1/1.

Where this shows up outside homework: halving a recipe that calls for 3/4 cup, adding up 1/2-inch and 3/8-inch drill bits to see if two holes overlap, or working out how much of a shift is left when you have done 2/3 of it and taken a 1/12 break.