CalConverters

Adding and Subtracting Fractions Calculator

Add or subtract 2 to 10 fractions at once with the least common denominator (LCD) method, showing every conversion, reduction, and mixed-number step.

Loading calculator…

How to use this calculator

Enter a numerator and denominator for each fraction. The first fraction is always added; every fraction after it has a + / − toggle so you can mix addition and subtraction freely. Use Add fraction to include up to 10 fractions, or remove a row with the × button (down to a minimum of two). The answer band updates live, and the steps below show the full least common denominator (LCD) method.

Tip: Enter a minus sign directly in a numerator (e.g. -3) to make that fraction negative — the calculator rewrites it using the standard sign rule automatically.

The LCD method: three steps

Unlike a two-fraction calculator that can cross-multiply, adding three or more fractions needs a shared denominator that all of them scale to cleanly. That's the least common denominator (LCD) — the least common multiple of every denominator in the expression.

  1. Find the LCD. Take the least common multiple of every denominator, exactly as entered.
  2. Convert each fraction. Multiply each fraction's numerator and denominator by whatever factor makes its denominator equal to the LCD.
  3. Add or subtract the numerators. With every fraction now over the same denominator, combine the numerators according to each fraction's sign, then reduce the result and convert to a mixed number if it's improper.

a1d1±a2d2±⋯±andn=(a1⋅Ld1)±(a2⋅Ld2)±…L\frac{a_1}{d_1} \pm \frac{a_2}{d_2} \pm \dots \pm \frac{a_n}{d_n} = \frac{\left(a_1 \cdot \frac{L}{d_1}\right) \pm \left(a_2 \cdot \frac{L}{d_2}\right) \pm \dots}{L}

where LL is the LCD of d1,d2,…,dnd_1, d_2, \dots, d_n.

Why you can't add the numerators and denominators directly

A fraction's denominator sets the size of the "slices" it's counting — 12\frac{1}{2} counts halves, 13\frac{1}{3} counts thirds. Those are different-sized pieces, so 12+13\frac{1}{2} + \frac{1}{3} can't become 1+12+3=25\frac{1+1}{2+3} = \frac{2}{5} (which is actually smaller than 12\frac{1}{2} alone, and clearly wrong). The LCD step rewrites every fraction in same-sized pieces (sixths, in this case) before the numerators are ever combined.

Tip: The LCD is taken from the denominators as entered, not from their reduced form. 23+46\frac{2}{3} + \frac{4}{6} uses LCD(3, 6) = 6, even though 46\frac{4}{6} reduces to 23\frac{2}{3}.

Handling negative fractions

Typing a minus sign into a numerator rewrites the expression using two rules: adding a negative fraction becomes subtracting its positive version, and subtracting a negative fraction becomes adding its positive version. Either way, every fraction shown in the steps ends up with a positive numerator and denominator — the sign moves entirely onto the + / − toggle.

Worked examples

Example 1 — three fractions: 12+13+14\frac{1}{2} + \frac{1}{3} + \frac{1}{4}

LCD(2, 3, 4) = 12. Converting: 612+412+312=1312\frac{6}{12} + \frac{4}{12} + \frac{3}{12} = \frac{13}{12}. Since 13 ≥ 12, convert to a mixed number: the answer is 1312=1112\frac{13}{12} = 1\frac{1}{12}.

Example 2 — mixed operations: 12+34−28\frac{1}{2} + \frac{3}{4} - \frac{2}{8}

LCD(2, 4, 8) = 8. Converting: 48+68−28=88\frac{4}{8} + \frac{6}{8} - \frac{2}{8} = \frac{8}{8}. The answer is the whole number 11.

Example 3 — a negative result: 12−34\frac{1}{2} - \frac{3}{4}

LCD(2, 4) = 4. Converting: 24−34=−14\frac{2}{4} - \frac{3}{4} = -\frac{1}{4}. The answer is −14-\frac{1}{4} — negative results are perfectly valid.

Precision note

Every computation uses arbitrary-precision integers (BigInt), not floating-point numbers — so the LCD, every converted numerator, and the final reduced fraction stay exact no matter how large the denominators are or how many fractions are combined.

Frequently asked questions

How do you add fractions that have different denominators?
Find the least common denominator (LCD) of every fraction, rewrite each one as an equivalent fraction over that LCD, then add the numerators. For 12+13\frac{1}{2} + \frac{1}{3}, the LCD is 6, so it becomes 36+26=56\frac{3}{6} + \frac{2}{6} = \frac{5}{6}.
Why can't you just add the numerators and denominators directly?
A fraction represents a share of a whole, and that whole is defined by the denominator. 12\frac{1}{2} and 13\frac{1}{3} are shares of differently sized wholes, so adding "1+1" over "2+3" would compare pieces that are not the same size — the LCD step makes every piece the same size first.
What is the LCD method, and how is it different from cross-multiplying?
The LCD method finds the least common multiple of all the denominators and scales every fraction to it, which stays efficient for any number of fractions. Cross-multiplying (used for exactly two fractions) multiplies every denominator together instead, which works but often produces a much larger, unreduced denominator.
Can this calculator add or subtract more than two fractions at once?
Yes. It handles 2 to 10 fractions in a single expression, mixing addition and subtraction freely, e.g. 12+34−28\frac{1}{2} + \frac{3}{4} - \frac{2}{8}.
How do you subtract a negative fraction?
Subtracting a negative fraction is the same as adding the positive version: 45−(−310)=45+310\frac{4}{5} - \left(-\frac{3}{10}\right) = \frac{4}{5} + \frac{3}{10}. The calculator applies this rewrite automatically whenever a numerator is entered as negative.
Why does 2/3 + 4/6 use an LCD of 6 instead of 3?
The LCD is taken from the denominators exactly as entered — 3 and 6 — giving LCD = 6, even though 46\frac{4}{6} could be simplified to 23\frac{2}{3} first. Simplifying beforehand would change which denominators are being compared, so the calculator scales from the literal input.
Does the calculator show mixed numbers?
Yes. Whenever the reduced sum is an improper fraction (numerator at least as large as the denominator), the calculator also converts it to a mixed number — for example 1312\frac{13}{12} becomes 11121\frac{1}{12}.

Related calculators