About

Fraction arithmetic errors cascade. A wrong common denominator in a structural load calculation or a medication dosage formula produces results that look plausible but are quantitatively wrong. This calculator finds the Least Common Denominator (LCD) using the Euclidean algorithm for GCD, performs the addition or subtraction, and reduces the result to lowest terms. It handles improper fractions and converts them to mixed numbers. All intermediate steps are exposed so you can verify each transformation. The tool assumes integer numerators and non-zero integer denominators. Negative values are permitted in any position.

Formulas

The core operation for adding or subtracting two fractions ab and cd requires a common denominator. The Least Common Denominator is computed from the Greatest Common Divisor.

GCD(a, b) =

{

a if b = 0GCD(b, a mod b) otherwise

LCM(b, d) = |b × d|GCD(b, d)

With the LCD L = LCM(b, d), each fraction is scaled:

ab ± cd = a × (L ÷ b) ± c × (L ÷ d)L

The result is then simplified by dividing both numerator and denominator by their GCD. If the absolute value of the resulting numerator is greater than or equal to the denominator, the fraction is also expressed as a mixed number: whole part w = floor(|n| ÷ d), remainder r = |n| mod d.

Where a = numerator of first fraction, b = denominator of first fraction, c = numerator of second fraction, d = denominator of second fraction, L = Least Common Denominator, n = resulting numerator, w = whole part of mixed number, r = remainder.

Reference Data

Operation	Example	LCD	Converted	Result	Simplified
Addition	12 + 13	6	36 + 26	56	56
Addition	25 + 310	10	410 + 310	710	710
Subtraction	34 − 16	12	912 − 212	712	712
Addition	58 + 78	8	58 + 78	128	1 12
Subtraction	79 − 29	9	79 − 29	59	59
Addition	14 + 14	4	14 + 14	24	12
Subtraction	56 − 14	12	1012 − 312	712	712
Addition	23 + 45	15	1015 + 1215	2215	1 715
Subtraction	1112 − 38	24	2224 − 924	1324	1324
Addition	715 + 1120	60	2860 + 3360	6160	1 160
Subtraction	12 − 34	4	24 − 34	−14	−14
Addition	05 + 37	35	035 + 1535	1535	37

Frequently Asked Questions

It computes the Greatest Common Divisor (GCD) of the two denominators using the Euclidean algorithm, then calculates LCM = |b × d| ÷ GCD(b, d). This LCM is the smallest positive integer that both denominators divide into evenly, giving you the Least Common Denominator.

When the absolute value of the numerator equals or exceeds the denominator, the calculator automatically converts the result to a mixed number. For example, 22/15 becomes 1 7/15. Both forms (improper and mixed) are displayed.

Yes. The calculator normalizes the sign so that any negative in the denominator is moved to the numerator. For instance, 3/(−4) is treated as (−3)/4. The final result always places the sign on the numerator.

Exposing each step - finding the LCD, scaling each fraction, combining numerators, and simplifying - lets you verify the math and learn the procedure. This is critical for educational use and for catching errors in manual work.

When denominators match, the LCD equals that shared denominator. The calculator skips the scaling step and directly adds or subtracts the numerators, then simplifies. For example, 5/8 + 7/8 = 12/8 = 3/2 = 1 1/2.

Yes. A numerator of zero is valid and simply produces zero or the other fraction as the result. A denominator of zero is rejected because division by zero is undefined. The tool shows an error toast if you attempt it.