Adding Fractions Calculator

Background.

This adding fractions calculator combines two fractions — even ones with completely different denominators — and shows every step a textbook would ask you to write out by hand: the least common denominator (LCD), each numerator rewritten over that LCD, the raw sum before reduction, and finally the simplified answer. Enter 1/4 + 1/6 and it does not just print '5/12' — it reports lcd = 12, scaledNumerator1 = 3 (because 1/4 = 3/12), scaledNumerator2 = 2 (because 1/6 = 2/12), sumNumeratorRaw = 5, and the final resultNumerator = 5 over resultDenominator = 12. That is the actual arithmetic a student is expected to reproduce on paper, returned as real, readable numbers rather than hidden inside a single black-box answer.

Quanta's general-purpose fraction calculator (math/fraction) also has an 'add' mode, but it takes a structurally different shortcut: it cross-multiplies to the product of the two denominators (b × d) as a guaranteed — but rarely smallest — common denominator, and it returns only the final simplified sum. It never computes the true least common denominator, and it declares no scaled-numerator outputs anywhere in its schema, so there is no way to read the 'convert both fractions to the same denominator, then add' steps from that calculator at all. This page is built specifically to show that missing work: the LCD via the least common multiple, each numerator scaled to match, and the sum before and after reduction — the actual method taught in every pre-algebra classroom under the heading 'adding fractions with unlike denominators.'

Why does the LCD matter, rather than just using the product of the denominators? Both approaches give a mathematically valid common denominator and both lead to the same final, fully-reduced answer — but the LCD keeps every intermediate number as small as possible. Adding 5/6 and 3/8 using the product 6 × 8 = 48 as the denominator works, but produces the raw fraction 58/48 that still needs a further reduction step. Using the least common denominator, lcm(6, 8) = 24, gets you to 29/24 directly — already in lowest terms, no further work needed. That difference is exactly why textbooks teach the LCD method: fewer, smaller numbers to track by hand, with less risk of an arithmetic slip along the way.

The underlying method has three clean steps, and this calculator performs and reports each one individually. First, find the least common denominator: the least common multiple of the two original denominators, computed via lcm(d1, d2) = (d1 ÷ gcd(d1, d2)) × d2, using the Euclidean algorithm to find the greatest common divisor efficiently. Second, rewrite each fraction with that shared denominator by multiplying its numerator by the same scale factor the denominator was multiplied by — n1 × (lcd ÷ d1) and n2 × (lcd ÷ d2) — a step justified by the fact that multiplying any fraction by k/k (which equals 1) never changes its value. Third, add the two now-comparable numerators directly, since they are counting the same-sized pieces, and simplify the result by dividing out the greatest common divisor of the sum and the LCD.

This calculator also handles the cases that trip up manual arithmetic: a negative numerator on either fraction (subtraction is simply addition of a negative fraction), a sum that cancels exactly to zero, and a sum that lands on or exceeds a whole number, in which case the mixed-number breakdown (wholePart and fractionalNumerator) is provided alongside the improper form. Whether you're checking homework on unlike-denominator addition, combining measurements in a recipe or a woodworking plan, or just want to see the least-common-denominator method applied transparently to your own numbers, this page and the calculator above cover the single operation of fraction addition completely, with primary-source citations to OpenStax, Euclid, and Knuth.

What is adding fractions calculator?

Adding two fractions n1/d1 and n2/d2 means finding a single fraction that represents their combined value. Unlike whole numbers, fractions with different denominators cannot be added by simply adding their numerators, because the denominators describe pieces of different sizes — a fourth and a sixth are not the same-sized slice, so '1 fourth plus 1 sixth' is not meaningfully '2' of anything until both are rewritten in terms of a shared, equal-sized piece. That shared piece size is the common denominator, and the smallest one that works — the one that keeps the resulting numbers as small as possible — is the least common denominator (LCD), which is mathematically identical to the least common multiple (LCM) of the two original denominators. Once both fractions are rewritten over the LCD, their numerators count the same-sized pieces and can be added directly: n1/d1 + n2/d2 becomes (n1 × d/d1)/d + (n2 × d/d2)/d = (n1 × d/d1 + n2 × d/d2)/d, where d is the LCD. The final step is simplifying that sum to lowest terms by dividing both the numerator and denominator by their greatest common divisor, found via the Euclidean algorithm. When the resulting fraction's magnitude is 1 or greater, it can equivalently be written as a mixed number — a whole-number part plus a proper fractional remainder.

How to use this calculator.

1. 01Enter the first fraction's numerator and denominator in the first pair of fields. The denominator must be a positive whole number.
2. 02Enter the second fraction's numerator and denominator in the second pair of fields. Same rule: the denominator must be a positive whole number.
3. 03Read the least common denominator (LCD) output — this is the smallest shared denominator both fractions are converted to before adding.
4. 04Check the two scaled-numerator outputs to see exactly what each original numerator becomes once its fraction is rewritten over the LCD.
5. 05Read the raw sum (before simplifying) to see the two scaled numerators added together, still over the LCD.
6. 06Read the primary result — the simplified result numerator and result denominator — for the final, fully-reduced answer.
7. 07Check the decimal value for a quick sense of magnitude, and the mixed-number outputs (isMixed, wholePart, fractionalNumerator) whenever the sum's magnitude is 1 or greater.
8. 08To add a negative fraction (which is the same as subtracting a positive one), simply enter a negative numerator — for example, to compute 3/4 − 1/2, enter n2 = −1, d2 = 2.

The formula.

The calculator performs the standard least-common-denominator method for adding fractions (OpenStax Prealgebra 2e §4.5), in three explicit, individually-reported steps. Step one: find the least common denominator, d = lcm(d1, d2). The least common multiple is computed from the greatest common divisor via the identity gcd(a, b) × lcm(a, b) = a × b, rearranged as lcm(d1, d2) = (d1 ÷ gcd(d1, d2)) × d2 to avoid any intermediate value larger than necessary (Knuth, TAOCP Vol. 2, §4.5.2). The GCD itself is found by the Euclidean algorithm (Euclid, Elements, Book VII, Proposition 2): repeatedly replace the larger of two numbers with the remainder of dividing it by the smaller, until the remainder is zero — the last non-zero remainder is the GCD. Step two: rewrite each fraction with denominator d by multiplying both its numerator and denominator by the same scale factor. Since d is a multiple of d1, the scale factor for the first fraction is exactly d ÷ d1, an integer; multiplying n1 by that integer gives the new, scaled numerator, and the fraction's value is unchanged because multiplying by (d ÷ d1) over (d ÷ d1) is multiplying by 1. The same reasoning scales the second fraction by d ÷ d2. Step three: because both fractions now describe pieces of the identical size 1/d, their numerators can be added directly — this is the entire justification for requiring a common denominator in the first place, since addition of counts only makes sense when the units being counted are the same size. The raw sum sits over d and is then reduced to lowest terms by dividing both the sum and d by their greatest common divisor, found by the same Euclidean algorithm used in step one. If the reduced result's magnitude is at least 1, it additionally admits a mixed-number form: the whole part is the floor of the decimal value, and the remaining fractional numerator is the reduced numerator minus the whole part times the reduced denominator.

A worked example.

Frequently asked questions.

References& sources.

1. [1]OpenStax. Prealgebra 2e, §4.5 'Add and Subtract Fractions with Different Denominators'. Rice University, peer-reviewed open-access textbook — the least-common-denominator method this calculator implements.
2. [2]OpenStax. Prealgebra 2e, §4.4 'Add and Subtract Fractions with Common Denominators'. Rice University — the same-denominator base case this calculator's LCD step reduces to when d1 = d2.
3. [3]Knuth, Donald E. The Art of Computer Programming, Volume 2: Seminumerical Algorithms, 3rd edition. Addison-Wesley, 1997. §4.5.2 gives the lcm(a,b) = (a/gcd(a,b))·b identity used to compute the LCD without unnecessary intermediate growth.
4. [4]Euclid. Elements, Book VII, Proposition 2 (c. 300 BCE). The Euclidean algorithm for the greatest common divisor, the basis of both the LCD computation and the final simplification step.
5. [5]NIST Digital Library of Mathematical Functions, §27.1 ('Functions of Number Theory') — the multiplicative identity gcd(a,b)·lcm(a,b) = a·b underlying the LCD computation.