sumivo

Sumivo · Math

Fraction Calculator

Add two fractions and simplify the sum. Use a common denominator, add the numerators, then divide both terms by their GCD. To reduce one fraction or mixed number without addition, use the Fraction Simplifier linked below.

01 / inputs

integer
integer
integer
integer

02 / result

SIMPLIFIED SUM

5/6

Result
5/6
Decimal value
0.83333333

Calculation trace

  1. 1/2 + 1/3 = 5/6
    0.83333333
    Add fractions
  2. 5/6 = 5/6
    0.83333333
    Simplify

WORKED EXAMPLES

These examples use the same denominator product and GCD reduction as the calculator. A smaller common denominator can shorten hand calculations without changing the simplified sum.

Same denominators: 1/4 + 1/4

  1. Common denominator: the engine uses 4 × 4 = 16, even though keeping 4 would also work.
  2. Convert and add: (1 × 4)/16 + (1 × 4)/16 = 4/16 + 4/16 = 8/16.
  3. Reduce: GCD(8, 16) = 8; (8 ÷ 8)/(16 ÷ 8) = 1/2.

Different denominators: 1/2 + 1/3

  1. Common denominator: 2 × 3 = 6.
  2. Convert and add: (1 × 3)/6 + (1 × 2)/6 = 3/6 + 2/6 = 5/6.
  3. Reduce: GCD(5, 6) = 1; (5 ÷ 1)/(6 ÷ 1) = 5/6, already in simplest form.

A sum that needs reducing: 1/6 + 1/3

  1. Common denominator: the engine uses 6 × 3 = 18. The least common denominator would be 6.
  2. Convert and add: (1 × 3)/18 + (1 × 6)/18 = 3/18 + 6/18 = 9/18.
  3. Reduce: GCD(9, 18) = 9; (9 ÷ 9)/(18 ÷ 9) = 1/2.

A negative fraction: −1/2 + 1/3

  1. Common denominator: 2 × 3 = 6; keep the negative sign on the first numerator.
  2. Convert and add: (−1 × 3)/6 + (1 × 2)/6 = −3/6 + 2/6 = −1/6.
  3. Reduce: GCD(|−1|, 6) = 1; (−1 ÷ 1)/(6 ÷ 1) = −1/6.
§

How it works

To add fractions, use a common denominator, add the numerators, then divide both terms by their greatest common divisor (GCD) to get the simplest form. This calculator uses the product of the denominators: a/b + c/d = (a·d + c·b) / (b·d). It finds the positive GCD using repeated remainders and moves any negative denominator sign to the numerator. The product is a valid common denominator, but is not always the least common denominator. The worked examples follow the same product-denominator steps as the engine.

Assumptions & limits

  • Enter two fractions as four integers. Negative numerators and negative denominators are accepted; zero denominators and non-integer values are rejected. A zero numerator is allowed.
  • This tool adds the two fractions you enter and shows the sum in lowest terms. To reduce one fraction or a mixed number without addition, use the related Fraction Simplifier.
  • The result keeps a positive denominator. A whole-number sum is displayed over 1, and a zero sum is displayed as 0/1. The decimal value is rounded to at most eight decimal places; it is not an exact repeating-decimal expansion.
  • This addition engine uses JavaScript number arithmetic, not arbitrary-precision integers. It does not enforce safe-integer limits or check intermediate products for overflow. Large inputs, products or sums can lose precision or fail to produce a usable result; do not rely on it for exact large-integer addition.
§

FAQ

How do you add fractions with different denominators?
Rewrite them with a common denominator, add the new numerators, and simplify. For 1/2 + 1/3, use 6: 3/6 + 2/6 = 5/6. The GCD of 5 and 6 is 1, so this sum is already in simplest form.
Why do fractions need a common denominator before adding?
The denominator tells you the size of each part. Halves and thirds are different-sized parts; rewriting both as sixths lets you count parts of the same size. If the denominators already match, you can add just the numerators and keep that denominator.
How do you simplify a fraction after adding?
Divide the sum numerator and denominator by their positive greatest common divisor. For 1/6 + 1/3, this engine gets 9/18, then divides both terms by GCD(9, 18) = 9 to give 1/2. To simplify a single fraction or mixed number, use the Fraction Simplifier linked below.
What is the difference between a common denominator and the least common denominator?
Any positive common multiple of the denominator magnitudes is a common denominator. The least common denominator (LCD) is their least common multiple (LCM). For denominators 6 and 3, the LCD is 6, while this engine uses the product 18; reducing the final sum gives the same answer for ordinary-sized inputs. Use the LCM Calculator linked below to find the LCD.
How do you add negative fractions?
Keep each sign when you multiply for the common denominator. For −1/2 + 1/3, use −3/6 + 2/6 = −1/6. This tool also accepts negative denominators: entering 1/−2 + 1/3 gives −1/6, with the sign on the final numerator.
Why is the sum of two fractions sometimes a whole number?
The numerator can be a multiple of the denominator, so reduction leaves a denominator of 1. For 1/2 + 1/2, the sum is 1, displayed here as 1/1. Opposite fractions such as 1/2 + (−1/2) give 0/1.
Why can’t you add the numerators and denominators separately?
Adding the denominators changes the size of the parts and gives the wrong value. For example, 1/2 + 1/2 = 1, whereas (1 + 1)/(2 + 2) = 2/4 = 1/2. Use a common denominator and add only the corresponding numerators.
Can this fraction calculator handle zero denominators or very large numbers?
Zero denominators are undefined and rejected. Very large integers are not guaranteed to be exact: this addition engine uses JavaScript numbers without safe-integer or intermediate-overflow guards. A displayed answer alone is not proof that a large-integer calculation is accurate.
§

Related calculators

§

SOURCES