Fraction Calculator
Enter two fractions or mixed numbers (e.g. 1 1/2 and 3/4) – the calculator works exactly, simplifies the answer and shows every step. It also reduces a single fraction and turns decimals, including repeating decimals, into fractions.
How it is calculated
How the fraction calculator works
A fraction a/b has a numerator a and a denominator b (b ≠ 0). The calculator only uses whole numbers of any length, so there are no rounding errors: 1/3 + 1/3 + 1/3 is exactly 1.
| Operation | Rule | Example |
|---|---|---|
| Add | a/b + c/d = (a·d + c·b)/(b·d), simpler with the least common denominator | 1/6 + 1/10 = 5/30 + 3/30 = 8/30 = 4/15 |
| Subtract | a/b − c/d = (a·d − c·b)/(b·d) | 1/3 − 1/2 = 2/6 − 3/6 = −1/6 |
| Multiply | a/b × c/d = (a·c)/(b·d) | 2/3 × 9/4 = 18/12 = 3/2 |
| Divide | a/b ÷ c/d = a/b × d/c (reciprocal) | 3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8 |
Example with a mixed number: 1 1/2 + 3/4
- Convert the mixed number: 1 1/2 = (1 × 2 + 1)/2 = 3/2.
- Least common denominator = LCM(2, 4) = 4.
- Rewrite: 3/2 = 6/4; 3/4 stays 3/4.
- Add the numerators: 6/4 + 3/4 = 9/4 (GCD(9, 4) = 1, already simplest form).
- Mixed number: 9 ÷ 4 = 2 remainder 1 → 2 1/4; as a decimal 2.25.
Simplifying with the Euclidean algorithm
A fraction is reduced by dividing numerator and denominator by their greatest common divisor (GCD). The calculator finds it with Euclid’s method: divide the larger number by the smaller, keep the remainder and repeat until the remainder is 0. For 12/18: 18 = 1 × 12 + 6, 12 = 2 × 6 + 0 → GCD = 6, so 12/18 = 2/3. The least common denominator is LCM(a, b) = a × b / GCD(a, b).
Decimal to fraction – including repeating decimals
Terminating decimal: count the decimal places, put the digits over the matching power of ten and simplify: 0.75 = 75/100 = 3/4.
Repeating decimal: numerator = all digits up to the end of the first repeating block minus the digits before the block; denominator = one 9 for each repeating digit followed by one 0 for each non-repeating decimal digit. Example 0.1(6) = 0.1666…: (16 − 1)/90 = 15/90 = 1/6. Likewise 0.(3) = 3/9 = 1/3 and 0.(9) = 9/9 = 1. Type the repeating digits in brackets or use the “Repeating digits” field.
Every answer is also shown as a decimal: fractions whose simplified denominator has no prime factors other than 2 and 5 terminate (9/4 = 2.25); all others repeat (1/7 = 0.142857142857…).
Frequently asked questions
How do I enter a mixed number?
Whole number, space, fraction: 1 1/2 for one and a half, -2 3/4 for negative two and three quarters. Whole numbers (5) and decimals (0.5) work too.
How do you add fractions with different denominators?
Rewrite both fractions with the least common denominator (the LCM of the denominators), then add the numerators: 1/6 + 1/10 → LCD 30 → 5/30 + 3/30 = 8/30 = 4/15.
How do you divide fractions?
Multiply by the reciprocal (“keep, change, flip”): 3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8 = 1 7/8. You cannot divide by zero.
How do I simplify a fraction to lowest terms?
Divide numerator and denominator by their greatest common divisor. 12/18: GCD = 6, so 2/3. The “Simplify a fraction” mode lists the steps of the Euclidean algorithm.
How do I convert 0.333… to a fraction?
The repeating block 3 has one digit, so the denominator is 9: 0.(3) = 3/9 = 1/3. In “Decimal → fraction” mode type 0.(3), or 0 plus 3 in the repeating digits field.
Why is this more accurate than a normal calculator?
It only works with whole-number numerators and denominators and simplifies exactly. A pocket calculator stores rounded decimals, so 1/3 becomes 0.3333333333 and three thirds can come out as 0.9999999999.
Sources and legal basis
- Euclid’s Elements, Book VII, Proposition 2 – greatest common measure (Euclidean algorithm), ed. D. E. Joyce, Clark University
- Common Core State Standards for Mathematics – Number & Operations—Fractions
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