What this tool does
Fractions break the moment the denominators differ: adding 1/4 to 2/3 by adding the top and bottom gives 3/7, which is wrong. This calculator finds a common denominator, writes out every step, and reduces the answer to lowest terms with the greatest common divisor. Mixed numbers, whole numbers and negative values all follow the same three rules, and every answer can be read as a fraction, a mixed number or a decimal. It is built for the common student cases where a wrong denominator quietly ruins an otherwise correct answer.
How it works
To add or subtract fractions the denominators must match, so the tool finds the least common denominator: the smallest number both denominators divide into. One quarter plus two thirds has an LCD of 12, so 1/4 becomes 3/12 and 2/3 becomes 8/12, and the sum is 11/12. A whole number joins the calculation as a fraction with denominator 1, which is how 3 plus 1/2 is handled.
Multiplication needs no common denominator: multiply the numerators and multiply the denominators, then reduce. Three quarters times two fifths gives 6/20, and because gcd(6, 20) is 2, the answer simplifies to 3/10. Division is the same operation with the second fraction flipped, so three quarters divided by two fifths becomes 3/4 times 5/2, which is 15/8, or 1 7/8 as a mixed number.
Mixed numbers are converted to improper fractions before any operation, so one and two thirds becomes 5/3, and converted back afterwards so the answer reads naturally. Reduction runs on every result using the Euclidean algorithm, which is why 6/20 never stays unsimplified. Decimals become exact fractions through their place value, so 0.75 is treated as 3/4 rather than an approximation, and each intermediate step is written out for checking.
Worked example
Adding 1/4 and 2/3, then multiplying and dividing 3/4 by 2/5.
- LCD of 4 and 3 = 12, so 1/4 = 3/12 and 2/3 = 8/12
- 3/12 + 8/12 = 11/12 (gcd(11, 12) = 1, so lowest terms)
- 3/4 × 2/5 = 6/20, and gcd(6, 20) = 2, so 6/20 = 3/10
- 3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8 = 1 7/8
1/4 + 2/3 = 11/12, 3/4 × 2/5 = 3/10, and 3/4 ÷ 2/5 = 1 7/8.
Accuracy and limitations
- Fractions with numerators or denominators beyond JavaScript's safe integer range (9,007,199,254,740,991) lose exactness.
- Repeating decimals are converted to fractions by place value, so an input that never terminates is approximated to the precision you type.
- The tool calculates arithmetic on simple and mixed fractions. It does not simplify algebraic expressions containing variables.
Frequently asked questions
- How do you add fractions with different denominators?
- Find the least common denominator, convert both fractions to it, then add only the numerators. One quarter plus two thirds becomes three twelfths plus eight twelfths, which is eleven twelfths. Never add the denominators together, because that is the most common fraction mistake.
- How do I simplify a fraction?
- Divide the numerator and the denominator by their greatest common divisor, then repeat until nothing divides both. Six twentieths has a GCD of 2, so dividing each by 2 gives three tenths. The Euclidean algorithm finds that GCD quickly even for large numbers.
- What does a mixed number calculator do?
- It converts numbers like one and two thirds into improper fractions, one and two thirds becoming five thirds, before calculating, then returns a mixed number when the result is greater than one. This keeps answers readable while the arithmetic stays exact. Whole numbers are treated the same way, as fractions with a denominator of 1.
- How do you convert a fraction to a decimal?
- Divide the numerator by the denominator. Three quarters is 3 divided by 4, which is exactly 0.75, while one third is approximately 0.333 because its digits repeat forever. The calculator shows the decimal next to the exact fraction so you can see where rounding begins.
- What is the difference between adding and multiplying fractions?
- Adding requires a common denominator first, while multiplying does not. Multiplication goes straight across, numerators times numerators and denominators times denominators, so three quarters times two fifths is six twentieths, which simplifies to three tenths.
- How do you divide fractions?
- Flip the second fraction and multiply instead. Three quarters divided by two fifths becomes three quarters times five halves, which is fifteen eighths, or one and seven eighths as a mixed number. Keeping the first fraction, changing the sign and flipping the second is the usual way to remember it.