Fraction Calculator
Add, subtract, multiply, or divide any two fractions. Get results as simplified fractions, mixed numbers, and decimals.
How Fraction Arithmetic Works
Fractions follow their own set of arithmetic rules that are different from whole number math. The reason comes down to what a fraction actually represents: a division that hasn't been carried out yet. When you write 3/4, you're saying "three divided by four" without actually doing the division. That delayed computation is what makes fractions powerful for exact arithmetic but also what makes them tricky to combine.
Adding fractions is the operation that confuses most people, and the confusion usually starts with denominators. You can't add 1/3 and 1/4 by adding the tops and bottoms separately — that would give you 2/7, which is wrong. The denominators need to match first. For 1/3 + 1/4, the least common denominator is 12: you rewrite the fractions as 4/12 + 3/12, and the answer is 7/12. Subtraction works identically, just replacing the plus sign.
Multiplication is actually the easiest operation with fractions. Multiply the numerators together, multiply the denominators together, and simplify. That's it. 2/3 × 5/7 = 10/21. No common denominators needed.
Division is multiplication in disguise. To divide by a fraction, flip it and multiply. So 3/4 ÷ 2/5 becomes 3/4 × 5/2 = 15/8, which simplifies to 1 7/8 as a mixed number.
Finding the Greatest Common Divisor
Simplifying fractions means dividing both the numerator and denominator by the largest number that goes evenly into both. That number is the greatest common divisor, or GCD. For small numbers, you can spot it by inspection: the GCD of 12 and 8 is 4, so 8/12 simplifies to 2/3.
For larger numbers, the Euclidean algorithm handles it efficiently. It works by repeatedly dividing and taking remainders. To find the GCD of 48 and 18: divide 48 by 18 to get a remainder of 12. Then divide 18 by 12 to get a remainder of 6. Then divide 12 by 6 to get a remainder of 0. When the remainder hits zero, the last non-zero remainder is the GCD — in this case, 6. So 18/48 simplifies to 3/8.
This algorithm is over 2,300 years old, described by Euclid around 300 BCE, and it's still used in modern computer programs because nothing faster has been found for this particular problem. The fraction calculator on this page uses it behind the scenes every time it simplifies a result.
Mixed Numbers and Improper Fractions
An improper fraction has a numerator larger than its denominator — like 17/5. A mixed number expresses the same value as a whole number plus a proper fraction: 3 2/5. Both represent exactly the same quantity, just written differently.
Converting from improper to mixed is straightforward division. Divide 17 by 5: you get 3 with a remainder of 2. So 17/5 = 3 remainder 2, or 3 2/5. Going the other direction, multiply the whole number by the denominator and add the numerator: 3 × 5 + 2 = 17, over the original denominator of 5.
School math tends to prefer mixed numbers because they're easier to visualize. Three and two-fifths of a pizza makes more intuitive sense than seventeen-fifths of a pizza. But in algebra, calculus, and most higher math, improper fractions are actually preferred because they're easier to work with in equations. There's no mathematical reason to convert either way — it's a matter of context and readability.
When Fractions Beat Decimals
Decimals are convenient, but they have a fundamental limitation that fractions don't share: some numbers can't be expressed exactly as terminating decimals. The fraction 1/3 becomes 0.333... repeating forever. In a decimal system, you're forced to round, and rounding introduces error. If you need to add 1/3 three times, fractions give you exactly 1. Decimals give you 0.999... which is technically equal to 1 in the limit, but can cause problems in programming and financial calculations where precision matters.
Cooking is a great everyday example. Recipes that call for 2/3 cup of flour or 3/4 teaspoon of salt are using fractions because measuring cups and spoons are designed around fractional divisions. Converting those to decimals — 0.667 cups or 0.75 teaspoons — doesn't help anyone holding a measuring cup marked in fractions.
In engineering and machining, fractions are the standard for bolt sizes, drill bits, and pipe fittings in the US. A 5/16-inch bolt is a 5/16-inch bolt. Converting it to 0.3125 inches doesn't add clarity and actually makes it harder to identify the right socket from a standard set. Fractions persist in practical fields because they often match the physical tools and measurements people use.
Fraction Arithmetic Rules
a/b + c/d = (ad + bc) / bd
Adding and subtracting fractions requires a common denominator. The quickest way to get one is to multiply the two denominators together, then adjust each numerator accordingly. For a/b + c/d, you multiply the first numerator by d and the second by b, giving (ad + bc) over the common denominator bd. Multiplication is simpler: just multiply straight across, numerator times numerator and denominator times denominator. Division flips the second fraction and then multiplies. After any operation, the result should be simplified by dividing both the numerator and denominator by their greatest common divisor.
Where:
- a/b = First fraction with numerator a and denominator b
- c/d = Second fraction with numerator c and denominator d
- GCD = Greatest common divisor, used to simplify the result
Example Calculations
Adding Fractions with Different Denominators
Calculate 3/4 + 2/3 step by step.
- Find a common denominator: LCD of 4 and 3 is 12
- Convert 3/4 to twelfths: 3/4 × 3/3 = 9/12
- Convert 2/3 to twelfths: 2/3 × 4/4 = 8/12
- Add the numerators: 9/12 + 8/12 = 17/12
- Convert to mixed number: 17 ÷ 12 = 1 remainder 5 → 1 5/12
- As a decimal: 17 ÷ 12 = 1.41667
The result 17/12 is already in its simplest form because 17 is a prime number and doesn't share any factors with 12.
Dividing Fractions
Calculate 5/6 ÷ 2/9 using the flip-and-multiply method.
- Write the division: 5/6 ÷ 2/9
- Flip the second fraction: 2/9 becomes 9/2
- Multiply: 5/6 × 9/2 = 45/12
- Find GCD of 45 and 12: GCD = 3
- Simplify: 45/12 ÷ 3/3 = 15/4
- As a mixed number: 3 3/4
- As a decimal: 3.75
Dividing by a fraction smaller than 1 always gives a result larger than the first fraction. Since 2/9 is less than 1, the quotient 15/4 (or 3.75) is larger than the original 5/6 (about 0.833).
Frequently Asked Questions
Division by zero is undefined in mathematics. A fraction with zero in the denominator would mean dividing something into zero groups, which doesn't produce a meaningful result. Calculators and programming languages treat it as an error. If you're getting a zero denominator in a calculation, it usually means something went wrong earlier in the problem setup.
Write the decimal over 1 and multiply top and bottom by 10 for each decimal place. For 0.75, that gives 75/100. Then simplify: GCD of 75 and 100 is 25, so 75/100 = 3/4. For repeating decimals like 0.333..., the fraction is 1/3. The pattern 0.142857142857... repeating is exactly 1/7.
A fraction represents part of a whole — 3/4 means three out of four equal pieces. A ratio compares two quantities: 3:4 means for every 3 of one thing, there are 4 of another. Mathematically they behave the same way in calculations, but the context and interpretation differ. A recipe calling for a 2:1 ratio of flour to sugar isn't saying you need two-thirds of a cup.
Yes. A negative fraction has exactly one negative sign, which can be placed on the numerator, the denominator, or in front of the fraction — all three are equivalent. -3/4, 3/-4, and -(3/4) all represent the same value: negative three-quarters. By convention, the negative sign is usually placed on the numerator or in front of the entire fraction.
Convert the whole number to a fraction by putting it over 1, then add normally. For 5 + 3/4: rewrite as 5/1 + 3/4. The common denominator is 4, so 5/1 becomes 20/4. Now add: 20/4 + 3/4 = 23/4, which equals 5 3/4. You'll notice the result is just the whole number with the fraction tacked on, which makes sense.