Fraction calculator
Free fraction calculator to add, subtract, multiply and divide fractions or mixed numbers. Get a reduced exact answer, decimal result and worked examples.
By Calcr · Updated
Fraction calculator: Calculate with exact fractions
Enter two fractions or mixed numbers and choose add, subtract, multiply or divide. A mixed number such as 1 1/2 means 1 + 1/2. The result includes a reduced exact fraction and a decimal approximation.
Formulas
For fractions a/b and c/d, addition is (ad + bc)/(bd), subtraction is (ad - bc)/(bd), multiplication is ac/(bd), and division is ad/(bc). Reduce the answer by the greatest common divisor of the numerator and denominator. A negative mixed number such as -1 1/2 means -(1 + 1/2).
Worked examples
- Unlike denominators: 2/3 + 5/12 = 8/12 + 5/12 = 13/12, or 1 1/12.
- Division: 3/4 divided by 9/10 = 3/4 * 10/9 = 30/36 = 5/6.
- Mixed number: 1 1/2 - 2/3 = 3/2 - 2/3 = 9/6 - 4/6 = 5/6.
Limits
Denominators cannot be zero, and division by a zero-valued fraction is undefined. Integer arithmetic preserves exact rational results within the tool's input limits. Each input is limited to 100 characters. Integers, finite decimals and properly formatted mixed numbers are accepted. A decimal such as 0.833333333333 is only an approximation to 5/6.
When to use this fraction calculator
Use this free fraction calculator when a quantity is naturally an exact part of a whole: recipe portions, lengths marked in fractional inches, or arithmetic exercises with rational numbers. Keeping 1/3 as a fraction avoids replacing it with a rounded decimal before later operations. Enter the numerator and denominator as written, choose the operation, and review the reduced fraction before using its decimal approximation.
Fractions still require matching units. Adding 1/2 metre to 1/4 metre gives 3/4 metre. Adding 1/2 metre to 1/4 centimetre without converting would mix different-sized wholes. A fraction names a count of equal parts of a stated unit; the denominator alone does not identify that unit. For a mixed number, separate the whole part from the fraction clearly, and do not type a decimal point where a space belongs.
Use another calculator if your input is an irrational quantity such as the square root of two, a trigonometric expression or a symbolic variable. A finite decimal is rational and can be converted exactly, but a few typed digits from an infinite decimal do not recover the original exact value. Entering 0.333 means exactly 333/1000, not exactly 1/3. The calculator cannot infer the digits that you omitted.
Example: increasing a recipe without decimal rounding
A recipe needs 3/4 cup of an ingredient for one batch. You want 1 1/2 batches. Choose multiplication and enter 3/4 and 1 1/2. Convert the mixed number to 3/2, then multiply: 3/4 * 3/2 = 9/8. The result is 1 1/8 cups, or 1.125 cups as a decimal. The exact fraction is convenient when a measuring cup has eighth-cup markings.
The increase is not 3/4 + 1/2, because the second number describes a number of batches rather than an extra amount in cups. Check the units in the sentence: cups per batch multiplied by batches gives cups. If you also scale other ingredients, apply the same 3/2 multiplier separately to each original quantity. Rounding 1.125 to 1.1 before measuring would change the intended ratio.
Example: measuring a difference with a negative result
Suppose a planned cut is 5/8 inch and the available piece is 3/4 inch. To calculate available length minus planned length, enter 3/4 - 5/8. Convert 3/4 to 6/8, then subtract to obtain 1/8 inch left over. Reversing the inputs gives -1/8, which records a signed difference rather than a physically negative piece of material.
The order matters for subtraction and division. When a result has an unexpected sign, check which quantity you intended to subtract from which. If a cutting process removes material along the cut, that width is a separate measurement; this arithmetic example does not automatically allow for it. You would need to include that additional amount in the original setup before deciding whether a real piece is long enough.
To check any reduced answer, multiply the reduced numerator and denominator by the same factor and confirm that the value remains unchanged. For example, 5/6 and 30/36 describe the same number. Reduction changes the representation, not the amount. A result denominator of one means the answer is an integer. A zero numerator means zero as long as the denominator is nonzero. Keep these simple checks in mind before trusting a complicated-looking input.
Check each denominator before submitting, including the fraction used as a divisor. The input 0/5 is a valid zero, but 5/0 is undefined, and dividing by 0/5 is also undefined. An error should lead you back to the operation and its inputs rather than to a guessed replacement value. For multiplication, estimate whether each factor is above or below one; multiplying a positive number by 1/2 should halve it. For addition, do not add denominators: 1/2 + 1/2 equals one, not 2/4. These small checks help detect a copied operation sign. If a result will be used in a measurement, also record the unit and whether the original quantities were exact or approximate. An exact fraction for approximate inputs does not make the measurements exact.
FAQ
Why does the denominator change when adding?
The fractions must describe equal-sized parts before their numerators can be combined.
Can an answer exceed one?
Yes. An improper fraction such as 13/12 is valid and can also be written as a mixed number.
Should I round before calculating?
Keep the fraction exact when possible. Rounding each input can change the final result.
Can a denominator be negative?
A negative denominator represents a negative fraction, provided it is not zero. The calculator normalizes the sign so the displayed denominator is positive. Thus 2/-3 and -2/3 have the same value, while -2/-3 is positive 2/3. For clarity, place the minus sign in front of the fraction when writing your own notes. Never remove both minus signs unless the numerator and denominator are both negative.
How is a negative mixed number interpreted?
The sign applies to the whole mixed quantity. Entering -1 1/2 means -(1 + 1/2), which equals -3/2. It does not mean -1 + 1/2, which would equal -1/2. If that distinction is easy to miss in a longer expression, convert the mixed number to an improper fraction first. Reviewing the decimal approximation, -1.5 in this example, is another way to catch a sign misunderstanding.