How do you add two fractions with different denominators?

Cross-multiply: a/b + c/d = ad + bc/bd. For 1/2 + 1/3, that's (1×3)+(2×1)/2×3 = 5/6 — no need to find the least common denominator first.

How do you subtract fractions?

The same cross-multiplication as addition, but subtract the cross products: a/b - c/d = ad - bc/bd. For 2/3 - 1/6, that's 12-3/18 = 9/18 = 1/2 after reducing.

How do you multiply two fractions?

Multiply straight across — numerator times numerator, denominator times denominator: a/b×c/d = ac/bd. No common denominator is needed for multiplication.

Why do you flip the second fraction when dividing?

Dividing by a number is the same as multiplying by its reciprocal. "Keep, Change, Flip": keep the first fraction, change ÷ to ×, and flip the second fraction upside down, so a/b÷c/d = a/b×d/c = ad/bc.

Can a denominator be zero?

No. A denominator of zero makes the fraction undefined, so this calculator reports an error whenever either input denominator is zero. When dividing, the second fraction's numerator also can't be zero, since flipping it would put a zero in the new denominator.

Why did I get a mixed number in the answer?

Whenever the reduced result is improper (its numerator, ignoring sign, is at least as large as the denominator) and the denominator isn't 1, the calculator also shows the equivalent mixed number. 9/4 becomes 2 1/4 because 9 ÷ 4 = 2 remainder 1.

Does this work with negative fractions?

Yes. Enter a minus sign on the numerator (or the denominator) of either fraction. One negative sign makes the fraction negative; two negative signs cancel out, exactly like signed whole numbers.

Fractions Calculator

Add, subtract, multiply, or divide two fractions with step-by-step cross-multiplication, automatic reduction, and mixed-number conversion.

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How to use this calculator

Enter a numerator and denominator for each of the two fractions, then choose an operation — addition, subtraction, multiplication, or division. The calculator applies the matching formula, shows the raw result before reducing, finds the greatest common factor (GCF), reduces the fraction, and converts to a mixed number when the result is improper.

Tip: Every input box accepts negative whole numbers too — type the minus sign directly in the numerator or denominator box, like -3 or -4.

The four formulas

This calculator uses cross-multiplication, so it never needs to find a least common denominator first — the raw result is always correct, just not always fully reduced.

Addition

$\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}$

Subtraction

$\frac{a}{b} - \frac{c}{d} = \frac{ad - bc}{bd}$

Multiplication — the simplest case, no common denominator needed:

$\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}$

Division

$\frac{a}{b} \div \frac{c}{d} = \frac{ad}{bc}$

Why dividing means multiplying by the reciprocal

Division by a fraction is defined as multiplication by its reciprocal (flip the numerator and denominator). The memory trick is Keep, Change, Flip:

Keep the first fraction as it is.
Change the ÷ sign to ×.
Flip the second fraction — swap its numerator and denominator.

$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc}$

Tip: Because dividing by

\frac{c}{d}

is the same as multiplying by

\frac{d}{c}

, the second fraction's numerator can never be zero going into a division — flipping a zero numerator would create a zero denominator.

After every operation: reduce, then convert to a mixed number

Find the GCF of the raw numerator and denominator, then divide both by it — this is the fully reduced fraction.
Convert to a mixed number if the reduced numerator's absolute value is at least the denominator (and the denominator isn't 1): divide with remainder, and the quotient becomes the whole part.

Worked examples

Example 1 — addition: $\frac{1}{2} + \frac{1}{3}$

$\frac{(1\times3)+(2\times1)}{2\times3} = \frac{5}{6}$ . GCF(5, 6) = 1, so the answer is $\frac{5}{6}$ — already in lowest terms.

Example 2 — addition needing reduction: $\frac{2}{4} + \frac{2}{6}$

$\frac{(2\times6)+(4\times2)}{4\times6} = \frac{20}{24}$ . GCF(20, 24) = 4, so $\frac{20\div4}{24\div4} = \frac{5}{6}$ . The answer is $\frac{5}{6}$ .

Example 3 — addition producing a mixed number: $\frac{7}{4} + \frac{1}{2}$

$\frac{(7\times2)+(4\times1)}{4\times2} = \frac{18}{8}$ . GCF(18, 8) = 2, so $\frac{9}{4}$ . Since 9 ≥ 4, convert: $9 \div 4 = 2$ remainder $1$ . The answer is $\frac{9}{4} = 2\frac{1}{4}$ .

Example 4 — division: $\frac{1}{2} \div \frac{1}{6}$

Keep, change, flip: $\frac{1}{2} \times \frac{6}{1} = \frac{6}{2}$ . GCF(6, 2) = 2, so $\frac{3}{1} = 3$ — a whole number.

Common results reference

Operation	Example	Answer
Add	1/2 + 1/3	5/6
Add	2/4 + 2/6	5/6
Subtract	2/3 − 1/6	1/2
Multiply	2/3 × 3/4	1/2
Divide	1/2 ÷ 1/6	3
Add (negative)	−1/2 + 1/3	−1/6

Precision note

Every computation uses arbitrary-precision integers (BigInt), not floating-point numbers — so the cross-multiplied result, the GCF, and the reduced fraction are always exact, no matter how large the numerators or denominators are.

Frequently asked questions

How do you add two fractions with different denominators?: Cross-multiply: $\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}$ . For $\frac{1}{2} + \frac{1}{3}$ , that's $\frac{(1\times3)+(2\times1)}{2\times3} = \frac{5}{6}$ — no need to find the least common denominator first.
How do you subtract fractions?: The same cross-multiplication as addition, but subtract the cross products: $\frac{a}{b} - \frac{c}{d} = \frac{ad - bc}{bd}$ . For $\frac{2}{3} - \frac{1}{6}$ , that's $\frac{12-3}{18} = \frac{9}{18} = \frac{1}{2}$ after reducing.
How do you multiply two fractions?: Multiply straight across — numerator times numerator, denominator times denominator: $\frac{a}{b}\times\frac{c}{d} = \frac{ac}{bd}$ . No common denominator is needed for multiplication.
Why do you flip the second fraction when dividing?: Dividing by a number is the same as multiplying by its reciprocal. "Keep, Change, Flip": keep the first fraction, change ÷ to ×, and flip the second fraction upside down, so $\frac{a}{b}\div\frac{c}{d} = \frac{a}{b}\times\frac{d}{c} = \frac{ad}{bc}$ .
Can a denominator be zero?: No. A denominator of zero makes the fraction undefined, so this calculator reports an error whenever either input denominator is zero. When dividing, the second fraction's numerator also can't be zero, since flipping it would put a zero in the new denominator.
Why did I get a mixed number in the answer?: Whenever the reduced result is improper (its numerator, ignoring sign, is at least as large as the denominator) and the denominator isn't 1, the calculator also shows the equivalent mixed number. $\frac{9}{4}$ becomes $2\frac{1}{4}$ because $9 \div 4 = 2$ remainder $1$ .
Does this work with negative fractions?: Yes. Enter a minus sign on the numerator (or the denominator) of either fraction. One negative sign makes the fraction negative; two negative signs cancel out, exactly like signed whole numbers.