Add, subtract, multiply or divide two fractions, simplified automatically.
Fractions are combined using standard rules (common denominator for +/-, cross-multiplication for x/÷), then simplified by dividing by the greatest common divisor (GCD).
Adding or subtracting fractions requires converting them to a common denominator first — you can't directly add 1/3 and 1/4 because they're measuring parts of differently-sized wholes (thirds and quarters aren't the same size piece), so both fractions first need to be rewritten using the same denominator (in this case 12, since it's the least common multiple of 3 and 4): 1/3 becomes 4/12 and 1/4 becomes 3/12, and now that they're measured in the same-sized units, 4/12 + 3/12 = 7/12 is straightforward. This step trips up more people than any other part of fraction arithmetic, precisely because it's the one operation where fractions don't behave like whole numbers.
Multiplication and division, by contrast, never require a common denominator at all — multiplying fractions just multiplies numerators together and denominators together (1/3 × 1/4 = 1/12), and dividing by a fraction is done by multiplying by its reciprocal (flipping the second fraction upside down): 1/3 ÷ 1/4 becomes 1/3 × 4/1 = 4/3. This asymmetry — addition/subtraction needing a common denominator, multiplication/division not needing one — is exactly why fraction arithmetic can feel inconsistent to learn: the underlying reason is that addition and subtraction require comparing like-sized units, while multiplication and division are fundamentally scaling operations that work correctly regardless of what denominator each fraction started with.
Addition and subtraction require comparing quantities measured in the same-sized units — you can't directly combine thirds and quarters without first converting both to a shared unit (like twelfths). Multiplication and division are scaling operations that work correctly on any denominators without needing them to match first.
Use the least common multiple (LCM) of the two denominators — the smallest number both denominators divide into evenly. For 1/3 and 1/4, the LCM of 3 and 4 is 12, so both fractions get converted to twelfths (4/12 and 3/12) before adding or subtracting.
Multiply the first fraction by the reciprocal (flipped version) of the second fraction. For example, 1/3 ÷ 1/4 becomes 1/3 × 4/1 = 4/3. This 'flip and multiply' rule works because dividing by a number is mathematically the same as multiplying by its reciprocal.
Divide both the numerator and denominator by their greatest common divisor (GCD). For example, 8/12 has a GCD of 4, so dividing both by 4 gives 2/3 — the simplest equivalent form of the same fraction.