Fraction Arithmetic Calculator

This fraction calculator takes two fractions and performs addition, subtraction, multiplication, and division, outputting the result in lowest terms. Enter the numerator and denominator of each fraction, then press calculate to see the result. It provides accurate results through mathematical operations and is useful for education.

Fraction Arithmetic Calculation Method

Addition

To add two fractions, first make their denominators equal by finding the least common multiple (LCM) of the denominators.
Adjust each fraction's numerator to the LCM, then add the adjusted numerators.
The resulting fraction consists of the sum of the adjusted numerators over the LCM.
Finally, divide by the greatest common divisor to reduce the result to its lowest terms.

Subtraction

Subtracting two fractions is similar to addition: first find the LCM to make the denominators equal.
Adjust each fraction's numerator to the LCM, then subtract the adjusted numerators.
The resulting fraction consists of the difference of the adjusted numerators over the LCM.
Finally, divide by the greatest common divisor to reduce the result to its lowest terms.

Multiplication

Multiplying two fractions is simple: multiply the first numerator by the second numerator to get the resulting numerator.
Multiply the first denominator by the second denominator to get the resulting denominator.
Finally, divide by the greatest common divisor to reduce the result to its lowest terms.

Division

Dividing two fractions is done by multiplying the first fraction by the reciprocal of the second (numerator and denominator swapped).
Multiply the first numerator by the second denominator to get the resulting numerator.
Multiply the first denominator by the second numerator to get the resulting denominator.
Finally, divide by the greatest common divisor to reduce the result to its lowest terms.

About This Calculator

Fraction arithmetic covers the four basic operations on rational numbers: adding, subtracting, multiplying and dividing. Adding and subtracting need a common denominator, while multiplying and dividing work directly on the numerators and denominators.

Enter the two fractions and the operation to get a fully simplified result, with the working shown step by step.

Each operation on fractions works differently: addition needs a common denominator, multiplication goes straight across, and division flips the second fraction. 2/3 + 3/4 becomes 8/12 + 9/12 = 17/12, or 1 and 5/12. Cancelling before multiplying keeps numbers small — 8/15 × 25/12 reduces to (8 × 5)/(3 × 12) = 10/9.

Key Formula:a/b + c/d = (ad + bc) / bd · (a/b) × (c/d) = ac/bd · (a/b) ÷ (c/d) = ad/bc

Frequently Asked Questions

How do I add fractions with different denominators?

Rewrite both with a common denominator — the least common multiple of the two — then add the numerators. For 1/3 + 1/4, use 12: 4/12 + 3/12 = 7/12.

Why do you flip the second fraction when dividing?

Dividing by a fraction is the same as multiplying by its reciprocal. 1/2 ÷ 3/4 becomes 1/2 × 4/3 = 4/6 = 2/3, because "how many 3/4 are in 1/2" equals two-thirds.

How do I simplify a fraction?

Divide the numerator and denominator by their greatest common divisor. For 18/24 the GCD is 6, so the simplified form is 3/4. Both forms name the same number.

Why does dividing by a fraction equal multiplying by its reciprocal?

Division asks how many times the divisor fits into the dividend. Dividing by 3/4 asks how many three-quarter pieces fit, and each whole contains 4/3 of them, so dividing by 3/4 is multiplying by 4/3.

Do I always need a common denominator?

Only for addition and subtraction. Multiplication and division work without one, and forcing a common denominator there merely enlarges the numbers and adds cancellation work.