Fraction Calculator

How to use this fraction calculator

This fraction calculator handles all four arithmetic operations — addition, subtraction, multiplication, and division — for any combination of proper fractions, improper fractions, mixed numbers, and whole numbers. Enter the whole number part (optional) plus numerator and denominator for each fraction, select your operation, and the result appears instantly with the full step-by-step solution.

The Simplify tab takes a single fraction and returns it in its lowest terms, showing the GCD used to reduce it. Both modes handle negative inputs and results correctly.

The calculator works with any fraction where the denominator is not zero. Results are always shown in simplified form, as a decimal, and as a mixed number when the result is an improper fraction.

How to add fractions with different denominators

Adding fractions with the same denominator is simple — just add the numerators and keep the denominator. Adding fractions with different denominators requires a shared base first. The method:

1. Find the LCD. The least common denominator is the least common multiple (LCM) of the two denominators. For 1/4 + 1/6, LCM(4, 6) = 12.
2. Convert both fractions. Multiply each fraction's numerator and denominator by the factor needed to reach the LCD. 1/4 → 3/12 (multiply by 3/3). 1/6 → 2/12 (multiply by 2/2).
3. Add numerators. 3/12 + 2/12 = 5/12.
4. Simplify. Check whether GCD(5, 12) = 1. It does, so 5/12 is already in lowest terms.

Another example: 2/3 + 3/4. LCM(3, 4) = 12. → 8/12 + 9/12 = 17/12. That's an improper fraction — convert to mixed: 17 ÷ 12 = 1 remainder 5 → 1 5/12.

The shortcut "cross-multiplication" method — multiply the first numerator by the second denominator, add the product of the second numerator and first denominator, over the product of the two denominators — always works but often produces larger numbers that need more simplification. The LCD method stays smaller.

How to subtract fractions

Subtraction follows exactly the same process as addition: find the LCD, convert both fractions, then subtract the numerators instead of adding them. The critical difference is handling negative results correctly.

Example: 3/4 − 5/6. LCM(4, 6) = 12. → 9/12 − 10/12 = −1/12. The result is negative because the second fraction was larger. This is correct and expected.

For mixed numbers: 2 1/3 − 1 3/4. Convert to improper fractions first: 2 1/3 = 7/3, 1 3/4 = 7/4. LCM(3, 4) = 12. → 28/12 − 21/12 = 7/12.

Watch for this common error: subtracting mixed numbers by subtracting whole parts and fraction parts separately only works when the fraction part of the first number is larger than the fraction part of the second. When it isn't (like 2 1/4 − 1 3/4), you need to borrow from the whole number. Converting to improper fractions first sidesteps this entirely.

How to multiply fractions

Multiplication is the most straightforward fraction operation — no common denominator is needed. Multiply numerator × numerator and denominator × denominator, then simplify.

Example: 3/5 × 4/7 = 12/35. GCD(12, 35) = 1, so this is already simplified.

Another: 2/3 × 3/4 = 6/12. GCD(6, 12) = 6. → 1/2.

Cross-cancellation shortcut: Before multiplying, you can cancel common factors diagonally. In 2/3 × 3/4, the 3 in the numerator of the second fraction and the 3 in the denominator of the first cancel to give 2/1 × 1/4 = 2/4 = 1/2. This keeps the numbers smaller.

For mixed numbers: always convert to improper fractions first. 1 1/2 × 2 2/3 = 3/2 × 8/3 = 24/6 = 4. The whole number 4 is the exact answer.

How to divide fractions

Dividing by a fraction is equivalent to multiplying by its reciprocal. The reciprocal of a/b is b/a. This gives the memorable rule: Keep, Change, Flip — keep the first fraction, change division to multiplication, flip the second fraction.

Example: 3/4 ÷ 1/2 = 3/4 × 2/1 = 6/4 = 3/2 = 1 1/2.

Example with mixed numbers: 2 1/2 ÷ 1 1/4 = 5/2 ÷ 5/4 = 5/2 × 4/5 = 20/10 = 2. So two and a half divided by one and a quarter equals exactly 2.

Why does this work? Division asks "how many times does the divisor fit into the dividend?" Dividing 3/4 by 1/2 asks how many halves fit into three-quarters. One half fits once completely (1/2 of 3/4 used = 3/8, which is the 3/4 remaining minus 3/8) — wait, more intuitively: 3/4 ÷ 1/2 = 1.5, meaning 1.5 halves fit into 3/4. The reciprocal relationship naturally represents this division as multiplication.

How to simplify fractions (reduce to lowest terms)

A fraction is in its simplest form — or lowest terms — when the numerator and denominator share no common factor other than 1. The method uses the greatest common divisor (GCD).

1. Find the GCD of the numerator and denominator using the Euclidean algorithm: repeatedly replace the larger number with the remainder when divided by the smaller, until the remainder is 0.
2. Divide both numerator and denominator by the GCD.

Example: Simplify 36/48. GCD(36, 48): 48 = 36×1 + 12 → 36 = 12×3 + 0 → GCD = 12. So 36/48 ÷ 12/12 = 3/4.

If you don't know the GCD directly, you can simplify in steps using any common factor you can see: 36/48 → divide by 2 → 18/24 → divide by 2 → 9/12 → divide by 3 → 3/4. Same result, just more steps. The GCD method does it in one step.

You can check: GCD(3, 4) = 1. Since no factor other than 1 divides both, 3/4 is in lowest terms.

Mixed numbers and improper fractions

A proper fraction has a numerator smaller than its denominator (e.g. 3/4). An improper fraction has a numerator equal to or larger than its denominator (e.g. 7/4 or 4/4). A mixed number combines a whole number with a proper fraction (e.g. 1 3/4).

Converting mixed number → improper fraction: Multiply the whole number by the denominator, add the numerator, keep the denominator. 2 3/5 = (2×5 + 3)/5 = 13/5.

Converting improper fraction → mixed number: Divide numerator by denominator. The quotient is the whole number, the remainder is the new numerator, denominator stays. 13/5: 13 ÷ 5 = 2 remainder 3 → 2 3/5.

This calculator accepts mixed number inputs in both modes — enter the whole number in the "whole" field and the fraction numerator/denominator in the fraction fields. If a result is improper, the calculator displays it as both an improper fraction and a mixed number for clarity.

Real-world uses for fraction calculations

Fractions appear constantly in contexts where division isn't clean or where precision beyond decimals matters:

Cooking and baking: Recipe scaling is pure fraction arithmetic. Doubling a recipe that calls for 2/3 cup of flour gives 4/3 = 1 1/3 cups. Halving a recipe calling for 3/4 teaspoon of baking powder gives 3/8 teaspoon. The fraction form preserves the exact measurement, avoiding decimal rounding that can throw off precise baking.

Woodworking and construction: Imperial measurements (inches) use fractions throughout. Adding 7/8" + 5/16": LCM(8, 16) = 16, so 14/16 + 5/16 = 19/16 = 1 3/16". Any carpenter working with fractional measurements needs this arithmetic constantly.

Probability: Fraction arithmetic is fundamental to probability calculations. If the probability of event A is 1/3 and event B is 1/4 (and they're independent), the probability of both is 1/3 × 1/4 = 1/12. The probability that at least one occurs is 1/3 + 1/4 − 1/12 = 4/12 + 3/12 − 1/12 = 6/12 = 1/2.

Finance: Interest rates, ratios, and percentages expressed as fractions combine through multiplication and addition. A portfolio allocation of 1/3 equities and 1/4 bonds with the remainder in cash: 1/3 + 1/4 = 7/12 invested, leaving 5/12 in cash. Exact fractional arithmetic avoids compounding decimal rounding errors in financial calculations.

Frequently asked questions