Stop struggling with fraction arithmetic — learn the rules, see worked examples, and solve any fraction problem in seconds with our free calculator.
Fractions are one of the first abstract math concepts people encounter, and they remain relevant well into adulthood. Whether you are adjusting a recipe that serves four to serve seven, calculating material cuts for a woodworking project, or helping your child with homework, fraction arithmetic shows up constantly. The rules are straightforward once you understand them — but the multi-step procedures (common denominators, cross-multiplying, simplifying) make errors easy to introduce.
This guide covers every fraction operation in depth, with clear formulas, worked examples, and a free fraction calculator that shows each step so you can both get the answer and understand the process.
🔢 Solve any fraction problem step by step
Open Fraction Calculator →Before diving into operations, let's establish the vocabulary:
When both fractions share the same denominator, simply add the numerators and keep the denominator.
2/7 + 3/7 = (2+3)/7 = 5/7
When denominators differ, find the least common denominator (LCD), convert each fraction, then add.
1/3 + 1/4 → LCD of 3 and 4 is 12
1/3 = 4/12, 1/4 = 3/12
4/12 + 3/12 = 7/12
This is where most mistakes happen. People either forget to find the LCD and just add straight across (wrong: 1/3 + 1/4 ≠ 2/7), or they convert one fraction but not the other. Our calculator handles the LCD automatically and shows every conversion step.
Subtraction follows the exact same rules as addition — same denominator means subtract numerators directly, different denominators require the LCD first.
5/6 − 1/4 → LCD is 12
5/6 = 10/12, 1/4 = 3/12
10/12 − 3/12 = 7/12
One pitfall: if the result is an improper fraction, you may want to convert it to a mixed number. For example, 7/4 = 1 3/4. Our calculator provides both forms.
Multiplication is actually the easiest fraction operation — no common denominators needed. Just multiply across: numerator times numerator, denominator times denominator.
2/5 × 3/7 = (2×3)/(5×7) = 6/35
Before multiplying, check if you can cross-simplify to keep numbers small. For example, 4/9 × 3/8: the 4 and 8 share a factor of 4, and the 9 and 3 share a factor of 3. Simplify first: 1/3 × 1/2 = 1/6. Much cleaner than multiplying 12/72 and then simplifying.
To divide fractions, flip the second fraction (find its reciprocal) and multiply. The old mnemonic "keep, change, flip" still works perfectly.
3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8 = 1 7/8
Why does this work? Dividing by a number is the same as multiplying by its inverse. Just as 10 ÷ 2 = 10 × 0.5, any fraction a/b ÷ c/d equals a/b × d/c. The math checks out every time.
Mixed numbers are common in real life (1 1/2 cups of flour, 3 3/4 inches), but fraction operations require improper fractions. The conversion is straightforward:
2 3/4 = (2 × 4 + 3) / 4 = 11/4
11/4 = 11 ÷ 4 = 2 remainder 3 → 2 3/4
Our calculator accepts mixed numbers directly and handles the conversion for you, then shows the result in both improper and mixed form.
Every fraction result should be reduced to its simplest form. To simplify, divide both the numerator and denominator by their greatest common divisor (GCD).
18/24 → GCD of 18 and 24 is 6
18 ÷ 6 = 3, 24 ÷ 6 = 4 → 3/4
For large numbers, finding the GCD by hand can be tedious. The Euclidean algorithm is the fastest method: repeatedly divide the larger number by the smaller, then divide the divisor by the remainder, until the remainder is 0. The last divisor is the GCD. Our calculator does this instantly.
Recipes are the classic fraction scenario. "This recipe serves 6, but I need to serve 4" — multiply every ingredient by 4/6 = 2/3. If the recipe calls for 1 1/2 cups of sugar, you need 1 1/2 × 2/3 = 3/2 × 2/3 = 1 cup. Our calculator handles mixed number multiplication seamlessly.
Cutting lumber, calculating material quantities, and reading tape measures all involve fractions. Need to cut a 2 3/4-inch piece from a 7 1/2-inch board? Subtract: 7 1/2 − 2 3/4 = 15/2 − 11/4 = 30/4 − 11/4 = 19/4 = 4 3/4 inches remaining.
Students encounter fractions from elementary school through algebra. A fraction calculator that shows steps is invaluable for learning — students can verify their work and understand where they went wrong, rather than just getting a final answer.
Find the least common denominator (LCD) of both fractions, convert each fraction to an equivalent fraction with the LCD, then add the numerators. For example, 1/3 + 1/4: LCD is 12, so 4/12 + 3/12 = 7/12.
Divide both the numerator and denominator by their greatest common divisor (GCD). For example, 12/18: GCD is 6, so 12÷6 / 18÷6 = 2/3. Our calculator simplifies automatically.
A mixed number combines a whole number and a fraction, like 2 3/4. To convert: multiply the whole number by the denominator, add the numerator, and place over the original denominator. So 2 3/4 = (2×4+3)/4 = 11/4.
Division of fractions is equivalent to multiplying by the reciprocal. Mathematically, a/b ÷ c/d = a/b × d/c. This works because dividing by a number is the same as multiplying by its inverse. For example, 3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8.
A proper fraction has a numerator smaller than its denominator (e.g., 3/5). An improper fraction has a numerator equal to or larger than its denominator (e.g., 7/4). Improper fractions can be converted to mixed numbers: 7/4 = 1 3/4.