Math
Fraction calculator
Every step shown: the common denominator, the unsimplified answer, the simplification and the decimal.
Why addition needs a common denominator
A fraction's denominator names the size of the pieces. Quarters and sixths are different sizes, so counting three quarters and five sixths together is like adding three apples to five oranges — the count is meaningless until both are expressed in the same unit.
Twelfths work for both, since 12 is divisible by 4 and by 6. Three quarters becomes nine twelfths and five sixths becomes ten twelfths, and now the pieces match: nineteen twelfths, or 1 7/12.
Multiplying the two denominators always gives a workable common denominator, but not always the lowest one. Dividing that product by their greatest common divisor gives the least common denominator, which keeps the numbers smaller.
Multiplying and dividing are easier
a⁄b ÷ c⁄d = a⁄b × d⁄c
Neither needs a common denominator. Multiplication goes straight across the top and bottom. Division flips the second fraction and multiplies, because dividing by a number is the same as multiplying by its reciprocal.
Students often find it counterintuitive that multiplying by a proper fraction makes a number smaller, while dividing by one makes it larger. It helps to read "÷ 1/2" as "how many halves fit in this", which makes 3 ÷ 1/2 = 6 obvious.
Simplifying
Divide the numerator and denominator by their greatest common divisor. For 18/24, the GCD is 6, giving 3/4. Doing this in stages works equally well — halving twice then dividing by three reaches the same place.
A fraction is in lowest terms when the numerator and denominator share no common factor other than 1. Checking this is quick for small numbers; for larger ones, Euclid's algorithm — repeatedly replacing the pair with the smaller number and the remainder — finds the GCD in a handful of steps.
Improper fractions and mixed numbers
An improper fraction has a numerator at least as large as its denominator, such as 19/12. A mixed number splits the same value into a whole part and a proper fraction: 1 7/12.
Mixed numbers read more naturally in measurements and recipes. Improper fractions are easier to compute with, which is why the calculator converts mixed input to improper form before doing anything and converts back only at the end. Doing arithmetic directly on mixed numbers is possible but error-prone, particularly with subtraction where borrowing from the whole part is needed.
Common questions
How do I convert a decimal to a fraction?
For a terminating decimal, write the digits over the appropriate power of ten and simplify: 0.375 is 375/1000, which reduces to 3/8. Repeating decimals need an algebraic trick — for 0.333…, let x equal the decimal, multiply by 10, subtract, and solve to get 1/3.
Why can a denominator never be zero?
Because a fraction means division, and division by zero has no defined answer. If 5/0 equalled some number n, then n × 0 would have to equal 5 — but anything times zero is zero. No such n exists.
How do I compare two fractions?
Cross-multiply: for a/b and c/d with positive denominators, compare a×d against c×b. For 3/4 and 5/7, that is 21 against 20, so 3/4 is larger. Converting both to decimals also works and is often quicker mentally.
What is the difference between a proper and improper fraction?
A proper fraction has a numerator smaller than its denominator and represents less than one. An improper fraction has a numerator at least as large as the denominator and represents one or more. Neither is wrong — improper fractions are usually preferred in algebra.