Fraction Calculator

How to Add and Subtract Fractions

Fractions need a common denominator before you can add or subtract their numerators. Find the least common denominator (LCD) of the two denominators, convert each fraction to an equivalent fraction with that denominator, then add or subtract the numerators while keeping the denominator the same.

$$\frac{a}{b} + \frac{c}{d} = \frac{a \times d + c \times b}{b \times d}$$

For example, 3/4 + 2/5: the LCD of 4 and 5 is 20, so 3/4 becomes 15/20 and 2/5 becomes 8/20, giving 15/20 + 8/20 = 23/20.

How to Multiply and Divide Fractions

Multiplying fractions is the simplest operation — multiply the numerators together and the denominators together, no common denominator needed:

$$\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}$$

Dividing fractions means multiplying by the reciprocal (flip the second fraction upside down, then multiply):

$$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$$

Always simplify the result afterward.

Worked Example: Multiplying Then Simplifying

$\frac{2}{3} \times \frac{3}{4} = \frac{2 \times 3}{3 \times 4} = \frac{6}{12}$, which simplifies to $\frac{1}{2}$ by dividing both numerator and denominator by their GCD of 6. Multiplying first and simplifying afterward always works, though it's sometimes faster to cancel common factors between numerators and denominators before multiplying — here, the 3 in the first fraction's denominator and the 3 in the second fraction's numerator cancel directly, giving $\frac{2}{4} = \frac{1}{2}$ immediately.

How to Simplify Fractions — the GCD Method

To simplify a fraction to its lowest terms, find the greatest common divisor (GCD) of the numerator and denominator, then divide both by it. The Euclidean algorithm finds the GCD efficiently: repeatedly replace the larger number with the remainder of dividing it by the smaller number, until the remainder is 0 — the last non-zero value is the GCD. It's named after the Greek mathematician Euclid, who described this method in Book VII of his "Elements" around 300 BCE, making it one of the oldest algorithms still in everyday use. For 48 and 18: $48 = 2 \times 18 + 12$, then $18 = 1 \times 12 + 6$, then $12 = 2 \times 6 + 0$ — the remainder hit zero, so the GCD is 6, and $\frac{48}{18}$ simplifies to $\frac{8}{3}$.

Mixed Numbers vs. Improper Fractions

A mixed number combines a whole number and a proper fraction, like 1 3/20. An improper fraction has a numerator larger than its denominator, like 23/20, expressing the same value without separating out the whole part. Improper fractions are easier to add, subtract, multiply, and divide; mixed numbers are often easier to read and understand intuitively — this calculator shows both.

A Brief History of Fraction Notation

Fractions appear in some of the earliest known mathematical writing — the Egyptian Rhind Papyrus, dating to around 1650 BCE, works almost entirely with "unit fractions" (fractions with a numerator of 1, like 1/3 or 1/7), representing other fractions as sums of these. Babylonian mathematicians used a base-60 (sexagesimal) system for fractions, a legacy that survives today in how we divide hours into 60 minutes and minutes into 60 seconds. The horizontal fraction bar familiar today traces back to Arabic mathematicians around the 12th century, building on earlier Hindu notation that stacked the numerator over the denominator without a dividing line; Italian mathematician Fibonacci helped popularize the bar notation in Europe after encountering it during his travels and study of Arabic mathematics, publishing it in his 1202 book "Liber Abaci."

Common Fraction Mistakes

Adding or subtracting numerators and denominators straight across without finding a common denominator first — turning 1/2 + 1/3 into the wrong 2/5 instead of the correct 5/6 — is probably the single most common fraction error. Forgetting to flip the second fraction when dividing (using the same multiplication rule instead of the reciprocal) is another frequent slip. Failing to simplify a final answer to lowest terms isn't technically wrong, but it's considered incomplete in most academic contexts and can make it harder to compare or combine with other fractions later.

Fraction Terms You Should Know

Numerator — the top number in a fraction, representing how many parts are being counted.

Denominator — the bottom number in a fraction, representing how many equal parts the whole is divided into.

Least Common Denominator (LCD) — the smallest number that both denominators divide into evenly, used to align fractions before adding or subtracting them.

Greatest Common Divisor (GCD) — the largest number that divides both the numerator and denominator evenly, used to reduce a fraction to its simplest form.

Reciprocal — a fraction flipped upside down (numerator and denominator swapped); multiplying by the reciprocal is equivalent to dividing.