Fraction Calculator

How fraction operations actually work

A fraction such as 3/4 represents three parts out of a whole divided into four equal pieces. Because the bottom number (the denominator) tells you how big each piece is, you cannot simply combine two fractions' numerators and denominators directly unless those pieces are the same size. Each of the four basic operations handles that difference in its own way, and understanding the logic behind each one makes it much easier to spot a wrong answer at a glance.

Adding and subtracting — find a common denominator first

To add or subtract two fractions a/b and c/d, you first need them expressed in pieces of the same size — a shared, or "common," denominator. The simplest reliable choice is the product of the two denominators (b × d), which always works even if it is not the smallest possible option. Cross-multiplying to reach that common denominator gives:

a/b + c/d = (a×d + c×b) / (b×d)  and  a/b − c/d = (a×d − c×b) / (b×d)

In words: scale each fraction up so both share denominator b×d, then add or subtract the new numerators and keep that shared denominator. The result is correct but often not in lowest terms, so the final step is always to simplify it.

Multiplying — straight across, no common denominator needed

Multiplication is the most direct of the four operations: multiply the numerators together to get the new numerator, and multiply the denominators together to get the new denominator — (a/b) × (c/d) = (a×c)/(b×d). There is no need to match denominators first, because multiplying fractions means taking "a part of a part," and the sizes of the original pieces do not need to agree for that to make sense.

Dividing — multiply by the reciprocal

Dividing by a fraction is defined as multiplying by its reciprocal (the fraction flipped upside down): (a/b) ÷ (c/d) = (a/b) × (d/c) = (a×d)/(b×c). This works because dividing by c/d asks "how many groups of size c/d fit into a/b," and multiplying by d/c — its multiplicative inverse — answers exactly that question. The one rule to watch for is that the numerator of the second fraction (c) can never be zero, since that would mean dividing by zero once you flip it into the denominator of the reciprocal.

Simplifying fractions and converting forms

Reducing to lowest terms with the GCD

A fraction is in "lowest terms" when its numerator and denominator share no common factor other than 1. To get there, find the greatest common divisor (GCD) of the numerator and denominator — most reliably with the Euclidean algorithm, which repeatedly replaces the larger number with the remainder of dividing it by the smaller until the remainder reaches zero — then divide both numerator and denominator by that GCD. For example, 18/24 has numerator-denominator GCD 6, so dividing both by 6 gives 3/4, and since gcd(3, 4) = 1, that is the simplest possible form.

Improper fractions and mixed numbers

An "improper" fraction has a numerator larger than its denominator, such as 19/12. To write it as a mixed number, divide the numerator by the denominator: the whole-number quotient becomes the leading whole number, and the remainder sits over the original denominator as the fractional part — 19 ÷ 12 = 1 remainder 7, so 19/12 becomes 1 7/12. To reverse the conversion, multiply the whole number by the denominator, add the numerator, and place that total back over the same denominator: 1 7/12 → (1 × 12) + 7 = 19, giving 19/12 again. This calculator performs that same conversion on whatever you type in, which is why each input field accepts an optional whole-number part.

Fractions as decimals

Any fraction can be converted to a decimal by dividing its numerator by its denominator. Some divisions terminate cleanly — 3/4 = 0.75 — while others repeat forever, such as 1/3 = 0.333… or 5/6 = 0.8333…. Because a repeating decimal cannot be written out in full, this calculator rounds the decimal equivalent to six places and notes when the value has been rounded, so you always know the fraction above it is the exact answer and the decimal beneath it is a convenient approximation.

Worked example — 3/4 + 5/6

Suppose you choose Add and enter 3/4 and 5/6. The calculator first finds a common denominator by multiplying the two denominators together: 4 × 6 = 24. Scaling each fraction up to twenty-fourths gives 3/4 = 18/24 and 5/6 = 20/24. Adding the new numerators while keeping the shared denominator produces 18/24 + 20/24 = 38/24. That sum is not yet in lowest terms, so the calculator finds gcd(38, 24) = 2 and divides both parts by it, leaving 19/12 as the simplified result. Because 19 is larger than 12, it converts to the mixed number 1 7/12 (19 ÷ 12 = 1 remainder 7), and dividing 19 by 12 gives a decimal equivalent of approximately 1.583333 (rounded, since the digits repeat). Enter the same two fractions and the Add operation into the calculator above to see each of these steps laid out for you in order.

How to read your results

• Simplified result — the exact answer to your operation, reduced to lowest terms with a positive denominator. This is the precise, unrounded value of the calculation.
• Mixed-number form — the same exact value rewritten as a whole number plus a proper fraction whenever the simplified result is improper (numerator larger than denominator); otherwise it mirrors the simplified fraction.
• Decimal equivalent — the result expressed as a decimal, rounded to six places for readability. A note appears whenever the true decimal repeats or runs longer than six digits, so you know it has been rounded.
• Step-by-step breakdown — the common-denominator (or product) step, the unsimplified intermediate fraction, the greatest common divisor used to reduce it, and the final simplified form, in the order the calculation actually happens.

Exact math, not an estimate

Unlike calculators that model real-world quantities — money, time, body measurements — this one performs pure arithmetic on ratios of whole numbers. Every step (cross- multiplying to find a common denominator, multiplying straight across, flipping a fraction to divide, and reducing with the greatest common divisor) is exact, so the simplified fraction and mixed-number form shown above are precise values you can rely on without qualification. The single place rounding appears is the optional decimal equivalent, which exists purely for convenience and is clearly labeled whenever the true value repeats or runs longer than the six digits shown.