Fraction Calculator
Add, subtract, multiply, or divide two fractions using exact integer arithmetic. Enter each fraction's numerator and denominator to get the simplified result, its decimal equivalent, and a mixed number.
How It's Calculated
Formula
\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}
\frac{a}{b} - \frac{c}{d} = \frac{ad - bc}{bd}
\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}
\frac{a}{b} \div \frac{c}{d} = \frac{ad}{bc}A fraction represents a part of a whole as two integers: a numerator (the part) over a denominator (the whole number of equal parts). This calculator takes two fractions, Fraction A and Fraction B, and applies one of four operations. Multiplying is direct — multiply the numerators together and the denominators together. Dividing a fraction by another is the same as multiplying by its reciprocal, which is why the divide formula flips fraction B's numerator and denominator before multiplying. Adding and subtracting are different: two fractions can only be combined directly when they share a denominator, so this calculator uses cross-multiplication to create a common denominator without needing to find the least common multiple — it multiplies fraction A's numerator by fraction B's denominator, fraction B's numerator by fraction A's denominator, and uses the product of both denominators (b×d) as the new shared denominator. Every result is then simplified: the greatest common divisor (GCD) of the numerator and denominator is found and both are divided by it, reducing the fraction to lowest terms. If the reduced denominator would be negative, the sign is moved to the numerator instead, so a result's denominator is always positive. All of this — cross-multiplication, multiplication, division, and simplification — is done with exact integer arithmetic, never floating-point approximation, so the fraction result itself is always exact. The one exception is the decimal equivalent shown alongside it, which is an approximation by definition (dividing a numerator by a denominator rarely produces a terminating decimal).
Worked Examples
Add: 3/4 + 3/4
- Cross-multiply for a common denominator: (3 × 4) + (3 × 4) = 12 + 12 = 24, over 4 × 4 = 16
- Result before simplifying: 24/16
- Simplify using GCD(24, 16) = 8: 24 ÷ 8 = 3, 16 ÷ 8 = 2 → 3/2
- Decimal equivalent: 3 ÷ 2 = 1.5
- Mixed number: 3/2 = 1 whole plus a remainder of 1/2 → 1 1/2
Divide: 1/2 ÷ 1/4
- Divide by multiplying by the reciprocal: (1 × 4) / (2 × 1) = 4/2
- Simplify using GCD(4, 2) = 2: 4 ÷ 2 = 2, 2 ÷ 2 = 1 → 2/1
- A denominator of 1 is a whole number, so the result is just 2
- Decimal equivalent: 2 ÷ 1 = 2
- Mixed number: a whole number with no remainder, shown as 2
Frequently Asked Questions
Why does adding fractions use cross-multiplication instead of finding the least common denominator (LCD)?
Cross-multiplication (using b×d as the shared denominator) always works and never requires finding the LCD first, which is what makes it reliable to compute directly. The trade-off is that the intermediate denominator can be larger than the true LCD would be — the simplification step at the end reduces it back down to lowest terms regardless, so the final answer is identical either way.
Why is a denominator of zero not allowed?
Dividing by zero is undefined — a fraction's denominator represents how many equal parts the whole is split into, and splitting something into zero parts has no meaning. Any fraction entered with a zero denominator is rejected before any calculation runs.
What happens if I try to divide by a fraction like 0/5?
Dividing by a fraction with a zero numerator is the same as dividing by zero (0/5 equals zero), which is undefined. This calculator rejects that combination rather than returning an incorrect or infinite result.
Why does a fraction like 0/8 just show as 0?
Zero is zero regardless of the denominator it was entered with, so the result is shown as the plain number 0 rather than a fraction like 0/1 — it's simpler and just as accurate.
How is the mixed number calculated from an improper fraction?
An improper fraction (numerator larger than denominator) is split into a whole-number part and a remaining proper fraction: divide the numerator by the denominator and keep only the whole-number portion, then whatever is left over (the remainder) becomes the new numerator over the same denominator. For example, 3/2 divides to a whole part of 1 with a remainder of 1, giving 1 1/2.