½Fraction Calculator

This fraction calculator handles every common fraction operation in one place: add, subtract, multiply, and divide fractions, simplify to lowest terms, convert between fractions and decimals, and work with mixed numbers and improper fractions. Results appear as a simplified fraction, mixed number, decimal value, and percentage so you always have the form you need.

Understanding Numerators, Denominators, and Types of Fractions

A fraction a/b consists of two parts. The numerator (a) counts the parts you have. The denominator (b) counts the equal parts the whole is divided into. So 3/4 means three out of four equal portions. When the numerator is smaller than the denominator, the fraction is a proper fraction (3/4, 2/7). When the numerator equals or exceeds the denominator, the fraction is an improper fraction (7/4, 9/3). Improper fractions can always be rewritten as mixed numbers, which combine a whole number with a proper fraction.

How to Add and Subtract Fractions with Different Denominators

Adding or subtracting fractions requires a common denominator because you can only combine parts of equal size. When denominators already match, add or subtract the numerators directly and keep the denominator. When denominators differ, follow these steps:

• Find the least common denominator (LCD), which is the least common multiple of the two denominators.
• Convert each fraction to an equivalent fraction with the LCD by multiplying numerator and denominator by the appropriate factor.
• Add or subtract the numerators and write the result over the LCD.
• Simplify the resulting fraction if possible.

Example: 1/4 + 1/6. The LCD of 4 and 6 is 12. Convert: 1/4 = 3/12 and 1/6 = 2/12. Add: 3/12 + 2/12 = 5/12. The general formula a/b + c/d = (ad + bc) / (bd) always works but may produce a fraction requiring further simplification. Using the LCD keeps intermediate numbers smaller and reduces simplification work.

Simplify Fractions Calculator: Reducing to Lowest Terms

A fraction is fully simplified when its numerator and denominator share no common factor other than 1, a condition called being coprime or in lowest terms. To simplify any fraction, divide both the numerator and denominator by their greatest common divisor (GCD).

Example: simplify 36/48. GCD(36, 48) = 12. Divide both: 36/12 = 3 and 48/12 = 4. The simplified fraction is 3/4. You can check your work: GCD(3, 4) = 1, confirming the fraction is fully reduced. Equivalent fractions represent the same value (3/4 = 6/8 = 9/12 = 36/48) but simplified form is easiest to interpret and compare.

Multiplying and Dividing Fractions

Fraction multiplication is straightforward: multiply the numerators together and the denominators together, then simplify. (2/3) x (3/5) = (2 x 3) / (3 x 5) = 6/15 = 2/5. A useful shortcut called cross-cancellation lets you cancel common factors before multiplying, keeping numbers smaller. For (2/3) x (3/5): the 3 in the numerator of the second fraction cancels with the 3 in the denominator of the first, leaving (2/1) x (1/5) = 2/5 directly.

To divide one fraction by another, multiply the first fraction by the reciprocal of the second. The reciprocal of c/d is d/c (flip the fraction). The memory aid "keep, change, flip" describes this: keep the first fraction, change division to multiplication, flip the second. So (3/4) / (2/5) = (3/4) x (5/2) = 15/8 = 1 and 7/8.

Fraction to Decimal Converter

Converting a fraction to a decimal is simply long division: divide the numerator by the denominator. 3/4 = 3 / 4 = 0.75. Some fractions produce terminating decimals (1/4 = 0.25, 1/8 = 0.125), while others produce repeating decimals (1/3 = 0.333..., 1/7 = 0.142857142857...). The repeating pattern always eventually appears because the possible remainders are limited to values less than the denominator.

To convert a terminating decimal back to a fraction, write the digits over the appropriate power of 10 and simplify. 0.625 = 625/1000 = 5/8 (dividing by GCD 125). The convert mode of this calculator performs this automatically.

Mixed Numbers and Improper Fractions

Mixed numbers such as 2 and 3/4 combine a whole number with a proper fraction. To convert to an improper fraction for calculation, multiply the whole number by the denominator and add the numerator: 2 and 3/4 = (2 x 4 + 3) / 4 = 11/4. To convert an improper fraction back to a mixed number, divide the numerator by the denominator: 11 / 4 = 2 remainder 3, giving 2 and 3/4. Always convert mixed numbers to improper fractions before applying any arithmetic operation, then convert the result back if a mixed number form is needed.

Practical Uses of Fraction Arithmetic

• Cooking: Scaling a recipe from 4 to 6 servings means multiplying every quantity by 6/4 = 3/2. A measurement of 2/3 cup becomes (2/3) x (3/2) = 1 cup.
• Construction: Adding 3/8 inch and 5/16 inch requires a common denominator of 16: 6/16 + 5/16 = 11/16 inch.
• Finance: Interest rates, price-to-earnings ratios, and budget allocations are all expressed as fractions or percentages derived from fraction arithmetic.
• Probability: The chance of two independent events both occurring is the product of their individual probabilities, each expressed as a fraction.