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) × (3/2) = 1 cup.
• Construction and carpentry: Adding 3/8 inch and 5/16 inch requires a common denominator of 16: 6/16 + 5/16 = 11/16 inch. Fractional measurements in imperial units make fraction arithmetic a daily reality in trades work.
• Finance: Interest rates, price-to-earnings ratios, and budget allocations are fractions or percentages derived from fraction arithmetic. A fund with a 0.75% expense ratio charges 3/4 of one percent of your balance each year.
• Probability: The chance of two independent events both occurring is the product of their individual probabilities. If one event has probability 1/4 and another has probability 1/3, both occurring has probability (1/4) × (1/3) = 1/12.

Common Mistakes in Fraction Arithmetic

Fraction errors are among the most common in math at every level. Knowing the mistakes helps you avoid them:

• Adding denominators instead of finding a common denominator: 1/3 + 1/4 ≠ 2/7. You must find the LCD (12), convert, then add: 4/12 + 3/12 = 7/12. The denominator of the sum is never the sum of the two denominators.
• Forgetting to simplify: 6/8 is a valid fraction but not in lowest terms. Always divide by the GCD at the end: GCD(6,8) = 2, so 6/8 = 3/4.
• Incorrectly applying "keep, change, flip" to multiplication: Flipping (reciprocal) only applies to division. For multiplication, multiply numerators by numerators and denominators by denominators directly.
• Not converting mixed numbers before calculating: You cannot add 2¾ + 1⅔ directly as written. Convert to improper fractions first: 11/4 + 5/3, then find the common denominator (12): 33/12 + 20/12 = 53/12 = 4 5/12.
• Misidentifying the LCD: The LCD of 4 and 6 is 12, not 24 (which is the product). Using 24 as the common denominator still gives the correct answer after simplification, but creates larger intermediate numbers. Finding the true LCD by computing the LCM keeps numbers manageable.

Fractions in Algebra: Working with Variables

Fraction arithmetic extends naturally to algebraic fractions where numerators and denominators contain variables. The same rules apply: to add x/3 + x/4, the LCD is 12, so the result is 4x/12 + 3x/12 = 7x/12. To multiply (x/2) × (3/x), multiply numerators and denominators: 3x / (2x) = 3/2 (the x cancels). The GCD of the numerator and denominator in an algebraic fraction may be a variable expression rather than just a number, which is why factoring algebraic expressions is a prerequisite for simplifying algebraic fractions.

Understanding numerical fraction arithmetic thoroughly is the foundation for working with rational expressions (algebraic fractions) in pre-algebra, algebra, and calculus. The operations are identical — only the values are more complex.

Fractions on a Number Line

A fraction's position on the number line provides an important visual check on whether your arithmetic is reasonable. Proper fractions (numerator less than denominator) lie between 0 and 1. A fraction closer to 1 has a numerator close to its denominator (7/8). A fraction closer to 0 has a very small numerator relative to its denominator (1/10). Improper fractions and mixed numbers lie to the right of 1.

When adding fractions, the result should lie between the two input values if they have the same sign, or between them if they have opposite signs. When multiplying two proper fractions, the result is always smaller than either factor — (2/3) × (3/4) = 1/2, which is smaller than both 2/3 and 3/4. This intuition acts as a quick sanity check: if your calculated result seems too large or is in the wrong direction on the number line, revisit the arithmetic.