Add, subtract, multiply, and divide fractions with step-by-step solutions showing LCD, GCD, and simplification.
Fraction arithmetic follows a systematic process. When adding or subtracting fractions with different denominators, you must first find the Least Common Denominator (LCD) — the smallest number that both denominators divide into evenly.
Finding the LCD using GCD: The LCD is calculated using the Greatest Common Divisor (GCD). The formula is: LCD = (denominator1 × denominator2) ÷ GCD(denominator1, denominator2).
Euclidean Algorithm for GCD: The GCD is found by repeatedly applying the division algorithm until the remainder is zero. Start with the larger number and divide by the smaller; then divide the previous divisor by the remainder, and repeat. The last non-zero remainder is the GCD.
Simplifying fractions: After performing the operation, simplify the result by dividing both the numerator and denominator by their GCD. This gives the fraction in lowest terms.
Mixed numbers: A mixed number (e.g., 2¾) combines a whole number with a fraction. Convert to an improper fraction by multiplying the whole number by the denominator and adding the numerator.
| Fraction | Decimal | Fraction | Decimal |
|---|---|---|---|
| 1/2 | 0.5 | 1/8 | 0.125 |
| 1/3 | 0.333… | 3/8 | 0.375 |
| 2/3 | 0.666… | 5/8 | 0.625 |
| 1/4 | 0.25 | 7/8 | 0.875 |
| 3/4 | 0.75 | 1/10 | 0.1 |
| 1/5 | 0.2 | 1/6 | 0.166… |
| 2/5 | 0.4 | 5/6 | 0.833… |
First, find the Least Common Denominator (LCD) of both denominators. Convert each fraction to an equivalent fraction with the LCD as the denominator. Then add the numerators while keeping the denominator the same. Finally, simplify the result by dividing both numerator and denominator by their GCD.
Find the Greatest Common Divisor (GCD) of the numerator and denominator using the Euclidean algorithm. Divide both the numerator and denominator by their GCD. The result is a fraction in lowest terms. For example, 12/18 simplifies to 2/3 because the GCD of 12 and 18 is 6.
A mixed number combines a whole number with a fraction, such as 2¾ or 3⅖. To convert a mixed number to an improper fraction, multiply the whole number by the denominator and add the numerator; this sum becomes the new numerator. For example, 2¾ = (2 × 4 + 3)/4 = 11/4.
When you divide by a fraction, you're asking "how many times does this fraction fit into the other?" This is equivalent to multiplying by its reciprocal (flipping the numerator and denominator). Mathematically, a ÷ (b/c) = a × (c/b) because division and multiplication are inverse operations.
The LCD is the smallest positive integer that is a multiple of both denominators. For example, the LCD of 1/4 and 1/6 is 12 because 12 is the smallest number divisible by both 4 and 6. Finding the LCD allows you to rewrite fractions with a common denominator so they can be added or subtracted.
This calculator implements standard mathematical procedures for fraction arithmetic: the Euclidean algorithm for GCD computation, the LCM formula, and mixed-number conversion. These are self-evident arithmetic principles taught in elementary and middle school mathematics.