Fraction Calculator

To convert a terminating decimal like 0.375 into a fraction, multiply it by a power of 10 to remove the decimal point. Since 0.375 has three decimal places, multiply by 1000 to get 375/1000. Then find the Greatest Common Factor (GCF) of 375 and 1000, which is 125. Divide both the numerator and denominator by 125 to get the simplified fraction 3/8. This method works for any terminating decimal: count the decimal places, write the number over the corresponding power of 10, and simplify. For repeating decimals like 0.333..., algebraic methods are needed — set x = 0.333..., multiply both sides by 10 to get 10x = 3.333..., subtract to get 9x = 3, so x = 1/3.

A fraction produces a terminating decimal only when the denominator (in its fully simplified form) has no prime factors other than 2 and 5. These are the prime factors of 10, which is the base of our decimal system. For example, 1/8 terminates because 8 = 2 cubed, and 1/20 terminates because 20 = 2 squared times 5. However, 1/3 repeats because 3 is a prime factor that is not 2 or 5. Similarly, 1/7 produces the repeating decimal 0.142857142857... because 7 is not a factor of any power of 10. The length of the repeating cycle is related to the smallest power of 10 that leaves a remainder of 1 when divided by the denominator.

The fastest method is to find the Least Common Multiple (LCM) of the two denominators and use it as the common denominator. For example, to add 2/9 + 5/12, compute the LCM of 9 and 12, which is 36. Convert each fraction: 2/9 becomes 8/36 (multiply by 4) and 5/12 becomes 15/36 (multiply by 3). Now add the numerators: 8 + 15 = 23, giving 23/36. Using the LCM rather than simply multiplying denominators (which would give 108) keeps the numbers smaller and often eliminates the need to simplify at the end. For two denominators a and b, the LCM equals (a times b) divided by their GCF.

Improper fractions (where the numerator is larger than the denominator, like 7/4) are better for calculations because they work directly with all four arithmetic operations without extra conversion steps. Mixed numbers (like 1 3/4) are better for communication and real-world contexts because they are easier to visualize and understand — saying a board is 5 3/8 inches long is more intuitive than 43/8 inches. As a rule of thumb, convert to improper fractions when you need to compute, then convert back to a mixed number when you need to present the result. In scientific and engineering contexts, decimals are often preferred over both formats.

To simplify a fraction, find the Greatest Common Factor (GCF) of the numerator and denominator, then divide both by that number. For example, to simplify 48/64, find the GCF of 48 and 64. Using the Euclidean algorithm: 64 divided by 48 leaves remainder 16, then 48 divided by 16 leaves remainder 0, so the GCF is 16. Divide both parts: 48/16 = 3 and 64/16 = 4, giving the simplified fraction 3/4. A fraction is fully simplified when the numerator and denominator share no common factors other than 1. This calculator performs simplification automatically and shows you the GCF used, so you can verify the result and learn the technique.