Fraction Calculator

Despite the prevalence of decimals in technology, fractions remain essential in cooking (¼ cup, ⅓ teaspoon, ½ pound), construction (where measurements are in eighths and sixteenths of an inch, and lumber is sold in fractional dimensional lumber like 2×4), music (time signatures like 3/4 and 6/8 that literally mean fractions of a whole measure), stock trading (where prices historically quoted in eighths and sixteenths before decimalization in 2001), and higher mathematics (where exact fractional values preserve precision that decimal approximations destroy). Fractions express exact values that decimals can only approximate — 1/3 equals the repeating decimal 0.333... which a computer or calculator truncates at some point, introducing small but accumulating errors.

This exactness matters in symbolic mathematics, engineering, and any field where rounding errors compound. A formula involving multiple 1/3 factors produces exact results when computed symbolically with fractions but accumulating rounding when computed with decimal approximations. Cultural persistence also matters: American cooking, American carpentry, and some international tire-size specifications remain thoroughly fractional. Students who avoid mastering fractions because "decimals are easier" find themselves handicapped in those domains, which is why fraction fluency remains a standard elementary-math objective despite decimalization pressure elsewhere.

Simplifying a fraction means dividing both the numerator and denominator by their greatest common divisor (GCD). For example, 12/18 simplifies to 2/3 by dividing both by 6 (the GCD of 12 and 18). A fraction is in "lowest terms" when no integer greater than 1 divides both numerator and denominator evenly. Computing the GCD uses the Euclidean algorithm, which repeatedly replaces the larger number with its remainder when divided by the smaller — for 12 and 18: 18 = 12×1 + 6, then 12 = 6×2 + 0, so GCD = 6.

When adding or subtracting fractions with different denominators, you must first convert them to a common denominator — the same "slice size" before you can combine them. The least common denominator (LCD) is the smallest number both original denominators divide evenly into, and using it minimizes the size of the numbers you work with. For 1/6 + 1/4, the LCD is 12 (smallest number both 6 and 4 divide evenly into), so 1/6 becomes 2/12 and 1/4 becomes 3/12, then 2/12 + 3/12 = 5/12. Multiplication and division don't require common denominators — you can multiply numerators and denominators directly, which is why those operations feel easier than addition and subtraction despite looking more complex algebraically.

Converting between fractions and decimals requires different techniques depending on the decimal type. To convert a fraction to a decimal, divide the numerator by the denominator — the result either terminates (like 1/4 = 0.25) or repeats (like 1/3 = 0.333...). A fraction terminates in decimal form if and only if its denominator (in lowest terms) has only 2 and 5 as prime factors; otherwise it produces a repeating pattern.

To convert a terminating decimal to a fraction, place the decimal digits over the appropriate power of ten and simplify: 0.375 = 375/1000 = 3/8 after dividing both by 125. For repeating decimals, algebraic techniques derive the exact fraction — to convert 0.666... to fraction form, let x = 0.666..., then 10x = 6.666..., so 10x − x = 9x = 6, giving x = 6/9 = 2/3. The calculator handles both directions automatically and always displays the simplified result, which is especially useful when working with repeating decimals that would be awkward to write out manually. Mixed numbers (like 1 ½) are another display format — internally, 1 ½ equals the improper fraction 3/2, and the calculator converts between formats as needed for readability.