Fraction Calculator

Fractions follow four rules, and three of them are easier than the one everybody remembers.

Multiplication is the simplest: multiply the numerators, multiply the denominators. 2/3 x 3/4 = 6/12 = 1/2. No common denominator is needed, which is the opposite of what most people half-remember from school.

Division is multiplication by the reciprocal: flip the second fraction and multiply. 1/2 / 1/4 = 1/2 x 4/1 = 4/2 = 2. This is also the clearest explanation of a fact that looks wrong: dividing by a number smaller than one makes the result bigger. "How many quarters fit in a half" is 2, and that is what the arithmetic says.

Addition and subtraction are the ones that need work, because you can only add parts of the same size. 1/3 + 1/6 needs both over a common denominator: 2/6 + 1/6 = 3/6 = 1/2. Any common denominator works, and the product of the two always is one, but the least common multiple keeps the numbers small. For 1/4 + 1/6, the LCM is 12 rather than 24, so you work with 3/12 + 2/12 instead of 6/24 + 4/24.

Then reducing. A fraction is in lowest terms when the numerator and denominator share no factor but 1, and getting there means dividing both by their greatest common divisor. 6/12 has a GCD of 6, so it reduces to 1/2. Euclid's algorithm finds the GCD by repeated remainders and is what every implementation uses, including this one.

Mixed numbers are the other source of errors, and the mistake is always about the sign. 2 1/3 means 2 + 1/3, so as an improper fraction it is 7/3. But -2 1/3 means -(2 + 1/3), which is -7/3, and not -2 + 1/3 = -5/3. The minus sign belongs to the whole quantity. This calculator reads the sign off the whole number and applies it to the total, which is the convention everywhere outside a few deliberately tricky textbook problems.

The reason to work in fractions rather than decimals at all is exactness. 1/3 cannot be written as a terminating decimal, so any decimal is an approximation, and approximations accumulate. In floating-point arithmetic 1/3 + 1/6 does not quite equal 0.5. It comes to 0.49999999999999994, which is harmless once but fatal in a chain of operations. Working in exact integers and reducing at every step, as this calculator does, never has that problem.

Which fractions terminate as decimals is a tidy result worth knowing: exactly those whose denominator in lowest terms has no prime factors other than 2 and 5. So halves, quarters, fifths, eighths, tenths and twentieths all terminate, and thirds, sevenths, ninths and elevenths all repeat. 1/7 is 0.142857 repeating, with a six-digit cycle, and no amount of decimal places will ever write it exactly.

Three more jobs sit under the operation menu. Simplify reduces one fraction to lowest terms and shows the greatest common factor found by Euclid's algorithm: 18/24 has a GCF of 6 and simplifies to 3/4. Compare says which of two fractions is larger by cross-multiplying, so 5/8 against 3/5 is 25 against 24 and 5/8 wins by 1/40. Rewrite over a denominator gives the equivalent fraction with the denominator you choose: 3/4 over 20 is 15/20.