½ Fraction Calculator

Addition and subtraction need a common denominator. The calculator uses the cross-multiplication form: a/b plus c/d equals (a x d plus c x b) divided by (b x d), and subtraction is the same with a minus sign in the numerator. Multiplication is simpler still, since a/b times c/d equals (a x c) over (b x d). Division inverts the second fraction, so a/b divided by c/d equals (a x d) over (b x c), which is the familiar instruction to multiply by the reciprocal.

Those rules usually produce a result that is not in lowest terms. The calculator then applies Euclid's algorithm to find the greatest common divisor of the numerator and the denominator: divide the larger by the smaller, replace the pair with the smaller number and the remainder, and repeat until the remainder is zero. The last non-zero value is the greatest common divisor, and dividing both parts by it gives the reduced fraction. The simplify mode runs the same routine on a single fraction and shows you the divisor it found.

Take 3/4 plus 5/6. Cross-multiplying gives a numerator of 3 x 6 plus 5 x 4, which is 18 plus 20, or 38. The denominator is 4 x 6, which is 24. The raw answer is therefore 38/24.

Now reduce it. Euclid's algorithm on 38 and 24: 38 divided by 24 leaves remainder 14; 24 divided by 14 leaves remainder 10; 14 divided by 10 leaves remainder 4; 10 divided by 4 leaves remainder 2; 4 divided by 2 leaves remainder 0. The greatest common divisor is 2. Dividing gives 38 divided by 2 equals 19, and 24 divided by 2 equals 12, so the reduced answer is 19/12.

Because 19 is larger than 12, this is an improper fraction. Twelve goes into 19 once with 7 left over, so the mixed number is 1 and 7/12. As a decimal, 19 divided by 12 is 1.583333, and as a percentage that is 158.33 per cent. All three forms describe the same quantity.

Which form you keep depends on the task. Improper fractions such as 19/12 are easier to feed into further calculations, while mixed numbers such as 1 and 7/12 are easier to picture when you are measuring a board or portioning a recipe. Decimals suit anything that ends up in a spreadsheet or on a digital scale.

Watch three recurring mistakes. The first is adding denominators, writing 3/4 plus 5/6 as 8/10; denominators are multiplied to form the common base, never added. The second is forgetting to convert a mixed number before doing arithmetic with it, since 1 and 7/12 must become 19/12 first. The third is treating a rounded decimal as exact when converting back to a fraction.

That last point deserves care. A repeating decimal typed as 0.333 converts to 333/1000, not to 1/3, because those are genuinely different numbers. When you know a value is a repeating decimal, enter the fraction you mean rather than a truncated decimal version of it.