Tool to reduce fractions in lowest term. A Fraction in Lowest Terms (Irreducible Fraction) is a reduced fraction in shich the numerator and the denominator are coprime (they do not share common factors)

Irreducible Fractions - dCode

Tag(s) : Arithmetics, Symbolic Computation

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A fraction can be written in different ways while keeping its value.

__Example:__ $ \frac{50}{100} = \frac{5}{10} = \frac{2}{4} = \frac{1}{2} = 0.5 $

A fraction in lowest form (irreducible fraction) is a fraction whose divisor (the number under the fraction bar) is the smallest possible integer. NB: the dividend (number above the fraction bar) must also be an integer.

The irreducible fraction is generally the preferred format for writing a fraction in the simplest form.

__Example:__ $ \frac{1}{2} $ is fraction in lowest form while $ \frac{2}{4} $ is not a fraction in lowest form.

To simplify a fraction $ a / b $ or $ frac{a}{b} $ composed of a numerator $ a $ and a denominator $ b $, find the greatest common divisor (GCD) of the numbers $ a $ and $ b $. The irreducible fraction is obtained by dividing the numerator and the denominator by the calculated GCD.

__Example:__ The fraction $ 12/10 $ has $ 12 $ for numerator and $ 10 $ for denominator. Calculate that $ GCD(12,10) = 2 $ and divide both the numerator $ 12/2 = 6 $ and the denominator $ 10/2 = 5 $, so the corresponding irreducible fraction is $ 6/5 $

dCode offers tools to calculate the GCD via, for example, Euclid's algorithm.

Use the aboce calculator form: enter the expressions / fractions and the simplifier will use formal calculations in order to keep variables and find the irreducible form of the division (simplification of the fraction in lowest term).

If the number has a **limited decimal development** then it only needs to be multiplied by the right power of 10, then simplify the fraction and solve the equation.

__Example:__ The number $ 0.14 $ is equivalent to $ 0.14/1 $, multiply by $ 10/10 (=1) $ until having no comma: $ 0.14/1 = 1.4/10 = 14/100 $ then simplify $ 14/100 = 7/50 $

If the number has a **non finite decimal expansion** then it is necessary to locate the repeating portion of the number after the repeating decimal point.

__Example:__ The number $ 0.166666666 ... $ where the $ 6 $ is repeated

If $ x $ is the decimal number, and $ n $ the size (number of digits) of the smallest repeated portion. To obtain a fraction, multiply $ x $ by $ 10^n $ and then subtract $ x $.

__Example:__ $ x = 0.1666666 ... $, the smallest repeated portion is $ 6 $, which has a single digit so that $ n = 1 $. Then compute $ 10^1 \ times x = 1.6666666 ... $ and $ 10x-x $. $$ 10x-x = 9x = 1.666666 ... - 0.1666666 ... = 1.5 \\ \iff 9x = 1.5 \\ \Rightarrow x = 1.5 / 9 = 15/90 = 1/6 $$ So $ 1/6 = 0.1666666 ... $

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