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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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)

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 obtained PGCD.

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 dCode calculator, enter the expressions / fractions and the simplifier will use formal calculations in order to keep variables and find the irreducible form.

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

By calling $ x $ the 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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