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Irreducible Fractions

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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)
Summary

Decimal to Fraction in Lowest Term Converter

How to make a fraction in lowest term?

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.

How to calculate and give the result under the lowest term form?

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.

How to make a fraction from a decimal number?

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 $1/6 = 0.1666666 ...$

Source code

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Tag(s) : Arithmetics, Symbolic Computation
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