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Irreducible Fractions

Tool to reduce a fraction 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)

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Irreducible Fractions -

Tag(s) : Arithmetics, Symbolic Computation, Mathematics

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# Irreducible Fractions

## Decimal to Fraction in Lowest Term Converter

Tool to reduce a fraction 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)

### 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: Fraction $$12/10$$, with $$12$$ the numerator and $$10$$ the 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 $$6/5$$

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

dCode uses 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: Consider the number $$0.14 = 0.14/1$$, multiply by $$10/10$$ 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: Consider the number $$0.166666666 ...$$ where the $$6$$ is repeated

We call $$x$$ 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 $$10x-x$$.

$$10x-x = 9x = 1.666666 ... - 0.1666666 ... = 1.5 \\ \iff 9x = 1.5 \\ \Rightarrow x = 1.5 / 9 = 15/90 = 1/6$$

Example: So $$1/6 = 0.1666666 ...$$