From decimal numbers to fractions

From exact numbers to fractions

In the numerator we write down the whole number without a comma and in the denominator we write down the number $1$ with as many zeros as digits has the decimal part.

Express the number $1,8182$ as a fraction. In the numerator we put the number without comma and in the denominator we write $1$ continued by $4$ zeros, because the decimal part has $4$ digits: $$\dfrac{18182}{10000}=\dfrac{9091}{5000}$$

Express the number $4,51$ as a fraction. $$\dfrac{451}{100}$$

From periodic numbers to fractions

if it is a pure periodical decimal, in the numerator we write down the whole number without a comma by subtracting its integer part. In the denominator we write down as many nines as digits has the period.

Express the number $1,\widehat{3}$ as a fraction: $$\dfrac{(13-1)}{9}=\dfrac{12}{9}=\dfrac{4}{3}$$

Express as a fraction the number $$19,\widehat{62}=\dfrac{(1962-19)}{99}=\dfrac{1943}{99}$$

The conversion from ultimately periodic decimal to fractions is done in a similar way. The idea is to turn the decimal into a pure periodic by multiplying it by the appropriate potency of $10$.

$$13,85\widehat{32}=\dfrac{1385,\widehat{32}}{100}=\dfrac{1}{100}\dfrac{138532-1385}{99}$$

$13,85\widehat{32}=\dfrac{137147}{9900}$