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.

Example

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: $\frac{18182}{10000} = \frac{9091}{5000}$

Example

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

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.

Example

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

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

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 $10$ .

$13, 85 \hat{32} = \frac{1385, \hat{32}}{100} = \frac{1}{100} \frac{138532 - 1385}{99}$

$13, 85 \hat{32} = \frac{137147}{9900}$