Converting repeating decimals to fractions

Converting a purely repeating decimal to fractions is an operation that we will use rather rarely. Nevertheless, for general knowledge , it is important to study this as well. We will begin by converting a pure repeating decimal into an ordinary fraction.

We have already said that if a period in a repeating decimal begins immediately after the decimal point, such a fraction is called a purely periodic.

To convert a purely repeating decimal into an ordinary fraction, write the period of the repeating decimal in the numerator of the ordinary fraction, and write some number of nines in the denominator of the ordinary fraction. In this case, the number of nines must be equal to the number of digits in the period of the repeating decimal .

As an example, consider a purely repeating decimal 0. (3) - zero integers and three in the period. Let's try to convert it into a fraction.

The rule states that the period of a repeating decimal should be written first in the numerator of an ordinary fraction.

So in the numerator we write the period of the decimal 0. (3) that is three:

And the denominator must contain some number of nines. In this case, the number of nines must be equal to the number of digits in the period of the repeating decimal 0. (3).

In the repeating decimal 0. (3) the period consists of one digit 3. So we write one nine in the denominator of the fraction:

The resulting fraction can be reduced by 3, then we get the following:

We got an ordinary fraction of .

Thus, when you translate the repeating decimal 0. (3) into an ordinary fraction, you get

Example 2. Convert the repeating decimal 0. (45) into an ordinary fraction.

Here the period is two digits 4 and 5. Write these two digits in the numerator of the fraction:

And in the denominator we write some number of nines. The number of nines must be equal to the number of digits in the period of the repeating decimal 0. (45).

In the repeating decimal 0. (45) the period consists of two digits 4 and 5. So we write two nines in the denominator of the fraction:

The resulting fraction can be reduced this fraction by 9, then we get the following:

Thus, when you convert the repeating decimal 0. (45) into an ordinary fraction, you get

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