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# Repeating Decimal to Fraction

In mathematics, a fraction is a value, which defines the part of a whole. In other words, the fraction is a ratio of two numbers. Whereas, the decimal is a number, whose whole number part and the fractional part is separated by a decimal point. The decimal number can be classified into different types, such as terminating and non-terminating decimals, repeating and non-repeating decimals. While solving many mathematical problems, the conversion of decimal to the fractional value is preferred, as we can easily simplify the fractional values. In this article, we are going to discuss how to convert repeating decimals to fractions in an easy way.

## Terminating and Non-Terminating Decimals

A terminating decimal is a decimal, that has an end digit. It is a decimal, which has a finite number of digits(or terms).

Example: 0.15, 0.86, etc.

Non-terminating decimals are the one that does not have an end term. It has an infinite number of terms.

Example: 0.5444444….., 0.1111111….., etc.

## Repeating and Non-Repeating Decimals

Repeating decimals are the one, which has a set of terms in decimal to be repeated uniformly.

Example: 0.666666…., 0.123123…., etc.

It is to be noted that the repeated term in decimal is represented by a bar on top of the repeated part. Such as

$$\begin{array}{l}0.333333….. = 0.\bar{3}\end{array}$$
.

Non-repeating decimals are the one that does have repeated terms.

Note:

Non-Terminating and non-repeating decimals are said to be an Irrational number. Eg.
$$\begin{array}{l}\sqrt{2} = 1.4142135……\end{array}$$
.

The square roots of all the terms (except perfect squares) are irrational numbers.

Non- Terminating and repeating decimals are Rational numbers and can be represented in the form of p/q, where q is not equal to 0.

## Repeating Decimals to Fraction Conversion

Let us now learn the conversion of repeating decimals into the fractional form. Now, we are going to discuss the two different cases of the repeating fraction.

### Case 1:  Fraction of the type $$\begin{array}{l}0.\overline{abcd}\end{array}$$

The formula to convert this type of repeating decimal to a fraction is given by:

$$\begin{array}{l}\overline{abcd}\end{array}$$
= Repeated term / Number of 9’s for the repeated term

Example 1:

Convert
$$\begin{array}{l}0.\overline{7}\end{array}$$
to the fractional form.

Solution:

Here, the number of repeated term is 7 only. Thus the number of times 9 to be repeated in the denominator is only once.

$$\begin{array}{l}0.\overline{7} = \frac{7}{9}\end{array}$$

Example 2:

Convert 0.125125125… to the fractional form.

Solution:

The decimal 0.125125125….. can be written as
$$\begin{array}{l}0.\overline{125}\end{array}$$
.

Here, 125 consists of three terms, and it is repeated in a continuous manner. Thus, the number of time 9 to be repeated in the denominator becomes three.

$$\begin{array}{l}0.\overline{125} = \frac{125}{999}\end{array}$$

### Case 2:  Fraction of the type $$\begin{array}{l}0.ab..\overline{cd}\end{array}$$

The formula to convert this type of repeating decimal to the fraction is given by:

$$\begin{array}{l}0.ab..\overline{cd} =\frac{(ab….cd…..) – ab……}{Number \; of \; time \; 9’s \; the \; repeating \; term \; followed \; by \; the \; number \; of \; times \; 0’s \; for \; the \; non-repeated \; terms }\end{array}$$

Example 3:

Convert
$$\begin{array}{l}0.12\overline{34}\end{array}$$
to the fractional form.

Solution:

In the given decimal number, 12 is a non-repeated decimal value, and 34 is in the repeating form. Thus denominator becomes 9900.

$$\begin{array}{l}0.12\overline{34} = \frac{1234 – 12 }{9900} = \frac{1222}{9900}\end{array}$$

Example 4:

Convert
$$\begin{array}{l}0.00\overline{69}\end{array}$$
in the fraction form.

Solution:

In the given decimal number, the number 00 is a non-repeated decimal value, and 69 is in the repeating form. Thus, the denominator becomes 9900.

$$\begin{array}{l}0.00\overline{69} = \frac{0069}{9900} = \frac{69}{9900}\end{array}$$

It is all about the conversion of repeating decimal to the fraction form. To learn more interesting topics in Maths, download BYJU’S – The Learning App and learn with ease.

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