This question aims to convert the given repeating decimal to a fraction.

A fraction regards the portion of a whole and is expressed as $\dfrac{a}{b}$ where $b$ must not be equal to zero. Contrary to the fraction, a decimal is a type of number incorporating a decimal point responsible for separating the whole number from the fractional part. Terminating/non-repeating or non-terminating/ repeating are two common types of decimal numbers.

The decimal form of a number that does not terminate until a certain number of digits is said to be repeating or non-terminating. On the other hand, terminating or non-repeating decimals have a finite number of terms after a decimal point. Usually, the common method to convert a decimal number to a fraction is that a decimal number is divided by $10$ to power the number of decimal places. However, in the case of non-terminating decimals, it is not possible to apply this rule because they have an infinite number of decimal places.

## Expert Answer

To convert the given non-terminating decimal to a fraction, suppose that:

$y=3.166…$

Since there is only one repeating digit, so multiply both sides by $10$:

$10y=31.66…$

Since, $9y=10y-y$

Therefore, $9y=31.66…-3.166…$

$9y=28.5$

Divide both sides by $9$ we get:

$y=\dfrac{28.5}{9}$

$y=\dfrac{285}{9\times 10}$

$y=\dfrac{285}{90}$

$y=\dfrac{19}{6}$

$y=3\dfrac{1}{6}$

## Example 1

Write the fractional form of $0.\overline{251}$.

### Solution

To convert the given non-terminating decimal to a fraction, suppose that:

$y=0.\overline{251}=0.251251…$

Since there are three repeating digits, multiply both sides by $1000$:

$1000y=251.251251…$

Since, $999y=1000y-y$

Therefore, $999y=251.251251…-0.251251…$

$999y=251$

Divide both sides by $999$ we get:

$y=\dfrac{251}{999}$

## Example 2

Write the fractional form of $0.34\overline{12}$.

### Solution

To convert the given non-terminating decimal to a fraction suppose that:

$y=0.34\overline{12}=0.341212…$

Since there are two repeating digits, so multiply both sides by $100$:

$100y=34.1212…$

Since, $99y=100y-y$

Therefore, $99y=34.1212…-0.341212…$

$99y=33.78$

Divide both sides by $99$ we get:

$y=\dfrac{33.78}{99}$

$y=\dfrac{3378}{99\times 100}$

$y=\dfrac{3378}{9900}$

## Example 3

Write the fractional form of $0.00\overline{12}$.

### Solution

To convert the given non-terminating decimal to a fraction suppose that:

$y=0.00\overline{12}=0.001212…$

Since there are two repeating digits, so multiply both sides by $100$:

$100y=0.1212…$

Since, $99y=100y-y$

Therefore, $99y=0.1212…-0.001212…$

$99y=0.12$

Divide both sides by $99$ we get:

$y=\dfrac{0.12}{99}$

$y=\dfrac{12}{99\times 100}$

$y=\dfrac{12}{9900}$