**Given below are some examples on repeating decimals.****Example 1****:** Convert 0.4646...

to fractions.

**Solution:**Let x = 0.4646

__Step 1:__ Multiply x by 100

Then 100x = 46.4646

__Step 2:__ Subtract x from 100x

100x - x = 46.4646 - 0.4646

99x = 46

__Step 3:__ Divide both sides by 99

x =

$\frac{46}{99}$ The fraction cannot be simplified any further.

Hence 0.4646.... =

$\frac{46}{99}$**Example 2****:** Convert 0.55555.........

to fraction.

**Solution:** __Step 1____:__ Consider x to be 0.55555..... ...1

In the given problem we see that only one digit is repeated so multiply the above equation by 10.

Now we get, 10x = 5.5555 ---2

__Step 2:__ Subtract equation 1 from equation 2

9x = 5

__Step 3____:__ $\rightarrow$ x =

$\frac{5}{9}$The fraction cannot be simplified, therefore

$\frac{5}{9}$ is in its simplest form.

Hence, 0.55555 =

$\frac{5}{9}$

**Example 3****:** Write $\frac{2}{3}$ as a repeating decimal.

**Solution:** Use long division method to divide 3 by 2.

Add a decimal point and atleast one zero.

Subtract 18 from 20, we are left with remainder 2. Add another zero, now we have another 20. Divide again. When we subtract 18 from 20, the remainder is 2 again. Add another zero, again we are left with 20. If we repeat this process infinity times also will left with same number.

Write the repeating decimals using a bar over the repeated digit.

This implies, $\frac{2}{3}$ = $0.\bar{6}$