How to express a repeating decimal number as a ratio of integers?

The non-terminating but repeating decimal expansion means that although the decimal representation has an infinite number of digits, there is a repetitive pattern to it.

Answer: To express a repeating decimals number as a ratio of integers we follow certain steps.

1) Assume the repeating decimal to be equal to some variable x
2) Write the number without using a bar and equal to x
3) Determine the number of digits having a bar on their heads or the number of digits before the bar for mixed recurring decimal.
4) If the repeating number is the same digit after decimal such as 0.2222... then multiply by 10, if repetition of the digits is in pairs of two numbers such as 0.7878... then multiply by 100 and so on.
5) Subtract the equation formed by step 2 and step 4.
6) Find the value of x in the simplest form.

Explanation:

Let's understand how are repeating decimals expressed as a ratio of integers through an example.

Let's take an example of a repeating number 0.6666...

Let , x = 0.666...    -------------- (1)

Multiplying 10 on both the sides we get,

10x = 6.666.. ----------------- (2)  (This has to be chosen in such a way that on subtracting we get rid of the decimal)

Subtracting the two equations

10x - x = 6.666 - 0.666

9x = 6

x = 6/9 or 2/3

Let's take another example to understand this

Let x = 0.6565...  --------------------- (1)

Multiplying 100 on both the sides

100x = 65.6565... -------------------- (2)

Subtracting the above equations

100x - x = 65.6565 - 0.6565

99x = 65

x = 65/99

Thus, the above repeating decimals has been expressed as a ratio of integers.