Question

Express the following recurring decimal in the form of a rational number `bb(("in fraction form" p/q)`:

`0.2bar(73)`

Answer 1

Given the recurring decimal `0.2bar(73)`, we want to express it as a rational number in the form `p/q`.

Step 1: Let x = 0.2737373...

Step 2: Separate the non-repeating and repeating parts:

The non-repeating part after the decimal point is 2.

The repeating part is “73”.

Step 3: Multiply by powers of 10 to shift decimals:

Multiply by 10 to move past the non-repeating part 10x = 2.737373...

Multiply by 1000 because repeating cycle length is 2 digits, plus 1 digit non-repeating, total 3 digits to move past one full cycle of non-repeating + repeating 1000x = 273.737373...

Step 4: Subtract the two equations:

1000x – 10x = 273.737373... – 2.737373... 

990x = 271

`x = 271/990`

Thus, the fraction form of `0.2bar(73)` is `271/990`.