Express \(0.\overline{52}\) in the form \(\frac{p}{q}\) where p and q are integers and q ≠ 0

Given:

Express 0.\(\overline{52}\) in the form \(\dfrac{p}{q}\), where p and q are integers and q ≠ 0.

Formula used:

Let x = 0.\(\overline{52}\)

Multiplying both sides by 100 to shift the decimal point two places to the right:

100x = 52.\(\overline{52}\)

Then, subtract the original equation from this new equation:

Calculations:

100x = 52.\(\overline{52}\)

x = 0.\(\overline{52}\)

⇒ 100x - x = 52.\(\overline{52}\) - 0.\(\overline{52}\)

⇒ 99x = 52

⇒ x = \(\dfrac{52}{99}\)

∴ The correct answer is option (2).