Show that 0.2353535... = \(0.2\overline{35}\) can be expressed in the form of p/q, where p and q are integers and q is not equal to zero.

The problem is based on the conversion of decimal numbers to a rational form.

Answer: 0.2353535... = \(0.2\overline{35}\) can be expressed as 233/990 i.e., in the form of p/q, where p and q are integers and q is not equal to zero.

Let's proceed with the conversion thereby establishing the proof of the given statement.

Explanation:

A rational number can have two types of decimal representations (expansions):

• Terminating
• Non-terminating but repeating

0.2353535... is a non-terminating but repeating decimal, it is denoted by \(0.2\overline{35}\).

let x = 0.2353535... … (1)

Multiplying both sides of (1) by 100:

100x = 235.353535... … (2)

Multiplying both sides of (2) by 10:

10x = 2.353535... … (3)

Now, subtract (3) from (2):

100x - 10x = (235.353535...) - (2.353535...)

990x = 233

x = 233/990

Therefore, 0.2353535... = \(0.2\overline{35}\) = 233/990 can be expressed in the rational form.

Thus, 0.2353535...= \(0.2\overline{35}\) can be expressed in the form of p/q, where p and q are integers and q is not equal to zero as 233/990.