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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.
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