Show that 0.235555......can be expressed in the form (p)/(q) where p and q are integers and q =/= 0.​

Question

Show that 0.235555......can be expressed in the form


`(p)/(q)`
where p and q are integers and q =/= 0.​

Answer 1

Answer and Step-by-step explanation:

Suppose x = 0.2355555... .

Multiply x by 100 and 1000:

100x = 23.5555....

1000x = 235.5555....

Let's subtract the first equation from the second:

1000x - 100x = 235.5555.... - 23.555.....

Because the decimal part for each goes on forever, we can simply cancel them out in our subtraction statement:

900x = 235 - 23 = 212

Divide both sides by 900:

x = 212/900 = 53/225

Thus, since x = 0.235555..., we know that:

0.23555.... = 53/225

Since p = 53 and q = 225 are integers, we have proven that this repeating decimal can be written as a fraction.

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