case a) the repeating decimal is 36
Let
[tex]x=0.1363636..[/tex]
Multiply x by a power of [tex]10[/tex], one that keeps the decimal part of the number the same:
[tex]1,000x=136.3636..[/tex]
[tex]10x=1.3636..[/tex]
Subtract [tex]10x[/tex] from [tex]\\1000x[/tex]
[tex]1,000x-10x=136.3636..-1.3636..=135[/tex]
The repeating decimals should cancel out
[tex]\\990x=135[/tex]
solve for x
Divide by [tex]990[/tex] both sides
[tex]990x/990=135/990[/tex]
[tex]x=135/990[/tex]
Simplify
Divide by [tex]45[/tex] both numerator and denominator
[tex]x=3/22[/tex]
therefore
the answer case a) is
The fraction in simplest form is [tex]3/22[/tex]
case b) the repeating decimal is 136
Let
[tex]x=0.136136136...[/tex]
Multiply x by a power of [tex]10[/tex], one that keeps the decimal part of the number the same:
[tex]1,000x=136.136136...[/tex]
Subtract [tex]x[/tex] from [tex]\\1000x[/tex]
[tex]1,000x-x=136.136136...-0.136136...=136[/tex]
The repeating decimals should cancel out
[tex]\\999x=136[/tex]
solve for x
Divide by [tex]999[/tex] both sides
[tex]999x/999=136/999[/tex]
[tex]x=136/999[/tex] -----> irreducible
therefore
the answer case b) is
The fraction in simplest form is [tex]136/999[/tex]
