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