Question

Which is 0.ModifyingAbove 54 with bar converted to a simplified fraction?
StartFraction 27 Over 50 EndFraction
StartFraction 54 Over 100 EndFraction
StartFraction 6 Over 11 EndFraction
StartFraction 54 Over 99 EndFraction

Answer 1

To convert \( 0.\overline{54} \) to a simplified fraction, we denote it as \( x = 0.\overline{54} \). This means:

\[
x = 0.54545454\ldots
\]

To eliminate the repeating decimal, we multiply both sides of the equation by 100 (since the repeating part is two digits long):

\[
100x = 54.54545454\ldots
\]

Now, we can set up an equation by subtracting the original \( x \) from this new equation:

\[
100x - x = 54.54545454\ldots - 0.54545454\ldots
\]

This simplifies to:

\[
99x = 54
\]

Next, we solve for \( x \):

\[
x = \frac{54}{99}
\]

Now, we simplify the fraction \( \frac{54}{99} \). We can find the greatest common divisor (GCD) of 54 and 99. The GCD is 9.

Now divide the numerator and denominator by 9:

\[
\frac{54 \div 9}{99 \div 9} = \frac{6}{11}
\]

Thus, \( 0.\overline{54} \) converted to a simplified fraction is:

\[
\frac{6}{11}
\]

Among the given options, the answer is:

\[
\frac{6}{11}
\]