Question

Convert 0.27¯¯¯¯¯
to a rational number in simplest form.(1 point)
Responses

311
Start Fraction 3 over 11 end fraction

127
Start Fraction 1 over 27 end fraction

27100
Start Fraction 27 over 100 end fraction

2799

Answer 1

To convert the repeating decimal \( 0.27\overline{27} \) into a rational number in simplest form, let \( x = 0.272727...\).

To eliminate the repeating part, we can multiply \( x \) by 100 (since the repeating part consists of two digits):

\[
100x = 27.272727...\quad (1)
\]
Now, we can subtract the original equation \( x = 0.272727... \) from this new equation:

\[
100x - x = 27.272727... - 0.272727...
\]

This simplifies to:

\[
99x = 27
\]

Now, to find \( x \):

\[
x = \frac{27}{99}
\]

Next, we simplify the fraction \( \frac{27}{99} \). The greatest common divisor (GCD) of 27 and 99 is 9.

So we divide both the numerator and the denominator by 9:

\[
x = \frac{27 \div 9}{99 \div 9} = \frac{3}{11}
\]

Thus, the repeating decimal \( 0.27\overline{27} \) can be expressed as the rational number in simplest form:

\[
\frac{3}{11}
\]

The correct response is:

**Start Fraction 3 over 11 end Fraction**