Question

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

2799
Start Fraction 27 over 99 end fraction

311
Start Fraction 3 over 11 end fraction

127
Start Fraction 1 over 27 end fraction

27100

Answer 1

To convert \( 0.27\overline{27} \) (where the "27" is the repeating part) to a rational number, you can use the following method:

Let \( x = 0.27\overline{27} \).

This can be expressed as:
\[ x = 0.272727... \]

To eliminate the repeating decimal, multiply both sides by 100 (since the repeating part has two digits):
\[ 100x = 27.272727... \]

Now, we have two equations:
1. \( x = 0.272727... \)
2. \( 100x = 27.272727... \)

Next, subtract the first equation from the second:
\[
100x - x = 27.272727... - 0.272727...
\]
\[
99x = 27
\]

Now, solve for \( x \):
\[
x = \frac{27}{99}
\]

We can simplify this fraction:
\[
\frac{27 \div 9}{99 \div 9} = \frac{3}{11}
\]

Thus, \( 0.27\overline{27} \) as a rational number in simplest form is:
\[
\frac{3}{11}
\]

So the correct response is:
**Start Fraction 3 over 11 end fraction**.