Question

Convert 0.27 to a rational number in simplest form. (1 point)
¯¯¯¯¯
27
100
27
99
3
11
1
27

Answer 1

To convert the repeating decimal \( 0.27 \overline{27} \) to a rational number, we can denote it as \( x = 0.272727...\).

To eliminate the repeating part, we can multiply \( x \) 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, we can subtract the first equation from the second:

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

This simplifies to:

\[
99x = 27
\]

Now, we can solve for \( x \):

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

Next, we simplify the fraction \( \frac{27}{99} \). Both the numerator and denominator can be divided by 9:

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

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

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