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

Question

¯¯¯¯¯
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:

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