Convert 0.27¯¯¯¯¯ (it's a repeating decimal)

Question

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

27/100
Start Fraction 27 over 100 end fraction

1/27
Start Fraction 1 over 27 end fraction

3/11
Start Fraction 3 over 11 end fraction

27/99

Answer 1

To convert the repeating decimal \( 0.27\overline{27} \) to a rational number, we can set up the equation as follows:

Let \( x = 0.272727...\).

To eliminate the repeating part, multiply \( x \) by 100 (since the repeat is two digits long):

\[ 100x = 27.272727..., \]

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 \( x \):

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

Next, simplify \( \frac{27}{99} \):

The greatest common divisor (GCD) of 27 and 99 is 9, so we can divide both the numerator and the denominator by 9:

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

Thus, the simplified rational number is

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

So the correct answer is:

3/11.