Convert 0.27¯¯¯¯¯

Question

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

27/99
Start Fraction 27 over 99 end fraction

27/100
Start Fraction 27 over 100 end fraction

3/11
Start Fraction 3 over 11 end fraction

1/27

Answer 1

Let \( x = 0.27\overline{27} \), which means \( x = 0.27272727\ldots \).

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

\[ 100x = 27.272727\ldots \]

Now we can set up the equations:

1. \( x = 0.272727\ldots \)
2. \( 100x = 27.272727\ldots \)

Next, we subtract the first equation from the second:

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

This simplifies to:

\[ 99x = 27 \]

Now, we solve for \( x \):

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

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

Dividing both the numerator and the denominator by 9 gives:

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

So, the simplified form of \( 0.27\overline{27} \) as a rational number is:

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

Thus, the correct response is:

3/11