Question

Convert 0.27 to a rational number in simplest form. (0.27 is a repeating decimal)

Answer 1

To convert the repeating decimal \(0.27\) (which can be written as \(0.272727...\)) into a rational number, we can follow these steps:

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

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

\[
100x = 27.272727...
\]

Now, we have two equations:

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

Next, we can subtract equation (1) from equation (2):

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

This simplifies to:

\[
99x = 27
\]

Now, dividing both sides by 99 gives us:

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

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

Now, we divide the numerator and the denominator by 9:

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

Thus, the repeating decimal \(0.27\) can be expressed as the rational number \(\frac{3}{11}\) in simplest form.