Question

Convert 0.72_^ to a rational number in the simplest form

Answer 1

To convert the repeating decimal \( 0.72\overline{72} \) (where "72" repeats indefinitely) into a rational number, we can use the following steps:

Let \( x = 0.727272\ldots \).

To eliminate the repeating part, we can multiply \( x \) by 100 (because the repeating part has 2 digits):

\[
100x = 72.727272\ldots
\]

Now we have two equations:

1. \( x = 0.727272\ldots \)
2. \( 100x = 72.727272\ldots \)

Next, we can subtract the first equation from the second:

\[
100x - x = 72.727272\ldots - 0.727272\ldots
\]

This simplifies to:

\[
99x = 72
\]

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

\[
x = \frac{72}{99}
\]

Next, we can simplify the fraction \( \frac{72}{99} \).

To do this, we find the greatest common divisor (GCD) of 72 and 99. The GCD is 9.

Now divide both the numerator and the denominator by their GCD:

\[
\frac{72 \div 9}{99 \div 9} = \frac{8}{11}
\]

So, the repeating decimal \( 0.72\overline{72} \) as a rational number in simplest form is:

\[
\frac{8}{11}
\]