Question

Convert 0.72 with a bar over 72 to a rational number and simplest form

Answer 1

To convert the repeating decimal \( 0.72\overline{72} \) to a rational number, we can set it equal to a variable:

Let \( x = 0.727272\ldots \)

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

\[
100x = 72.727272\ldots
\]

Now, we see that the right-hand side can also be expressed in terms of \( x \):

\[
100x = 72 + x
\]

Next, we solve for \( x \) by isolating it:

\[
100x - x = 72
\]
\[
99x = 72
\]
\[
x = \frac{72}{99}
\]

Now, we simplify \( \frac{72}{99} \). To do this, we find the greatest common divisor (GCD) of 72 and 99.

The prime factorization of 72 is \( 2^3 \times 3^2 \), and the prime factorization of 99 is \( 3^2 \times 11 \). The GCD is \( 3^2 = 9 \).

Now, we simplify:

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

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

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