Question

Convert 0.¯¯72
to a rational number in simplest form.

Answer 1

To convert the repeating decimal \( 0.\overline{72} \) to a rational number, we can follow these steps:

Let \( x = 0.\overline{72} \).

This means:
\[
x = 0.72727272\ldots
\]

To eliminate the repeating part, we can multiply \( x \) by \( 100 \) (since the repeat is 2 digits long):
\[
100x = 72.727272\ldots
\]

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, divide both sides by \( 99 \):
\[
x = \frac{72}{99}
\]

Now, we need to simplify \( \frac{72}{99} \). We find the greatest common divisor (GCD) of \( 72 \) and \( 99 \).

The prime factorization of \( 72 \) is:
\[
72 = 2^3 \times 3^2
\]

The prime factorization of \( 99 \) is:
\[
99 = 3^2 \times 11
\]

The GCD can be found as follows: the common factor is \( 3^2 \), which is \( 9 \).

Now we simplify \( \frac{72}{99} \) by dividing both the numerator and the denominator by their GCD:
\[
\frac{72 \div 9}{99 \div 9} = \frac{8}{11}
\]

Thus, the rational number in simplest form that corresponds to \( 0.\overline{72} \) is:
\[
\boxed{\frac{8}{11}}
\]