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 set it equal to a variable.

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

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

Next, to eliminate the repeating part, we can multiply by 100 (since the repeating block "72" has two digits): \[ 100x = 72.727272\ldots \]

Now we can set up an equation: \[ 100x = 72.727272\ldots \ x = 0.727272\ldots \]

Next, we subtract the second equation from the first: \[ 100x - x = 72.727272\ldots - 0.727272\ldots \ 99x = 72 \]

Now, we can solve for \( x \): \[ x = \frac{72}{99} \]

Next, we need to simplify \( \frac{72}{99} \). We can find the greatest common divisor (GCD) of 72 and 99. The GCD is 9.

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

Thus, the repeating decimal \( 0.\overline{72} \) as a rational number in simplest form is: \[ \boxed{\frac{8}{11}} \]