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:
- \( x = 0.727272\ldots \)
- \( 100x = 72.727272\ldots \)
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: \[ \boxed{\frac{8}{11}} \]
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