To convert the repeating decimal \(0.72\ldots\) (which can be represented as \(0.727272...\)) into a rational number, we can use the following method:
Let \(x = 0.727272...\).
Next, we can multiply both sides of this equation by 100, which is a power of 10 that shifts the decimal point two places to the right:
\[ 100x = 72.727272... \]
Now, we can see that the decimal part \(0.727272...\) on the right-hand side is the same as our original equation for \(x\):
\[ 100x = 72 + x \]
Now we can isolate \(x\) by subtracting \(x\) from both sides:
\[ 100x - x = 72 \] \[ 99x = 72 \]
Now, we solve for \(x\) by dividing both sides by 99:
\[ x = \frac{72}{99} \]
Next, we simplify \(\frac{72}{99}\). We can 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 of 72 and 99 is \(3^2 = 9\).
Now we divide both the numerator and the denominator by their GCD:
\[ \frac{72 \div 9}{99 \div 9} = \frac{8}{11} \]
Thus, the simplest form of \(0.727272...\) as a rational number is:
\[ \boxed{\frac{8}{11}} \]
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