To convert the repeating decimal \( 0.72\overline{72} \) to a rational number, we will use a method that involves setting up an equation.
Let \( x = 0.727272\ldots \).
Now, to eliminate the repeating part, we multiply \( x \) by 100 (since the repeating part has 2 digits):
\[ 100x = 72.727272\ldots \]
Next, we can subtract the original \( x \) from this equation:
\[ 100x - x = 72.727272\ldots - 0.727272\ldots \] \[ 99x = 72 \]
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 prime factorizations are as follows:
- \( 72 = 2^3 \times 3^2 \)
- \( 99 = 3^2 \times 11 \)
The common factor is \( 3^2 = 9 \).
Now we divide both the numerator and the denominator by their GCD, which is 9:
\[ \frac{72 \div 9}{99 \div 9} = \frac{8}{11} \]
Thus, the simplifying of \( 0.72\overline{72} \) results in:
\[ \boxed{\frac{8}{11}} \]
2 answers