To convert the repeating decimal \( 0.454545454545\ldots \) (which has the repeating block "45") into a rational number, we can use a variable to represent the decimal and then manipulate it algebraically.
Let's define: \[ x = 0.454545454545\ldots \]
Since the repeating part "45" has 2 digits, we can multiply \( x \) by 100 (which shifts the decimal point 2 places to the right): \[ 100x = 45.454545454545\ldots \]
Now we have two equations:
- \( x = 0.454545454545\ldots \)
- \( 100x = 45.454545454545\ldots \)
Next, we can subtract the first equation from the second: \[ 100x - x = 45.454545454545\ldots - 0.454545454545\ldots \] This simplifies to: \[ 99x = 45 \]
Now, we can solve for \( x \) by dividing both sides by 99: \[ x = \frac{45}{99} \]
Next, we need to simplify the fraction \( \frac{45}{99} \). To do this, we can find the greatest common divisor (GCD) of 45 and 99. The prime factorization of both numbers is:
- \( 45 = 3^2 \times 5 \)
- \( 99 = 3^2 \times 11 \)
The GCD of 45 and 99 is \( 9 \) (since \( 3^2 \) is the common factor). We can now simplify \( \frac{45}{99} \) by dividing the numerator and the denominator by their GCD: \[ \frac{45 \div 9}{99 \div 9} = \frac{5}{11} \]
Thus, the repeating decimal \( 0.454545454545\ldots \) as a rational number in simplest form is: \[ \boxed{\frac{5}{11}} \]
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