4. Convert.454545454545. (repeating decimal) to a rational number in simplest form. Show your work and make sure I understand what ur explaining

Answer 1

To convert the repeating decimal \( 0.454545454545... \) (which we can denote as \( x = 0.\overline{45} \)) into a fraction, follow these steps:

Step 1: Assign the repeating decimal to a variable.

Let: \[ x = 0.454545454545... \]

Step 2: Multiply by a power of 10.

Since the repeating part "45" has two digits, we multiply both sides of the equation by \( 100 \) (which is \( 10^2 \)): \[ 100x = 45.454545454545... \] This step helps us to align the decimals.

Step 3: Set up a system of equations.

Now we have two equations:

1. \( x = 0.454545454545... \)
2. \( 100x = 45.454545454545... \)

Step 4: Subtract the two equations.

Now, we will subtract the first equation from the second: \[ 100x - x = 45.454545454545... - 0.454545454545... \] This simplifies to: \[ 99x = 45 \]

Step 5: Solve for \( x \).

Now, divide both sides by \( 99 \): \[ x = \frac{45}{99} \]

Step 6: Simplify the fraction.

To simplify \( \frac{45}{99} \), we need to find the greatest common divisor (GCD) of \( 45 \) and \( 99 \).

1. Finding the GCD:

   • The factors of \( 45 = 3^2 \times 5 \).
   • The factors of \( 99 = 3^2 \times 11 \).
   • The GCD is \( 9 \) (which is \( 3^2 \)).

2. Dividing the numerator and denominator by \( 9 \): \[ \frac{45 \div 9}{99 \div 9} = \frac{5}{11} \]

Conclusion

Therefore, the repeating decimal \( 0.454545454545... \) can be expressed as the fraction \( \frac{5}{11} \).

So: \[ 0.454545454545... = \frac{5}{11} \]