Question

4. Convert.454545454545. (repeating decimal) to a rational number in simplest form. Show your work

Answer 1

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

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:
\[
\boxed{\frac{5}{11}}
\]