Certainly! Let's convert the repeating decimal 0.4545... into a rational number in its simplest form. Here are the detailed steps:
1. Let [tex]\( x \)[/tex] be the repeating decimal:
[tex]\[ x = 0.4545\ldots \][/tex]
2. Multiply both sides of the equation by 100 to shift the decimal point two places to the right:
[tex]\[ 100x = 45.4545\ldots \][/tex]
3. Set up the original equation and the new equation together:
[tex]\[ x = 0.4545\ldots \][/tex]
[tex]\[ 100x = 45.4545\ldots \][/tex]
4. Subtract the first equation from the second equation to eliminate the repeating part:
[tex]\[ 100x - x = 45.4545\ldots - 0.4545\ldots \][/tex]
[tex]\[ 99x = 45 \][/tex]
5. Solve for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{45}{99} \][/tex]
6. Reduce the fraction to its simplest form:
- To simplify [tex]\(\frac{45}{99}\)[/tex], we need to find the greatest common divisor (GCD) of 45 and 99.
- The GCD of 45 and 99 is 9.
- Divide both the numerator and the denominator by their GCD (9):
[tex]\[ \frac{45 \div 9}{99 \div 9} = \frac{5}{11} \][/tex]
Therefore, [tex]\( 0.4545\ldots \)[/tex] expressed as a rational number in its lowest form is:
[tex]\[ \frac{5}{11} \][/tex]
