Let's tackle each part of the question step-by-step:
### Part (a)
Convert [tex]\(0.05\)[/tex] to a fraction.
1. Realize that [tex]\(0.05\)[/tex] is a finite decimal.
2. Since [tex]\(0.05\)[/tex] has two decimal places, it can be written as [tex]\(\frac{5}{100}\)[/tex].
3. Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor (GCD), which is 5:
[tex]\[
\frac{5 \div 5}{100 \div 5} = \frac{1}{20}
\][/tex]
Therefore, [tex]\(0.05\)[/tex] as a fraction is [tex]\(\frac{1}{20}\)[/tex].
### Part (b)
Convert [tex]\(0.2\dot{5}\)[/tex] (which is [tex]\(0.25555\ldots\)[/tex]) to a fraction.
1. Let's set up the repeating decimal. Let [tex]\(x = 0.25555\ldots\)[/tex].
2. Realize that the repeating part starts after the "2":
[tex]\[
x = 0.25555\ldots
\][/tex]
3. To eliminate the repeating part, multiply [tex]\(x\)[/tex] by 10 so that the repeating part lines up:
[tex]\[
10x = 2.5555\ldots
\][/tex]
4. Now, we have two equations:
[tex]\[
\begin{cases}
x = 0.25555\ldots & \text{(i)} \\
10x = 2.5555\ldots & \text{(ii)}
\end{cases}
\][/tex]
5. Subtract equation (i) from equation (ii):
[tex]\[
10x - x = 2.5555\ldots - 0.25555\ldots
\][/tex]
[tex]\[
9x = 2.3
\][/tex]
6. Solve for [tex]\(x\)[/tex] by dividing both sides by 9:
[tex]\[
x = \frac{2.3}{9}
\][/tex]
7. As a fraction, [tex]\(2.3\)[/tex] needs to be converted into an improper fraction first. Recall that [tex]\(2.3 = 2 + \frac{3}{10} = \frac{23}{10}\)[/tex]:
[tex]\[
x = \frac{\frac{23}{10}}{9} = \frac{23}{10} \times \frac{1}{9} = \frac{23}{90}
\][/tex]
Thus, the fraction for [tex]\(0.2\dot{5} \)[/tex] in its simplest form is [tex]\(\frac{23}{90}\)[/tex].
