Certainly! Let's work through the problem step by step.
1. Convert [tex]\(0.\dot{4}\)[/tex] to a fraction:
The repeating decimal [tex]\(0.\dot{4}\)[/tex] can be expressed as:
[tex]\[
0.\dot{4} = \frac{4}{9}
\][/tex]
2. Convert [tex]\(0.0\dot{7}\)[/tex] to a fraction:
The repeating decimal [tex]\(0.0\dot{7}\)[/tex] can be expressed as follows. Let [tex]\(x = 0.0\dot{7}\)[/tex].
[tex]\( x = 0.07777\ldots \)[/tex]
To find [tex]\(x\)[/tex] as a fraction, let's multiply [tex]\(x\)[/tex] by 10 to make the repeating part move one decimal place to the left:
[tex]\[
10x = 0.77777\ldots
\][/tex]
Now multiply by an additional 10:
[tex]\[
100x = 7.7777\ldots
\][/tex]
Next, subtract the first equation from the second:
[tex]\[
100x - 10x = 7.7777\ldots - 0.7777\ldots
\][/tex]
[tex]\[
90x = 7
\][/tex]
[tex]\[
x = \frac{7}{90}
\][/tex]
So,
[tex]\[
0.0\dot{7} = \frac{7}{90}
\][/tex]
3. Add the fractions [tex]\(\frac{4}{9}\)[/tex] and [tex]\(\frac{7}{90}\)[/tex]:
To add these fractions, we first need a common denominator. The least common multiple (LCM) of 9 and 90 is 90. Convert [tex]\(\frac{4}{9}\)[/tex] to an equivalent fraction with the denominator 90:
[tex]\[
\frac{4}{9} = \frac{4 \times 10}{9 \times 10} = \frac{40}{90}
\][/tex]
Now, we add the two fractions:
[tex]\[
\frac{4}{9} + \frac{7}{90} = \frac{40}{90} + \frac{7}{90} = \frac{40 + 7}{90} = \frac{47}{90}
\][/tex]
4. Simplify the fraction (if necessary):
The fraction [tex]\(\frac{47}{90}\)[/tex] is already in its simplest form because 47 is a prime number and does not have any common factors with 90.
Thus,
[tex]\[
0.\dot{4} + 0.0\dot{7} = \frac{47}{90}
\][/tex]
