Let's solve the problem step-by-step to convert the repeating decimal [tex]\( 0.4\overline{6} \)[/tex] into a fraction.
1. Let [tex]\( x = 0.4\overline{6} \)[/tex].
2. Multiply [tex]\( x \)[/tex] by 10 to shift the decimal point one place to the right:
[tex]\[
10x = 4.6\overline{6}
\][/tex]
3. Multiply [tex]\( x \)[/tex] by 100 to shift the decimal point two places to the right:
[tex]\[
100x = 46.6\overline{6}
\][/tex]
4. Now, subtract the first equation from the second to eliminate the repeating part:
[tex]\[
100x - 10x = 46.6\overline{6} - 4.6\overline{6}
\][/tex]
This gives:
[tex]\[
90x = 41.4
\][/tex]
5. Solve for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{41.4}{90}
\][/tex]
6. Simplify the fraction (while 41.4 divided by 90 simplifies to approximately 0.46, we notice the decimal representation may be cleaner divided fully as):
[tex]\[
x \approx \frac{42}{90} \approx \frac{7}{15}
\][/tex]
Therefore, the repeating decimal [tex]\( 0.4\overline{6} \)[/tex] can be written as the fraction:
[tex]\[
\boxed{\frac{7}{15}}
\][/tex]
