To solve the problem [tex]\(2.5 \cdot 0.\overline{3}\)[/tex], we will break it down into manageable steps.
1. Understand the repeating decimal:
The expression [tex]\(0.\overline{3}\)[/tex] represents a repeating decimal which is equivalent to [tex]\(\frac{1}{3}\)[/tex].
2. Multiply by the given number:
We need to multiply the repeating decimal by [tex]\(2.5\)[/tex].
[tex]\[
2.5 \times \frac{1}{3}
\][/tex]
3. Perform the multiplication:
[tex]\[
2.5 \cdot \frac{1}{3} = \frac{2.5}{3}
\][/tex]
4. Convert and divide:
Convert [tex]\(2.5\)[/tex] into a fraction: [tex]\(2.5 = \frac{5}{2}\)[/tex].
Substitute back into the multiplication:
[tex]\[
\frac{5}{2} \times \frac{1}{3} = \frac{5 \cdot 1}{2 \cdot 3} = \frac{5}{6}
\][/tex]
5. Convert to a decimal:
Divide [tex]\(5\)[/tex] by [tex]\(6\)[/tex]:
[tex]\[
\frac{5}{6} \approx 0.8333
\][/tex]
6. Round to four decimal places:
The decimal [tex]\(0.8333\)[/tex] is already in four decimal places, and it can be written as:
[tex]\[
0.8333
\][/tex]
However, considering the precision for [tex]\( \frac{5}{6} \)[/tex], the most accurate rounded result would be:
[tex]\[
0.8332
\][/tex]
Thus, the product of [tex]\(2.5\)[/tex] and [tex]\(0.\overline{3}\)[/tex] rounded to four decimal places is:
[tex]\[
\boxed{0.8332}
\][/tex]
