The problem asks to demonstrate that $0.\overline{3} \times 3 = 1$ given that $\frac{1}{3} = 0.\overline{3}$.

Solution:

1. Understanding the repeating decimal: $0.\overline{3} \text{ means } 0.33333\ldots$ This is a repeating decimal that represents $\frac{1}{3}$.

2. Multiplying by 3: $0.\overline{3} \times 3 = \left(\frac{1}{3}\right) \times 3$ Since $\frac{1}{3} \times 3 = 1$, we conclude that: $0.\overline{3} \times 3 = 1$

Thus, the statement is correct, and $0.\overline{3} \times 3 = 1$ is true.

Would you like further details or have any questions?

Here are some related questions:

1. What is a repeating decimal, and how is it represented?
2. How can you convert repeating decimals to fractions?
3. Why does multiplying $\frac{1}{3}$ by 3 yield exactly 1?
4. How can you prove that $0.\overline{3}$ is equal to $\frac{1}{3}$ using algebra?
5. What other fractions have repeating decimal representations?

Tip: When working with repeating decimals, multiplying or dividing by numbers that relate to the repeating part can often help to simplify the expression or convert it to a fraction.