(6) $1\frac{2}{3}\times \frac{2}{5}=$

Step 1 ：Convert the mixed number $1\frac{2}{3}$ to an improper fraction. The formula to convert a mixed number to an improper fraction is $\text{{whole number}}\times \text{{denominator}}+\text{{numerator}}$. So, $1\frac{2}{3}$ becomes $1\times 3+2=5$, so the improper fraction is $\frac{5}{3}$.

Step 2 ：Next, multiply the two fractions $\frac{5}{3}$ and $\frac{2}{5}$. The formula to multiply fractions is $\text{{numerator1}}\times \text{{numerator2}}$ over $\text{{denominator1}}\times \text{{denominator2}}$. So, $\frac{5}{3}\times \frac{2}{5}=\frac{5\times 2}{3\times 5}$.

Step 3 ：Simplify the result. The numerator and denominator have a common factor of 5, so we can simplify the fraction by dividing both the numerator and the denominator by 5. So, $\frac{5\times 2}{3\times 5}=\frac{2}{3}$.

Step 4 ：$\boxed{{\displaystyle \frac{2}{3}}}$ is the final answer.