(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 ：\(\boxed{\frac{2}{3}}\) is the final answer.