1. The problem is to multiply the fractions $\frac{4}{5}$, $\frac{9}{10}$, and $\frac{1}{5}$.\n\n2. The formula for multiplying fractions is: $$\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}$$\n\n3. Multiply the numerators: $4 \times 9 \times 1 = 36$.\n\n4. Multiply the denominators: $5 \times 10 \times 5 = 250$.\n\n5. So, the product is $$\frac{36}{250}$$.\n\n6. Simplify the fraction by dividing numerator and denominator by their greatest common divisor (GCD). The GCD of 36 and 250 is 2.\n\n7. Divide numerator and denominator by 2: $$\frac{36 \div 2}{250 \div 2} = \frac{18}{125}$$.\n\n8. The simplified product of the fractions is $$\frac{18}{125}$$.