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 to multiply the numerators together and multiply the denominators together: $$\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}$$\n\n3. Applying this to our fractions: $$\frac{4}{5} \times \frac{9}{10} \times \frac{1}{5} = \frac{4 \times 9 \times 1}{5 \times 10 \times 5}$$\n\n4. Calculate the numerator: $4 \times 9 \times 1 = 36$.\n\n5. Calculate the denominator: $5 \times 10 \times 5 = 250$.\n\n6. So the product is $$\frac{36}{250}$$.\n\n7. Simplify the fraction by dividing numerator and denominator by their greatest common divisor (GCD). The GCD of 36 and 250 is 2.\n\n8. Divide numerator and denominator by 2: $$\frac{36 \div 2}{250 \div 2} = \frac{18}{125}$$.\n\n9. The fraction $\frac{18}{125}$ cannot be simplified further.\n\nFinal answer: $$\frac{18}{125}$$.