1. **State the problem:** Calculate the exact value of $$4 \frac{1}{5} - \left(1 \frac{1}{9} \times 3\right)$$. 2. **Convert mixed numbers to improper fractions:** $$4 \frac{1}{5} = \frac{4 \times 5 + 1}{5} = \frac{21}{5}$$ $$1 \frac{1}{9} = \frac{1 \times 9 + 1}{9} = \frac{10}{9}$$ 3. **Calculate the multiplication inside the parentheses:** $$1 \frac{1}{9} \times 3 = \frac{10}{9} \times 3 = \frac{10}{9} \times \frac{3}{1} = \frac{30}{9}$$ Simplify $$\frac{30}{9}$$ by dividing numerator and denominator by 3: $$\frac{\cancel{30}^{10}}{\cancel{9}^{3}} = \frac{10}{3}$$ 4. **Subtract the fractions:** $$\frac{21}{5} - \frac{10}{3}$$ Find common denominator: $$15$$ Rewrite fractions: $$\frac{21}{5} = \frac{21 \times 3}{5 \times 3} = \frac{63}{15}$$ $$\frac{10}{3} = \frac{10 \times 5}{3 \times 5} = \frac{50}{15}$$ Subtract: $$\frac{63}{15} - \frac{50}{15} = \frac{63 - 50}{15} = \frac{13}{15}$$ 5. **Final answer:** $$\boxed{\frac{13}{15}}$$