1. **State the problem:** We need to find the value of $y$ when $x=\frac{4}{5}$ in the formula $$y = \frac{4}{x} + \sqrt{x} + 0.2 - 5x.$$\n\n2. **Write down the formula:** $$y = \frac{4}{x} + \sqrt{x} + 0.2 - 5x.$$\n\n3. **Substitute $x=\frac{4}{5}$ into the formula:** $$y = \frac{4}{\frac{4}{5}} + \sqrt{\frac{4}{5}} + 0.2 - 5 \times \frac{4}{5}.$$\n\n4. **Simplify each term:**\n- $$\frac{4}{\frac{4}{5}} = 4 \times \frac{5}{4} = 5.$$\n- $$\sqrt{\frac{4}{5}} = \frac{\sqrt{4}}{\sqrt{5}} = \frac{2}{\sqrt{5}} = \frac{2\sqrt{5}}{5}.$$\n- $$0.2$$ is already simplified.\n- $$5 \times \frac{4}{5} = 4.$$\n\n5. **Put it all together:** $$y = 5 + \frac{2\sqrt{5}}{5} + 0.2 - 4.$$\n\n6. **Combine like terms:** $$5 - 4 + 0.2 = 1.2.$$\n\n7. **Final expression:** $$y = 1.2 + \frac{2\sqrt{5}}{5}.$$\n\n8. **Approximate the value:**\n- $$\sqrt{5} \approx 2.236.$$\n- $$\frac{2 \times 2.236}{5} = \frac{4.472}{5} = 0.8944.$$\n- $$y \approx 1.2 + 0.8944 = 2.0944.$$\n\n**Answer:** $$y \approx 2.094.$$