1. **State the problem:** We are given the original ratio of workers to supervisors as 8:2. 2. **Express the original numbers:** Let the number of workers be $8x$ and the number of supervisors be $2x$ for some positive number $x$. 3. **After adding supervisors:** 4 supervisors are added, so the new number of supervisors is $2x + 4$. 4. **New ratio given:** The new ratio of workers to supervisors is 4:1, so $$\frac{8x}{2x + 4} = \frac{4}{1}$$ 5. **Solve the equation:** Cross-multiply: $$8x = 4(2x + 4)$$ $$8x = 8x + 16$$ 6. **Simplify:** Subtract $8x$ from both sides: $$8x - 8x = 16$$ $$0 = 16$$ 7. **Check for error:** The equation $0=16$ is impossible, so re-examine the problem. 8. **Reconsider the ratio:** The original ratio 8:2 simplifies to 4:1. The new ratio is also 4:1 after adding supervisors, which suggests the number of workers remains the same but supervisors increase. 9. **Set up the correct equation:** Since the new ratio is 4:1, the number of workers is 4 times the new number of supervisors: $$8x = 4(2x + 4)$$ 10. **Solve again:** $$8x = 8x + 16$$ $$8x - 8x = 16$$ $$0 = 16$$ 11. **Conclusion:** The problem as stated leads to a contradiction, meaning the ratio cannot be 4:1 after adding 4 supervisors if the original ratio is 8:2. 12. **Alternative approach:** Assume original ratio is 8:2, which simplifies to 4:1. Adding 4 supervisors changes the ratio to 4:1 again, which is impossible unless the number of workers also changes. 13. **Re-express the problem:** Let original workers be $8x$, supervisors $2x$. After adding 4 supervisors, ratio is 4:1: $$\frac{8x}{2x + 4} = 4$$ 14. **Solve for $x$:** $$8x = 4(2x + 4)$$ $$8x = 8x + 16$$ $$0 = 16$$ 15. **No solution:** The problem as stated is inconsistent. **Final answer:** There is no number of workers that satisfies the given conditions because the ratios contradict each other.