1. **Problem statement:** We have two branches of a company with 100 and 80 employees respectively. The arithmetic means of their monthly salaries are 4570 and 6750 respectively. We need to find the overall arithmetic mean salary of all employees combined. 2. **Formula used:** The combined arithmetic mean $\bar{x}$ for two groups is given by the weighted average formula: $$\bar{x} = \frac{n_1 \times \bar{x}_1 + n_2 \times \bar{x}_2}{n_1 + n_2}$$ where $n_1, n_2$ are the number of employees in each branch and $\bar{x}_1, \bar{x}_2$ are their respective means. 3. **Substitute the values:** $$n_1 = 100, \quad \bar{x}_1 = 4570$$ $$n_2 = 80, \quad \bar{x}_2 = 6750$$ 4. **Calculate numerator:** $$100 \times 4570 = 457000$$ $$80 \times 6750 = 540000$$ 5. **Sum numerator:** $$457000 + 540000 = 997000$$ 6. **Sum denominator:** $$100 + 80 = 180$$ 7. **Calculate combined mean:** $$\bar{x} = \frac{997000}{180} = 5538.89$$ **Final answer:** The arithmetic mean salary of all employees combined is approximately $5538.89$.