1. **State the problem:** We have three groups of workers: skilled, semi-skilled, and unskilled. Total workers = 45. 2. **Define variables:** Let $x$ = number of skilled workers, $y$ = number of semi-skilled workers, $z$ = number of unskilled workers. 3. **Write equations from the problem:** - Total workers: $$x + y + z = 45$$ - Total wage bill: $$50000x + 30000y + 20000z = 1500000$$ - Unskilled workers are 5 fewer than semi-skilled: $$z = y - 5$$ 4. **Substitute $z$ in the first two equations:** - $$x + y + (y - 5) = 45$$ - Simplify: $$x + 2y - 5 = 45$$ - Add 5 to both sides: $$x + 2y = 50$$ 5. **Substitute $z$ in the wage equation:** - $$50000x + 30000y + 20000(y - 5) = 1500000$$ - Expand: $$50000x + 30000y + 20000y - 100000 = 1500000$$ - Combine like terms: $$50000x + 50000y - 100000 = 1500000$$ - Add 100000 to both sides: $$50000x + 50000y = 1600000$$ 6. **Simplify wage equation by dividing both sides by 10000:** - $$\cancel{50000}x + \cancel{50000}y = \frac{1600000}{10000}$$ - $$5x + 5y = 160$$ 7. **Divide entire equation by 5:** - $$\cancel{5}x + \cancel{5}y = \frac{160}{5}$$ - $$x + y = 32$$ 8. **Now solve the system:** - From step 4: $$x + 2y = 50$$ - From step 7: $$x + y = 32$$ 9. **Subtract the second equation from the first:** - $$(x + 2y) - (x + y) = 50 - 32$$ - $$x + 2y - x - y = 18$$ - $$y = 18$$ 10. **Find $x$ using $x + y = 32$:** - $$x + 18 = 32$$ - $$x = 14$$ 11. **Find $z$ using $z = y - 5$:** - $$z = 18 - 5 = 13$$ **Final answer:** - Skilled workers ($x$) = 14 - Semi-skilled workers ($y$) = 18 - Unskilled workers ($z$) = 13