A local workplace requires one member of the board of directors for every 8 executives. there are 5 managers for every executive and 8 workers for every manager. The average salary paid to directors is $100,000, $75,000 to executives, $50,000 to managers and $32,000 to workers. If the facility has 1,107 employees, what is the total labour cost?

To find the total labor cost for the facility with 1,107 employees, we need to determine the number of directors, executives, managers, and workers. We'll use the given ratios and solve step by step.

Given ratios and salaries:
1. One director for every 8 executives.
2. Five managers for every executive.
3. Eight workers for every manager.
4. Salaries:
   - Director: $100,000
   - Executive: $75,000
   - Manager: $50,000
   - Worker: $32,000

Let:
- $ D $ = Number of directors
- $ E $ = Number of executives
- $ M $ = Number of managers
- $ W $ = Number of workers

From the given ratios, we have:
\[ D = \frac{E}{8} \]
\[ M = 5E \]
\[ W = 8M = 8(5E) = 40E \]

The total number of employees is:
\[ D + E + M + W = 1,107 \]

Substituting the ratios:
\[ \frac{E}{8} + E + 5E + 40E = 1,107 \]

Combining like terms:
\[ \frac{E}{8} + 46E = 1,107 \]

To simplify, multiply the entire equation by 8 to clear the fraction:
\[ E + 368E = 8,856 \]
\[ 369E = 8,856 \]
\[ E = \frac{8,856}{369} \]
\[ E = 24 \]

Now, find the number of directors, managers, and workers:
\[ D = \frac{E}{8} = \frac{24}{8} = 3 \]
\[ M = 5E = 5 \times 24 = 120 \]
\[ W = 40E = 40 \times 24 = 960 \]

Verify the total number of employees:
\[ D + E + M + W = 3 + 24 + 120 + 960 = 1,107 \]

Now, calculate the total labor cost:
\[ \text{Total cost} = (D \times \$100,000) + (E \times \$75,000) + (M \times \$50,000) + (W \times \$32,000) \]
\[ \text{Total cost} = (3 \times 100,000) + (24 \times 75,000) + (120 \times 50,000) + (960 \times 32,000) \]

Breaking it down:
\[ 3 \times 100,000 = 300,000 \]
\[ 24 \times 75,000 = 1,800,000 \]
\[ 120 \times 50,000 = 6,000,000 \]
\[ 960 \times 32,000 = 30,720,000 \]

Summing these amounts:
\[ 300,000 + 1,800,000 + 6,000,000 + 30,720,000 = 38,820,000 \]

Thus, the total labor cost is:
\[ \boxed{38,820,000} \]

Would you like more details or have any questions?

1. How would the total labor cost change if the number of executives doubled?
2. What would be the impact on total labor cost if the average salary of managers increased by $10,000?
3. How many more workers would be required if the ratio of workers to managers changed to 10:1?
4. What is the total labor cost if the facility has 2,214 employees with the same ratios?
5. If the average salary for directors increased by 20%, how would the total labor cost be affected?
6. How does changing the ratio of directors to executives to 1:10 impact the number of directors?
7. What would be the new total labor cost if the number of managers increased by 10%?
8. How would the total labor cost change if workers' salaries were reduced by 5%?

**Tip:** When working with ratios, always verify the total count to ensure the calculations align with the given total.

Learn how to calculate the total labor cost for a facility with 1,107 employees using ratios and salaries of directors, executives, managers, and workers. Includes step-by-step solution and breakdown of costs. Ideal for advanced high school students studying ratios and arithmetic.

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