1. **Problem Statement:** We need to analyze the cost behavior of XYZ Furniture's Production and Assembly departments using the High-Low method based on monthly production units and total costs. 2. **High-Low Method Formula:** The High-Low method estimates variable cost per unit and fixed costs using the highest and lowest activity levels: $$\text{Variable Cost per Unit} = \frac{\text{Cost at High Activity} - \text{Cost at Low Activity}}{\text{Units at High Activity} - \text{Units at Low Activity}}$$ $$\text{Fixed Cost} = \text{Total Cost} - (\text{Variable Cost per Unit} \times \text{Units})$$ 3. **Step 1: Identify High and Low Activity Levels for Each Department** - Production units high: 500 (June), low: 150 (January) - Assembly units high: 600 (June), low: 300 (January) 4. **Step 2: Calculate Total Variable Cost Increase per 100 Units** Total cost difference between June and January: $$11,500,000 - 4,800,000 = 6,700,000$$ Production units difference: $$500 - 150 = 350$$ Assembly units difference: $$600 - 300 = 300$$ 5. **Step 3: Calculate Variable Cost per Unit Increase Rate** Variable costs increase by 10% for every additional 100 units in either department. Let $v_p$ and $v_a$ be the base variable costs per unit for Production and Assembly respectively. The variable cost per unit for Production at 350 units increase is: $$v_p \times (1 + 0.10 \times \frac{350}{100}) = v_p \times 1.35$$ Similarly, for Assembly at 300 units increase: $$v_a \times (1 + 0.10 \times \frac{300}{100}) = v_a \times 1.30$$ 6. **Step 4: Set up equations for total variable cost increase** The total variable cost increase from January to June is: $$6,700,000 = 350 \times v_p \times 1.35 + 300 \times v_a \times 1.30$$ 7. **Step 5: Use fixed costs information** Fixed costs total: $$1,200,000 + 800,000 = 2,000,000$$ Total cost in January (lowest activity) is 4,800,000, so variable cost in January is: $$4,800,000 - 2,000,000 = 2,800,000$$ Variable cost in January is: $$150 \times v_p + 300 \times v_a = 2,800,000$$ 8. **Step 6: Solve the system of equations:** $$\begin{cases} 150 v_p + 300 v_a = 2,800,000 \\ 350 \times 1.35 v_p + 300 \times 1.30 v_a = 6,700,000 \end{cases}$$ Simplify second equation: $$472.5 v_p + 390 v_a = 6,700,000$$ 9. **Step 7: Solve for $v_p$ and $v_a$** Multiply first equation by 1.3 to align $v_a$ terms: $$195 v_p + 390 v_a = 3,640,000$$ Subtract from second equation: $$(472.5 - 195) v_p = 6,700,000 - 3,640,000$$ $$277.5 v_p = 3,060,000$$ $$v_p = \frac{3,060,000}{277.5} = 11,027.03$$ Substitute $v_p$ back: $$150 \times 11,027.03 + 300 v_a = 2,800,000$$ $$1,654,054.05 + 300 v_a = 2,800,000$$ $$300 v_a = 1,145,945.95$$ $$v_a = \frac{1,145,945.95}{300} = 3,819.82$$ 10. **Step 8: Total fixed costs** Given fixed costs are: - Production: 1,200,000 - Assembly: 800,000 Total fixed costs = 2,000,000 11. **Step 9: Estimate total cost for 375 Production units and 475 Assembly units** Calculate variable cost multipliers: Production increase from base 150 units: $$\frac{375 - 150}{100} = 2.25$$ Variable cost per unit for Production: $$11,027.03 \times (1 + 0.10 \times 2.25) = 11,027.03 \times 1.225 = 13,505.08$$ Assembly increase from base 300 units: $$\frac{475 - 300}{100} = 1.75$$ Variable cost per unit for Assembly: $$3,819.82 \times (1 + 0.10 \times 1.75) = 3,819.82 \times 1.175 = 4,487.82$$ Total variable cost: $$375 \times 13,505.08 + 475 \times 4,487.82 = 5,064,405 + 2,131,715 = 7,196,120$$ Total cost: $$7,196,120 + 2,000,000 = 9,196,120$$ 12. **Step 10: Impact of 20% increase in production units** New units: Production: $$375 \times 1.20 = 450$$ Assembly: $$475 \times 1.20 = 570$$ Calculate new variable cost per unit: Production increase from base 150: $$\frac{450 - 150}{100} = 3$$ $$11,027.03 \times (1 + 0.10 \times 3) = 11,027.03 \times 1.30 = 14,335.14$$ Assembly increase from base 300: $$\frac{570 - 300}{100} = 2.7$$ $$3,819.82 \times (1 + 0.10 \times 2.7) = 3,819.82 \times 1.27 = 4,851.18$$ Total variable cost: $$450 \times 14,335.14 + 570 \times 4,851.18 = 6,450,813 + 2,765,174 = 9,215,987$$ Total cost: $$9,215,987 + 2,000,000 = 11,215,987$$ **Summary:** - Variable cost per unit Production: $11,027.03$ - Variable cost per unit Assembly: $3,819.82$ - Total fixed costs: $2,000,000$ - Estimated total cost at 375 Production and 475 Assembly units: $9,196,120$ - Estimated total cost after 20% increase in production units: $11,215,987$