Overview
Analysis of a business decision comparing hand loom versus power loom production methods to determine the break-even price of coal that makes both equally cost-effective.
Production Methods Comparison
| Method | Worker Wage | Fabric Output | Additional Costs |
|---|
| Hand Loom | 12 pounds | 17 yards | None |
| Power Loom | 12 pounds | 81 yards | 16 pounds of coal (price unknown) |
Problem Setup
- Goal is to find the coal price (p) where both methods produce fabric at equal cost per yard.
- Hand loom cost efficiency: 17 yards per 12 pounds spent.
- Power loom cost efficiency: 81 yards per (12 + 16p) pounds spent.
- Need to determine when these ratios are equal to identify the break-even point.
Mathematical Solution
- Set up equation: 17/12 = 81/(12 + 16p).
- Use cross multiplication to isolate the variable p.
- Rearrange to: 12 + 16p = (12/17) × 81.
- Subtract 12 from both sides: 16p = (12/17) × 81 - 12.
- Divide both sides by 16: p = [(12/17) × 81 - 12] / 16.
- Calculate final result: p = 2.82 pounds per pound of coal (rounded to two decimals).
Decision Rule
- If coal price is below 2.82 pounds, use power loom (more cost-effective).
- If coal price is above 2.82 pounds, use hand loom (more economical).
- This break-even analysis helps businesses optimize production costs by choosing the appropriate method.
Historical Context
- Relates to virtuous cycle theory during Industrial Revolution.
- Low energy prices relative to labor costs made steam engine adoption profitable.
- First time mechanization became economically viable at scale.
- Shift in energy-labor cost ratio ignited industrial transformation and changed production methods permanently.