1. **State the problem:** We need to calculate productivity measures for Carl's Carpet Cleaning Company based on the data for two months ago and last month. 2. **Given data:** - Price per carpet cleaned = $119 - Labor cost per hour = $11.25 - Solution cost per gallon = $8.99 Two months ago: - Carpets cleaned = 63 - Labor hours = 130 - Solution gallons = 115 Last month: - Carpets cleaned = 75 - Labor hours = 126 - Solution gallons = 93 --- 3. **Single-factor productivity in terms of labor dollars (two months ago):** Formula: $$\text{Single-factor productivity} = \frac{\text{Output in dollars}}{\text{Input cost}}$$ Output in dollars = number of carpets cleaned \(\times\) price per carpet Input cost = labor hours \(\times\) labor cost per hour Calculate output two months ago: $$63 \times 119 = 7497$$ Calculate labor cost two months ago: $$130 \times 11.25 = 1462.5$$ Single-factor productivity: $$\frac{7497}{1462.5} = 5.1265 \approx 5.13$$ Interpretation: For every dollar spent on labor two months ago, Carl obtained about $5.13 in sales. --- 4. **Percentage change in productivity in terms of solution gallons:** Calculate productivity for two months ago and last month as number of carpets cleaned per gallon of solution. Two months ago: $$\frac{63}{115} = 0.5478$$ Last month: $$\frac{75}{93} = 0.8065$$ Percentage change: $$\frac{0.8065 - 0.5478}{0.5478} \times 100 = 47.12\%$$ Interpretation: Productivity in terms of carpets cleaned per gallon of solution increased by 47.12%. --- 5. **Last month's multi-factor productivity (MFP) in terms of dollars:** MFP formula: $$\text{MFP} = \frac{\text{Output in dollars}}{\text{Total input cost in dollars}}$$ Calculate output last month: $$75 \times 119 = 8925$$ Calculate total input cost last month: Labor cost: $$126 \times 11.25 = 1417.5$$ Solution cost: $$93 \times 8.99 = 836.07$$ Total input cost: $$1417.5 + 836.07 = 2253.57$$ MFP last month: $$\frac{8925}{2253.57} = 3.96$$ Interpretation: Each dollar of inputs produced $3.96 in revenue last month. --- 6. **Percentage change in multi-factor productivity from two months ago to last month:** Calculate total input cost two months ago: Labor cost: $$130 \times 11.25 = 1462.5$$ Solution cost: $$115 \times 8.99 = 1033.85$$ Total input cost two months ago: $$1462.5 + 1033.85 = 2496.35$$ MFP two months ago: $$\frac{7497}{2496.35} = 3.00$$ Percentage change: $$\frac{3.96 - 3.00}{3.00} \times 100 = 32.00\%$$ Interpretation: Multi-factor productivity improved by 32.00% from two months ago to last month. --- **Final answers:** - Two months ago single-factor productivity (labor dollars): **5.13** - Percentage change in productivity (solution gallons): **47.12%** - Last month's multi-factor productivity (dollars): **3.96** - Percentage change in multi-factor productivity (dollars): **32.00%**