When fully burned, natural gas produces approximately 1,000 BTU of heat per cubic foot of fuel, and propane produces about...
GMAT Two Part Analysis : (TPA) Questions
When fully burned, natural gas produces approximately 1,000 BTU of heat per cubic foot of fuel, and propane produces about 2,500 BTU of heat per cubic foot of fuel. For a furnace in which either natural gas or propane may be burned as fuel, the efficiency of a given fuel is the usable heat energy produced when the fuel is burned in the furnace, expressed as a percentage of the total heat energy produced when the fuel is burned. Kaiser will purchase a furnace whose efficiency with respect to either natural gas burned alone or propane burned alone is 90%.
In the table select for natural gas cost per cubic foot and for propane cost per cubic foot the values that are jointly consistent with the information given for which the fuel cost per BTU of usable heat energy produced by this furnace would be approximately the same for each fuel burned alone. Make only two selections, one in each column.
Phase 1: Owning the Dataset
Visual Representation
We'll use a comparison table to organize the fuel properties:
| Property | Natural Gas | Propane |
| BTU per cubic foot | 1,000 | 2,500 |
| Furnace efficiency | 90% | 90% |
| Usable BTU per cubic foot | 900 | 2,250 |
| Cost per cubic foot | ? | ? |
| Cost per usable BTU | Must be equal | Must be equal |
Key Relationships
- Usable BTU = Total BTU × Efficiency
- Natural gas: \(1,000 \times 0.90 = 900\) usable BTU per cubic foot
- Propane: \(2,500 \times 0.90 = 2,250\) usable BTU per cubic foot
Phase 2: Understanding the Question
We need to find costs where both fuels have the same cost per BTU of usable heat energy.
For equal cost per usable BTU:
- Natural gas: Cost per usable BTU = \(\mathrm{(Cost\ per\ cubic\ foot)} \div 900\)
- Propane: Cost per usable BTU = \(\mathrm{(Cost\ per\ cubic\ foot)} \div 2,250\)
Setting these equal:
\(\mathrm{(Natural\ gas\ cost)} \div 900 = \mathrm{(Propane\ cost)} \div 2,250\)
Cross-multiplying:
\(\mathrm{Natural\ gas\ cost} \times 2,250 = \mathrm{Propane\ cost} \times 900\)
\(\mathrm{Natural\ gas\ cost} \times 2.5 = \mathrm{Propane\ cost}\)
Key Insight: The propane cost per cubic foot must be exactly 2.5 times the natural gas cost per cubic foot.
Phase 3: Finding the Answer
Let's check each natural gas cost to see if 2.5 times that value appears in our choices:
If natural gas = $0.0035:
- Propane should be: \(\$0.0035 \times 2.5 = \$0.00875\)
- Is $0.00875 in our choices? No, continue.
If natural gas = $0.0070:
- Propane should be: \(\$0.0070 \times 2.5 = \$0.0175\)
- Is $0.0175 in our choices? Yes! ✓
? Stop here - we found our answer.
Phase 4: Solution
Verification: Let's confirm these values give equal cost per usable BTU:
- Natural gas: \(\$0.0070 \div 900 = \$0.00000778\) per usable BTU
- Propane: \(\$0.0175 \div 2,250 = \$0.00000778\) per usable BTU ✓
Final Answer:
- Natural gas cost per cubic foot: $0.0070
- Propane cost per cubic foot: $0.0175