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04 Apr 2024, 14:16
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B
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Difficulty:
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Question Stats:
73% (02:01) correct
27%
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For a 30-mile trip, the fuel consumption of a certain car was 15 miles per gallon for the first 10 miles and 45 miles per gallon for the rest of the trip. What was the car's fuel consumption for the trip, in miles per gallon?
A. 25
B. 27
C. 30
D. 33
E. 35
A. 25
B. 27
C. 30
D. 33
E. 35
04 Apr 2024, 15:25
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delphsan
Amount of fuel used for the first 10 miles = \(\frac{10}{15}\)
Amount of fuel used for the next 20 miles = \(\frac{20}{45}\)
Total Amount of fuel used = \(\frac{10}{15} + \frac{20}{45} = \frac{50}{45}\)
The car's fuel consumption for the trip, in miles per gallon = \(\frac{\text{Distance Travelled}}{ \text{Fuel Consumed}}\)
The car's fuel consumption for the trip, in miles per gallon = \(\frac{30}{\frac{50}{45}} = 27\)
Option B
04 Sep 2024, 06:59
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Ankit_Raturi
I thought the same
Response via Chat GPT
- It's a great intuition to think about using a weighted average, but there's a subtle reason why it doesn't work directly in this context. The key issue lies in the fact that fuel efficiency is a rate (miles per gallon), and when calculating an overall rate, the correct approach requires taking into account the actual fuel consumed, not just the distances.
### Why Weighted Average Doesn't Work Here
Let's break it down:
1. **Fuel Efficiency as a Rate**:
- Fuel efficiency (miles per gallon) is a rate, not a direct quantity.
- To combine rates (like 15 mpg and 45 mpg), you need to consider the actual amount of fuel used, not just the distances.
2. **What Weighted Average Assumes**:
- A weighted average assumes that you're combining like quantities (e.g., distances, times, etc.) in a way that directly correlates.
- If you were averaging the distances themselves, you could use a weighted average.
3. **The Correct Calculation**:
- The correct way to combine different fuel efficiencies is to calculate the total fuel used and then divide the total distance by this total fuel, as we did earlier.
### Example of Misleading Weighted Average
If you incorrectly tried to use a weighted average:
- First part of the trip: 15 mpg for 10 miles.
This gives 35 mpg, which is incorrect because this calculation assumes that the fuel efficiencies can be directly averaged, ignoring the different amounts of fuel used at different efficiencies.
Conclusion
The proper method requires considering the total fuel consumed, not just the distances driven, because fuel efficiency is not a linear function that can be averaged by simple weighted average when mixing different rates. This is why we calculate the total fuel used and then compute the overall efficiency, resulting in 27 mpg instead of 35 mpg.
