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When David drove from his home to his parents' home, was his average speed between \(35\) miles per hour and \(50\) miles per hour?
We need to determine: "Was David's average speed between 35 and 50 miles per hour?"
This is a yes/no question. To answer it definitively, we need to either:
This problem involves compounded uncertainty. When both distance AND time have uncertainty from rounding, the resulting speed calculation has even greater uncertainty. This amplified uncertainty is often the key to DS problems with rounded values.
Statement 1: "To the nearest 100 miles, the distance was 300 miles."
This means the actual distance could be anywhere from 250 to just under 350 miles—a 100-mile range of uncertainty!
Without time information, let's see what's possible:
Since we can get both YES and NO answers, Statement 1 alone is NOT sufficient.
[STOP - Not Sufficient!] This eliminates choices A and D.
Important: We now forget Statement 1 completely and analyze Statement 2 independently.
Statement 2: "To the nearest hour, it took David 8 hours."
This means the actual time could be anywhere from 7.5 to just under 8.5 hours.
Without distance information, let's explore possibilities:
Again, we can construct both YES and NO scenarios. Statement 2 alone is NOT sufficient.
[STOP - Not Sufficient!] This eliminates choice B.
Now we use both pieces of information:
Here's where it gets interesting. When we have uncertainty in both the numerator (distance) and denominator (time) of our speed calculation, we need to consider the extreme cases:
Slowest possible speed:
Fastest possible speed:
Our possible speed range is approximately \([29.4, 46.7]\) mph. This range includes:
Since we can still get both YES and NO answers depending on the actual values, even both statements together are NOT sufficient.
[STOP - Not Sufficient!] This eliminates choice C.
Even with both statements combined, the uncertainty in distance and time creates a range of possible speeds that spans both inside and outside the target range of 35-50 mph. We cannot give a definitive yes or no answer.
The correct answer is E: Statements (1) and (2) together are NOT sufficient.
When dealing with rounded values in ratios, remember that uncertainty in both parts creates amplified uncertainty in the result. This often leads to insufficient information for definitive yes/no answers about ranges.