When David drove from his home to his parents' home, was his average speed between 35 miles per hour and...

Understanding the Question

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:

• Calculate David's exact average speed (\(\mathrm{distance} ÷ \mathrm{time}\)), OR
• Have enough information to know with certainty whether his speed falls inside or outside the 35-50 mph range

Key Insight

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.

Analyzing Statement 1

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!

What We Know and Don't Know

• We know: Distance is somewhere in the range \([250, 350)\) miles
• We don't know: Anything about how long the trip took

Testing with Examples

Without time information, let's see what's possible:

• If the trip took 4 hours: Speed ≈ \(300/4 = 75\) mph → Answer: NO (above 50 mph)
• If the trip took 7 hours: Speed ≈ \(300/7 = 43\) mph → Answer: YES (within 35-50 mph)
• If the trip took 10 hours: Speed ≈ \(300/10 = 30\) mph → Answer: NO (below 35 mph)

Since we can get both YES and NO answers, Statement 1 alone is NOT sufficient.

[STOP - Not Sufficient!] This eliminates choices A and D.

Analyzing Statement 2

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.

What We Know and Don't Know

• We know: Time is somewhere in the range \([7.5, 8.5)\) hours
• We don't know: Anything about the distance traveled

Testing with Examples

Without distance information, let's explore possibilities:

• If David drove 200 miles: Speed ≈ \(200/8 = 25\) mph → Answer: NO (below 35 mph)
• If David drove 320 miles: Speed ≈ \(320/8 = 40\) mph → Answer: YES (within 35-50 mph)
• If David drove 500 miles: Speed ≈ \(500/8 = 62.5\) mph → Answer: NO (above 50 mph)

Again, we can construct both YES and NO scenarios. Statement 2 alone is NOT sufficient.

[STOP - Not Sufficient!] This eliminates choice B.

Combining Both Statements

Now we use both pieces of information:

• Distance: between 250 and just under 350 miles
• Time: between 7.5 and just under 8.5 hours

The Compounded Uncertainty Effect

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:

• Minimum distance ÷ Maximum time = \(250 \text{ miles} ÷ 8.5 \text{ hours} ≈ 29.4\) mph
• This is below 35 mph → Answer would be NO

Fastest possible speed:

• Maximum distance ÷ Minimum time = \(350 \text{ miles} ÷ 7.5 \text{ hours} ≈ 46.7\) mph
• This is within 35-50 mph → Answer would be YES

The Critical Finding

Our possible speed range is approximately \([29.4, 46.7]\) mph. This range includes:

• Speeds below 35 mph (like 30 or 33 mph) → Answer: NO
• Speeds within 35-50 mph (like 40 or 45 mph) → Answer: YES

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.

The Answer: E

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.

Key Takeaway

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.