Question

If Paula drove the distance from her home to her college at an average speed that was greater than 70 kilometers per hour, did it take her less than 3 hours to drive this distance?
(1) The distance that Paula drove from her home to her college was greater than 200 kilometers.
(2) The distance that Paula drove from her home to her college was less than 205 kilometers.

• A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
• B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
• C. BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient.
• D. EACH statement ALONE is sufficient to answer the question asked.
• E. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

Hint:

When dealing with inequalities in rate-time-distance problems, test the boundary conditions. To find the maximum or minimum value of a fraction like Time = Distance/Speed, consider the maximum value of the numerator and the minimum value of the denominator, and vice-versa.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
This problem involves the relationship between distance, speed (rate), and time. Let \(D\) be the distance, \(S\) be the speed, and \(T\) be the time.
We are given that Paula's average speed \(S>70\) km/h.
The question asks: Is \(T<3\) hours?
Step 2: Key Formula or Approach:
The formula connecting the variables is \(D = S \times T\), which can be rearranged to \(T = \frac{D}{S}\).
The question is: Is \(\frac{D}{S}<3\)?
Step 3: Detailed Explanation:
Analyze Statement (1): The distance was greater than 200 kilometers (\(D>200\)).
We have \(D>200\) and \(S>70\). Let's test some values to see if we get a consistent answer.

Case 1 (Answer is "Yes"): Let the distance \(D = 201\) km and the speed \(S = 75\) km/h. Both conditions are met. Then the time is \(T = \frac{201}{75} = 2.68\) hours. Since \(2.68<3\), the answer is "Yes".

Case 2 (Answer is "No"): Let the distance \(D = 215\) km and the speed \(S = 71\) km/h. Both conditions are met. Then the time is \(T = \frac{215}{71} \approx 3.02\) hours. Since \(3.02>3\), the answer is "No".

Since we can get both "Yes" and "No" answers, statement (1) is not sufficient.
Analyze Statement (2): The distance was less than 205 kilometers (\(D<205\)).
We have \(D<205\) and \(S>70\). To determine if \(T<3\), we should look at the maximum possible value of \(T\). The time \(T = \frac{D}{S}\) is maximized when the numerator (\(D\)) is as large as possible and the denominator (\(S\)) is as small as possible.

The maximum possible value for \(D\) is just under 205.

The minimum possible value for \(S\) is just over 70.

Let's calculate the value at the boundary: \(T_{max} = \frac{205}{70} \approx 2.928\) hours.
Since \(D<205\) and \(S>70\), the actual time \(T\) must be less than this value: \(T<\frac{205}{70}\).
So, \(T<2.928\). Since any possible value for \(T\) is less than 2.928, it must also be less than 3.
The answer to the question is always "Yes". Therefore, statement (2) is sufficient.
Step 4: Final Answer:
Statement (2) alone is sufficient to answer the question, while statement (1) is not.