Question

On a recent trip, Pat drove 10 miles. What was Pat's average speed?
(1) Pat drove 6 miles at an average speed of 12 miles per hour and then drove the remaining 4 miles at an average speed of 10 miles per hour.
(2) Pat drove a total of 0.9 hours.

• A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
• B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
• C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
• D. EACH statement ALONE is sufficient.
• E. Statements (1) and (2) TOGETHER are NOT sufficient.

Hint:

Remember that average speed is NOT the average of the speeds. It is always Total Distance / Total Time. A statement is sufficient if it allows you to calculate the Total Time, given that the Total Distance is already known.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The question asks for the average speed of a trip. The total distance is given as 10 miles.
Step 2: Key Formula or Approach:
The formula for average speed is:
\[ \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} \]
To find the average speed, we need to determine the total time of the trip.
Step 3: Detailed Explanation:
Analyzing Statement (1):
This statement breaks the trip into two segments and provides the distance and average speed for each. We can calculate the time taken for each segment.
Time for segment 1: \(t_1 = \frac{\text{Distance}_1}{\text{Speed}_1} = \frac{6 \text{ miles}}{12 \text{ mph}} = 0.5 \text{ hours}\).
Time for segment 2: \(t_2 = \frac{\text{Distance}_2}{\text{Speed}_2} = \frac{4 \text{ miles}}{10 \text{ mph}} = 0.4 \text{ hours}\).
The total time for the trip is \(T = t_1 + t_2 = 0.5 + 0.4 = 0.9 \text{ hours}\).
Now we can calculate the average speed:
\[ \text{Average Speed} = \frac{10 \text{ miles}}{0.9 \text{ hours}} = \frac{100}{9} \text{ mph} \]
Since we can find a unique value, statement (1) alone is sufficient.
Analyzing Statement (2):
This statement directly gives the total time for the trip: Total Time = 0.9 hours.
We can use this to calculate the average speed:
\[ \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{10 \text{ miles}}{0.9 \text{ hours}} = \frac{100}{9} \text{ mph} \]
Since we can find a unique value, statement (2) alone is sufficient.
Step 4: Final Answer:
Both statements, independently, provide enough information to calculate the average speed. Therefore, each statement alone is sufficient.