Question

A yellow taxi cab went from Downtown to the Beachside and back at an average speed of 2/3 miles per hour. If the distance from Beachside to Downtown is 1 mile, and the trip back took half as much time as the trip there, what was the average speed of the yellow taxi cab on the way to Beachside?

• A. 1/3
• B. 1/2
• C. 3/4
• D. 2/3
• E. 3/2

Hint:

Be careful with average speed problems. You cannot simply average the speeds of the two legs of a journey unless the time taken for each leg is identical. Always work with total distance and total time.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
This problem involves the relationship between speed, distance, and time. The key is to understand that average speed is calculated as total distance divided by total time, not as the average of the speeds of the individual parts of the journey.
Step 2: Key Formula or Approach:
The fundamental formula is: \[ \text{Speed} = \frac{\text{Distance}}{\text{Time}} \] This can be rearranged to find time: \( \text{Time} = \frac{\text{Distance}}{\text{Speed}} \).
For the entire trip: \( \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} \).
Step 3: Detailed Explanation:
First, let's define our variables:
\(d = 1\) mile (one-way distance).
Total Distance \(D_{total} = d + d = 1 + 1 = 2\) miles.
Average Speed \(S_{avg} = 2/3\) mph.
Let \(t_1\) be the time from Downtown to Beachside.
Let \(t_2\) be the time from Beachside back to Downtown.
We are given \(t_2 = \frac{1}{2} t_1\).
Using the average speed formula, we can find the total time for the round trip: \[ T_{total} = \frac{D_{total}}{S_{avg}} = \frac{2 \text{ miles}}{2/3 \text{ mph}} = 2 \times \frac{3}{2} = 3 \text{ hours} \] The total time is also the sum of the individual trip times: \[ T_{total} = t_1 + t_2 \] \[ 3 = t_1 + t_2 \] Now substitute the relationship \(t_2 = \frac{1}{2} t_1\) into this equation: \[ 3 = t_1 + \frac{1}{2} t_1 \] \[ 3 = \frac{3}{2} t_1 \] Solve for \(t_1\), the time taken on the way to Beachside: \[ t_1 = 3 \times \frac{2}{3} = 2 \text{ hours} \] The question asks for the average speed on the way to Beachside (\(S_1\)). The distance for this trip is 1 mile and we just found the time is 2 hours. \[ S_1 = \frac{\text{Distance}}{\text{Time}} = \frac{d}{t_1} = \frac{1 \text{ mile}}{2 \text{ hours}} = \frac{1}{2} \text{ mph} \] Step 4: Final Answer
The average speed of the yellow taxi cab on the way to Beachside was 1/2 miles per hour.