Question

Clarissa spent all day on a sightseeing trip in Britain. Starting from her hotel, Clarissa boarded a bus, which traveled at an average speed of 15 miles per hour through a 30 mile section of the countryside. The bus then stopped for lunch in London before continuing on a 3 hour tour of the city's sights at a speed of 10mph. Finally, the bus left the city and drove 40 miles straight back to the hotel. Clarissa arrived at her hotel exactly 2 hours after leaving London. What was the bus's average rate, approximately, for the entire journey?

• A. 8
• B. 14
• C. 21
• D. 25
• E. 30

Hint:

Always use the formula: Average Speed = Total Distance / Total Time. Avoid the common mistake of simply averaging the given speeds (e.g., (15+10)/2). This only works if the time spent at each speed is identical.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The average speed (or rate) is not the average of the speeds of different parts of a journey. It is calculated by dividing the total distance traveled by the total time taken for the journey.
Step 2: Key Formula or Approach:
\[ \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} \] We need to calculate the distance and time for each segment of the trip and then sum them up. The lunch stop is mentioned but no duration is given, so it's typically ignored in the total time calculation for the journey itself.
Step 3: Detailed Explanation:
Let's break the journey into three parts:
Part 1: Countryside
- Distance \(d_1\) = 30 miles
- Speed \(s_1\) = 15 mph
- Time \(t_1 = \frac{d_1}{s_1} = \frac{30}{15} = 2\) hours
Part 2: City Tour
- Time \(t_2\) = 3 hours
- Speed \(s_2\) = 10 mph
- Distance \(d_2 = s_2 \times t_2 = 10 \times 3 = 30\) miles
Part 3: Return to Hotel
- Distance \(d_3\) = 40 miles
- Time \(t_3\) = 2 hours
Now, let's calculate the totals for the entire journey:
Total Distance
\[ D_{total} = d_1 + d_2 + d_3 = 30 + 30 + 40 = 100 \text{ miles} \] Total Time
\[ T_{total} = t_1 + t_2 + t_3 = 2 + 3 + 2 = 7 \text{ hours} \] Finally, calculate the average speed:
\[ \text{Average Speed} = \frac{D_{total}}{T_{total}} = \frac{100}{7} \] \[ \text{Average Speed} \approx 14.2857 \text{ mph} \] The question asks for the approximate rate. The value 14.2857 is closest to 14.
Step 4: Final Answer:
The bus's approximate average rate for the entire journey was 14 mph.