Question

A scooter rider covers a certain distance at a uniform speed of \(45\text{ km}\) per hour and retraces the same path at a uniform speed of \(55\text{ km}\) per hour to reach the starting point. His average speed during the whole journey is:

• A. \(49.5\text{ km/hr}\)
• B. \(50\text{ km/hr}\)
• C. \(50.5\text{ km/hr}\)
• D. \(51\text{ km/hr}\)

Hint:

Since the harmonic mean of two positive, unequal numbers is always strictly less than their arithmetic mean:
\[ \text{Arithmetic Mean} = \frac{45 + 55}{2} = 50\text{ km/hr} \]
The average speed must be strictly less than 50.
Looking at the options, only \(49.5\text{ km/hr}\) is less than 50, allowing you to identify the correct answer immediately.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
When an object travels equal distances at different speeds, its average speed over the entire journey is not the simple arithmetic mean of those speeds.
Instead, it is calculated as the harmonic mean of the individual speeds.

Step 2: Key Formula or Approach:
If an object covers a distance at speed \(v_1\) and returns over the same distance at speed \(v_2\), the average speed \(V_{\text{avg}}\) is:
\[ V_{\text{avg}} = \frac{2 v_1 v_2}{v_1 + v_2} \]

Step 3: Detailed Explanation:
We are given:
- Speed during the onward journey, \(v_1 = 45\text{ km/hr}\)
- Speed during the return journey, \(v_2 = 55\text{ km/hr}\)
Since the rider travels the same path back, the distance covered in both directions is equal.
Therefore, we use the harmonic mean formula to calculate the average speed:
\[ V_{\text{avg}} = \frac{2 \times 45 \times 55}{45 + 55} \]
First, calculate the product in the numerator:
\[ 2 \times 45 = 90 \]
\[ 90 \times 55 = 4950 \]
Next, calculate the sum in the denominator:
\[ 45 + 55 = 100 \]
Substitute these values back into our formula:
\[ V_{\text{avg}} = \frac{4950}{100} = 49.5\text{ km/hr} \]
The average speed of the scooter rider over the entire journey is \(49.5\text{ km/hr}\).
This matches the first option.

Step 4: Final Answer:
Therefore, the correct option is (A).