On the second leg of a certain journey, a person traveled at an average speed 50 percent greater than the average speed at which the person traveled the first leg of the journey, and for an amount of time 50 percent greater than the amount of time spent on the first leg of the journey. The person's average speed for the entire journey was what percent greater than the person's average speed on the first leg of the journey ?

Options

A. `22.5%`
B. `25%`
C. `30%`
D. `50%`

Explanation

To solve the problem step by step, we will define the variables and calculate the average speeds for both legs of the journey and the entire journey.

### Step 1: Define Variables
Let:
- \( r \) = average speed during the first leg of the journey
- \( t \) = time spent on the first leg of the journey

### Step 2: Calculate Distance for the First Leg
Using the formula for distance:
\[ \text{Distance}_1 = r \times t = rt \]

### Step 3: Calculate Speed and Time for the Second Leg
The average speed during the second leg is 50% greater than the first leg:
\[ \text{Speed}_2 = 1.5r \]
The time spent on the second leg is also 50% greater than the first leg:
\[ \text{Time}_2 = 1.5t \]

### Step 4: Calculate Distance for the Second Leg
Using the formula for distance again:
\[ \text{Distance}_2 = \text{Speed}_2 \times \text{Time}_2 = (1.5r) \times (1.5t) = 2.25rt \]

### Step 5: Calculate Total Distance and Total Time
Now, we can find the total distance and total time for the entire journey:
\[ \text{Total Distance} = \text{Distance}_1 + \text{Distance}_2 = rt + 2.25rt = 3.25rt \]
\[ \text{Total Time} = \text{Time}_1 + \text{Time}_2 = t + 1.5t = 2.5t \]

### Step 6: Calculate Average Speed for the Entire Journey
Using the formula for average speed:
\[ \text{Average Speed}_{\text{total}} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{3.25rt}{2.5t} \]
Cancelling \( t \) from the numerator and denominator:
\[ \text{Average Speed}_{\text{total}} = \frac{3.25r}{2.5} = 1.3r \]

### Step 7: Calculate the Percentage Increase in Average Speed
To find the percentage increase in average speed compared to the first leg:
\[ \text{Percentage Increase} = \left( \frac{\text{Average Speed}_{\text{total}} - \text{Average Speed}_{\text{leg 1}}}{\text{Average Speed}_{\text{leg 1}}} \right) \times 100 \]
Substituting the values:
\[ \text{Percentage Increase} = \left( \frac{1.3r - r}{r} \right) \times 100 = \left( \frac{0.3r}{r} \right) \times 100 = 30\% \]

### Final Answer
The person's average speed for the entire journey was **30% greater** than the person's average speed on the first leg of the journey.