`(1)/(3)rd` part of a certain journey is covered with the speed of 25 km/hr `(1)/(2)nd` part of the journey is covered with the speed of 45 km/hr and the remaining part is covered with the speed of 37.5 km/hr. What is the average speed (in km/hr) for the whole journey?

Options

A. `37.08`
B. `34.61`
C. 43
D. 56

Answer

To find the average speed for the entire journey, we can follow these steps:

### Step 1: Define the Total Distance
Let the total distance of the journey be \( D \). We can express this distance in terms of a variable \( x \). For simplicity, let’s assume:
\[ D = 12x \]

### Step 2: Determine the Distances for Each Part of the Journey
- The first part (1/3 of the journey) is:
\[ \text{Distance}_1 = \frac{1}{3}D = \frac{1}{3}(12x) = 4x \]
- The second part (1/2 of the journey) is:
\[ \text{Distance}_2 = \frac{1}{2}D = \frac{1}{2}(12x) = 6x \]
- The remaining part (the rest of the journey) is:
\[ \text{Distance}_3 = D - \left(\text{Distance}_1 + \text{Distance}_2\right) = 12x - (4x + 6x) = 2x \]

### Step 3: Determine the Speeds for Each Part of the Journey
- Speed for the first part:
\[ \text{Speed}_1 = 25 \text{ km/hr} \]
- Speed for the second part:
\[ \text{Speed}_2 = 45 \text{ km/hr} \]
- Speed for the third part:
\[ \text{Speed}_3 = 37.5 \text{ km/hr} \]

### Step 4: Calculate the Time Taken for Each Part of the Journey
Using the formula \( \text{Time} = \frac{\text{Distance}}{\text{Speed}} \):
- Time for the first part:
\[ \text{Time}_1 = \frac{\text{Distance}_1}{\text{Speed}_1} = \frac{4x}{25} \]
- Time for the second part:
\[ \text{Time}_2 = \frac{\text{Distance}_2}{\text{Speed}_2} = \frac{6x}{45} \]
- Time for the third part:
\[ \text{Time}_3 = \frac{\text{Distance}_3}{\text{Speed}_3} = \frac{2x}{37.5} \]

### Step 5: Calculate the Total Time for the Journey
Now, we sum the times for each part:
\[ \text{Total Time} = \text{Time}_1 + \text{Time}_2 + \text{Time}_3 \]
Substituting the values:
\[ \text{Total Time} = \frac{4x}{25} + \frac{6x}{45} + \frac{2x}{37.5} \]

### Step 6: Simplify the Total Time Expression
To simplify, we need a common denominator. The least common multiple of 25, 45, and 37.5 is 225. We convert each term:
1. For \( \frac{4x}{25} \):
\[ \frac{4x}{25} = \frac{4x \times 9}{225} = \frac{36x}{225} \]
2. For \( \frac{6x}{45} \):
\[ \frac{6x}{45} = \frac{6x \times 5}{225} = \frac{30x}{225} \]
3. For \( \frac{2x}{37.5} \):
\[ \frac{2x}{37.5} = \frac{2x \times 6}{225} = \frac{12x}{225} \]

Now, summing these:
\[ \text{Total Time} = \frac{36x + 30x + 12x}{225} = \frac{78x}{225} \]

### Step 7: Calculate the Average Speed
Average speed is given by the formula:
\[ \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} \]
Substituting the values:
\[ \text{Average Speed} = \frac{12x}{\frac{78x}{225}} = 12x \times \frac{225}{78x} = \frac{12 \times 225}{78} \]

### Step 8: Simplify the Average Speed
Calculating:
\[ \text{Average Speed} = \frac{2700}{78} \]
Dividing:
\[ \text{Average Speed} = 34.615 \text{ km/hr} \approx 34.61 \text{ km/hr} \]

### Final Answer
Thus, the average speed for the whole journey is approximately:
\[ \text{Average Speed} \approx 34.61 \text{ km/hr} \]

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