One-fourth of a certain journey is covered at the rate of 25 km/h, one-third at the rate of 30 km/h and the rest at 50 km/h. Find the average speed for the whole journey.

Options

A. 600/53 km/h
B. 1200/53 km/h
C. 1800/53 km/h
D. 1600/53 km/h

Answer

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

### Step 1: Define the Total Distance
Let the total distance of the journey be \( D \) kilometers.

### Step 2: Calculate the Distances for Each Segment
- The distance covered at 25 km/h is \( \frac{1}{4}D \).
- The distance covered at 30 km/h is \( \frac{1}{3}D \).
- The remaining distance is covered at 50 km/h.

To find the remaining distance, we first need to calculate how much distance has already been covered:
- Distance covered at 25 km/h: \( \frac{1}{4}D \)
- Distance covered at 30 km/h: \( \frac{1}{3}D \)

The remaining distance can be found by subtracting these distances from the total distance \( D \).

### Step 3: Calculate the Remaining Distance
To find the remaining distance, we first need a common denominator for \( \frac{1}{4}D \) and \( \frac{1}{3}D \). The least common multiple (LCM) of 4 and 3 is 12.

- Convert \( \frac{1}{4}D \) to twelfths:
\[ \frac{1}{4}D = \frac{3}{12}D \]

- Convert \( \frac{1}{3}D \) to twelfths:
\[ \frac{1}{3}D = \frac{4}{12}D \]

Now, we can find the remaining distance:
\[ \text{Remaining distance} = D - \left(\frac{3}{12}D + \frac{4}{12}D\right) = D - \frac{7}{12}D = \frac{5}{12}D \]

### Step 4: Calculate the Time Taken for Each Segment
Now we can calculate the time taken for each segment of the journey.

1. **Time for the first segment (at 25 km/h)**:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} = \frac{\frac{1}{4}D}{25} = \frac{D}{100} \text{ hours} \]

2. **Time for the second segment (at 30 km/h)**:
\[ \text{Time} = \frac{\frac{1}{3}D}{30} = \frac{D}{90} \text{ hours} \]

3. **Time for the third segment (at 50 km/h)**:
\[ \text{Time} = \frac{\frac{5}{12}D}{50} = \frac{D}{120} \text{ hours} \]

### Step 5: Calculate the Total Time Taken
Now, we sum the time taken for all segments:
\[ \text{Total time} = \frac{D}{100} + \frac{D}{90} + \frac{D}{120} \]

To add these fractions, we need a common denominator. The LCM of 100, 90, and 120 is 1800.

- Convert each term:
\[ \frac{D}{100} = \frac{18D}{1800}, \quad \frac{D}{90} = \frac{20D}{1800}, \quad \frac{D}{120} = \frac{15D}{1800} \]

Now, add them:
\[ \text{Total time} = \left(\frac{18D + 20D + 15D}{1800}\right) = \frac{53D}{1800} \text{ hours} \]

### Step 6: Calculate the Average Speed
The average speed is given by the formula:
\[ \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{D}{\frac{53D}{1800}} = \frac{1800}{53} \text{ km/h} \]

### Final Answer
The average speed for the whole journey is:
\[ \frac{1800}{53} \text{ km/h} \approx 33.96 \text{ km/h} \]