A person travelled the forst half of the distance from her office at 24 km/h ,exactly half of the remaining distance at 20 km / h and the rest at 15 km/h .Her average speed (km/h) in entire journey was :

Options

A. 20
B. `18(1)/(3)`
C. `19(2)/(3)`
D. 18

Answer

To find the average speed of the person during her entire journey, we can break down the problem step by step.

### Step 1: Define the total distance
Let's assume the total distance traveled by the person is \( D \) kilometers.

### Step 2: Calculate the distances for each segment
1. The first half of the distance is \( \frac{D}{2} \).
2. The remaining distance after the first half is also \( \frac{D}{2} \). Half of this remaining distance is \( \frac{D}{4} \).
3. The rest of the distance after traveling half of the remaining distance is also \( \frac{D}{4} \).

### Step 3: Calculate the time taken for each segment
1. **Time for the first half**:
- Speed = 24 km/h
- Distance = \( \frac{D}{2} \)
- Time \( T_1 = \frac{\text{Distance}}{\text{Speed}} = \frac{\frac{D}{2}}{24} = \frac{D}{48} \) hours

2. **Time for the second segment (half of the remaining distance)**:
- Speed = 20 km/h
- Distance = \( \frac{D}{4} \)
- Time \( T_2 = \frac{\text{Distance}}{\text{Speed}} = \frac{\frac{D}{4}}{20} = \frac{D}{80} \) hours

3. **Time for the last segment (the rest of the distance)**:
- Speed = 15 km/h
- Distance = \( \frac{D}{4} \)
- Time \( T_3 = \frac{\text{Distance}}{\text{Speed}} = \frac{\frac{D}{4}}{15} = \frac{D}{60} \) hours

### Step 4: Calculate the total time taken
Total time \( T = T_1 + T_2 + T_3 \):
\[ T = \frac{D}{48} + \frac{D}{80} + \frac{D}{60} \]

### Step 5: Find a common denominator to add the times
The least common multiple (LCM) of 48, 80, and 60 is 240. We convert each term:
- \( \frac{D}{48} = \frac{5D}{240} \)
- \( \frac{D}{80} = \frac{3D}{240} \)
- \( \frac{D}{60} = \frac{4D}{240} \)

Now, adding these:
\[ T = \frac{5D + 3D + 4D}{240} = \frac{12D}{240} = \frac{D}{20} \text{ hours} \]

### Step 6: Calculate the average speed
Average speed \( V \) is given by:
\[ V = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{D}{\frac{D}{20}} = 20 \text{ km/h} \]

Thus, the average speed for the entire journey is **20 km/h**.