A person travels half of the distance at the speed of x km/h and remaining half of the distance at 4x km/h. Find the value of 'x' if the average speed is 36.8 km/h?

Options

A. A)21
B. B)25
C. C)24
D. D)23

Answer

To solve the problem, we need to find the value of 'x' given that a person travels half of the distance at a speed of 'x' km/h and the other half at a speed of '4x' km/h, with an average speed of 36.8 km/h.

### Step-by-Step Solution:

1. **Define the Distance**:
Let the total distance be 'D'. Therefore, half of the distance is \( \frac{D}{2} \).

2. **Calculate Time for Each Half**:
- Time taken to travel the first half at speed \( x \) km/h:
\[ t_1 = \frac{\frac{D}{2}}{x} = \frac{D}{2x} \]
- Time taken to travel the second half at speed \( 4x \) km/h:
\[ t_2 = \frac{\frac{D}{2}}{4x} = \frac{D}{8x} \]

3. **Total Time**:
The total time \( T \) taken for the journey is:
\[ T = t_1 + t_2 = \frac{D}{2x} + \frac{D}{8x} \]
To combine these, find a common denominator (which is \( 8x \)):
\[ T = \frac{4D}{8x} + \frac{D}{8x} = \frac{5D}{8x} \]

4. **Average Speed Formula**:
The average speed \( V_{avg} \) is given by the formula:
\[ V_{avg} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{D}{T} \]
Substituting the expression for \( T \):
\[ V_{avg} = \frac{D}{\frac{5D}{8x}} = \frac{8x}{5} \]

5. **Set Average Speed Equal to Given Value**:
We know from the problem that the average speed is 36.8 km/h:
\[ \frac{8x}{5} = 36.8 \]

6. **Solve for 'x'**:
Cross-multiply to solve for \( x \):
\[ 8x = 36.8 \times 5 \]
Calculate \( 36.8 \times 5 = 184 \):
\[ 8x = 184 \]
Divide both sides by 8:
\[ x = \frac{184}{8} = 23 \]

### Final Answer:
The value of \( x \) is **23 km/h**.

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