If a person travels in a car at a speed of 30 km/h then he would reach his destination on time . He covers half journey in 4/5 th time. What should be his speed for the remaining part of the journey so that he reaches his destination on time ?

Answer

To find the speed required for the remaining part of the journey so that the person reaches his destination on time, we can follow these steps:

### Step 1: Define the total distance and time
Let the total distance to the destination be \( D \) km. The speed of the car is given as 30 km/h.

### Step 2: Calculate the total time to reach the destination
The total time \( T \) to reach the destination can be calculated using the formula:
\[ T = \frac{D}{\text{Speed}} = \frac{D}{30} \text{ hours} \]

### Step 3: Determine the distance covered in the first half of the journey
Since the person covers half of the journey, the distance for the first half is:
\[ \text{Distance for half journey} = \frac{D}{2} \]

### Step 4: Calculate the time taken for the first half of the journey
The time taken for the first half of the journey is given as \( \frac{4}{5} \) of the total time:
\[ \text{Time for half journey} = \frac{4}{5} \times T = \frac{4}{5} \times \frac{D}{30} = \frac{4D}{150} = \frac{2D}{75} \text{ hours} \]

### Step 5: Calculate the remaining time for the second half of the journey
The remaining time \( T_{\text{remaining}} \) for the second half of the journey is:
\[ T_{\text{remaining}} = T - \text{Time for half journey} = \frac{D}{30} - \frac{2D}{75} \]
To subtract these fractions, we need a common denominator. The least common multiple of 30 and 75 is 150:
\[ T = \frac{D}{30} = \frac{5D}{150}, \quad \text{Time for half journey} = \frac{2D}{75} = \frac{4D}{150} \]
Now, substituting these values:
\[ T_{\text{remaining}} = \frac{5D}{150} - \frac{4D}{150} = \frac{1D}{150} \text{ hours} \]

### Step 6: Calculate the distance for the second half of the journey
The distance for the second half of the journey is also \( \frac{D}{2} \).

### Step 7: Determine the required speed for the second half of the journey
The speed \( S \) required for the second half of the journey can be calculated using the formula:
\[ S = \frac{\text{Distance}}{\text{Time}} = \frac{\frac{D}{2}}{\frac{D}{150}} = \frac{D}{2} \times \frac{150}{D} = \frac{150}{2} = 75 \text{ km/h} \]

### Conclusion
The speed required for the remaining part of the journey so that the person reaches his destination on time is **75 km/h**.

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