We are given that a person travels by car at 30 km/h to reach the destination in time. However, for the first half of the journey, the person covers it in 4/5th of the total time. We need to determine the speed at which the person should travel for the rest of the journey to still arrive on time.

Steps:

1. Define variables:

   • Let the total distance of the journey be $D$ (in kilometers).
   • The total time to complete the journey is $T$ (in hours).
   • The speed for the first part of the journey is 30 km/h.

2. For the first part of the journey:

   • The person covers half the distance in $\frac{4}{5}$ of the total time.
   • The distance for the first half is $\frac{D}{2}$.
   • The time taken for the first part is $\frac{4}{5} T$.

3. Time taken for the first part using speed:

   • The formula for time is $\text{Time} = \frac{\text{Distance}}{\text{Speed}}$.
   • For the first part, the time taken should be $\frac{D}{2} \div 30 = \frac{D}{60}$.

4. Equating the two expressions for the time of the first part: $\frac{D}{60} = \frac{4}{5} T$ Solving for $T$: $T = \frac{5D}{240} = \frac{D}{48}$

5. For the second part of the journey:

   • The remaining distance is also $\frac{D}{2}$.
   • The time left for the second part is $T - \frac{4}{5}T = \frac{1}{5}T$.

6. Substitute $T$ into the remaining time: $\frac{1}{5}T = \frac{1}{5} \times \frac{D}{48} = \frac{D}{240}$

7. Find the required speed for the second part:

   • The speed needed for the second part of the journey is $\text{Speed} = \frac{\text{Distance}}{\text{Time}}$.
   • The required speed for the second part is: $\text{Speed} = \frac{\frac{D}{2}}{\frac{D}{240}} = 120 \text{ km/h}$

Final Answer:

The person should travel at 120 km/h for the second part of the journey to reach on time.

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Would you like further clarification or details on any of the steps? Here are 5 related questions to explore:

1. How does the speed-time-distance relationship work in a typical journey?
2. How would this change if the person covered a different fraction of the journey in the first part?
3. What if the person had taken longer than the given time for the first part—how would that impact the necessary speed?
4. How can we generalize this problem to apply to different scenarios with varying speeds and times?
5. Can we model this problem using algebraic inequalities to show how the person’s time and speed are related?

Tip: Always break down a problem involving time, speed, and distance into smaller parts. Relating the fractions of the journey and time can help simplify solving for unknown variables!