Calculating Speed to Reach Station On Time

This problem involves understanding the relationship between speed, distance, and time, specifically when dealing with scenarios where a target time (train departure) is missed by a certain margin.

Let's define the variables:

• Let \(D\) be the distance from the man's starting point to the station in km.
• Let \(T\) be the exact time required to reach the station on time in hours.

The fundamental formula connecting these is: $$ \text{Distance} = \text{Speed} \times \text{Time} $$ which can be rearranged to find time or speed.

Analyzing the Given Scenarios

Scenario 1: Speed = 4 kmph

The man travels at 4 kmph and misses the train by 1 hour. This means the time taken is 1 hour more than the required time \(T\).

• Speed \(S_1 = 4\) kmph
• Time taken \(t_1 = T + 1\) hours
• Distance \(D = S_1 \times t_1 = 4 \times (T + 1)\)
• So, the equation is: $$ D = 4T + 4 \quad (Equation\ 1) $$

Scenario 2: Speed = 5 kmph

The man increases his speed to 5 kmph but still misses the train by 24 minutes. First, convert 24 minutes to hours: $$ 24 \text{ minutes} = \frac{24}{60} \text{ hours} = \frac{2}{5} \text{ hours} = 0.4 \text{ hours} $$

The time taken in this scenario is 0.4 hours more than the required time \(T\).

• Speed \(S_2 = 5\) kmph
• Time taken \(t_2 = T + 0.4\) hours
• Distance \(D = S_2 \times t_2 = 5 \times (T + 0.4)\)
• So, the equation is: $$ D = 5T + 2 \quad (Equation\ 2) $$

Solving for Distance (D) and Required Time (T)

We have two equations for the same distance \(D\):

Equation 1: \(D = 4T + 4\)

Equation 2: \(D = 5T + 2\)

Since both expressions equal \(D\), we can set them equal to each other:

Now, we solve for \(T\):

So, the required time to reach the station exactly on time is 2 hours.

Now, substitute the value of \(T = 2\) hours back into either Equation 1 or Equation 2 to find the distance \(D\).

Using Equation 1:

The distance to the station is 12 km.

Calculating the Speed Needed to be On Time

We need to find the speed \(S_3\) required to travel the distance \(D = 12\) km in exactly the required time \(T = 2\) hours.

Therefore, the man should have travelled at a speed of 6 kmph to reach the station exactly on time.

Revision Table: Speed, Time, and Distance Summary

Additional Information: Distance-Speed-Time Problems

Distance-speed-time problems are common in quantitative aptitude tests. They often involve scenarios with different speeds, times, or distances. Key concepts include:

• Understanding the formula: \(D = S \times T\).
• Converting units consistently (e.g., minutes to hours, kmph to m/s).
• Handling situations where times are relative (e.g., arriving early or late compared to a scheduled time). Setting up equations based on these relative times is crucial.
• Solving simultaneous equations when two different scenarios provide relationships between unknown variables like distance and required time.
• These problems can sometimes be solved using ratios if one variable is constant (like distance in this case), but the algebraic method using equations is generally more versatile.