Ram starts from Delhi towards Goa. After sometime he realises that he will cover only 75% of the distance in the scheduled time and he therefore doubles his speed immediately and thus manages to reach Goa exactly on time. Find the time after which Ram changed his speed, given that he could have been late by 3 hours if he had not changed his speed:

Answer

To solve the problem, we need to analyze the situation step by step. Let's define some variables first:

- Let \( D \) be the total distance from Delhi to Goa.
- Let \( S \) be Ram's initial speed.
- Let \( T \) be the scheduled time to cover the distance \( D \).

### Step 1: Determine the distance covered in the scheduled time
In the scheduled time \( T \), Ram would have covered the entire distance \( D \). Thus, we can express this as:
\[ D = S \times T \]

### Step 2: Calculate the distance covered before changing speed
According to the problem, Ram realizes that he will only cover 75% of the distance in the scheduled time. Therefore, the distance he would cover at his initial speed \( S \) in time \( t \) is:
\[ \text{Distance covered} = 0.75D \]

### Step 3: Determine the remaining distance
The remaining distance to Goa after covering 75% is:
\[ \text{Remaining distance} = D - 0.75D = 0.25D \]

### Step 4: Calculate the time left for the journey
If Ram had continued at his initial speed \( S \), he would have been late by 3 hours. This means the time he would have taken to cover the remaining distance at speed \( S \) is:
\[ \text{Time taken at speed } S = \frac{0.25D}{S} \]

Since he would be late by 3 hours, we can express this as:
\[ \frac{0.25D}{S} = T - t + 3 \]
Where \( t \) is the time he has already traveled.

### Step 5: Express the time taken after changing speed
After realizing he would be late, Ram doubles his speed to \( 2S \). The time taken to cover the remaining distance \( 0.25D \) at the new speed is:
\[ \text{Time taken at speed } 2S = \frac{0.25D}{2S} = \frac{0.125D}{S} \]

### Step 6: Set up the equation
Now, we can set up the equation using the total time:
\[ t + \frac{0.125D}{S} = T \]

### Step 7: Substitute \( D \) in terms of \( S \) and \( T \)
From our earlier equation \( D = S \times T \), we substitute \( D \):
\[ t + \frac{0.125(S \times T)}{S} = T \]
This simplifies to:
\[ t + 0.125T = T \]

### Step 8: Solve for \( t \)
Rearranging gives:
\[ t = T - 0.125T \]
\[ t = 0.875T \]

### Step 9: Conclusion
Thus, the time after which Ram changed his speed is:
\[ t = 0.875T \]