Raju travels at half of his usual speed and reaches his office 24 minutes late than his usual time. If Raju travels at his usual speed, then what will be the time taken by him to reach his office?

Options

A. 64 minutes
B. 24 minutes
C. 48 minutes
D. 56 minutes

Answer

To solve the problem, we need to establish the relationship between speed, time, and distance. Let's denote:

- Raju's usual speed as \( S \) (in km/h).
- The time taken by Raju to reach his office at his usual speed as \( T \) (in minutes).
- The distance to his office as \( D \) (in km).

### Step 1: Establish the relationship between speed, time, and distance

Using the formula for distance:
\[ D = S \times T \]

### Step 2: Determine the time taken at half speed

When Raju travels at half of his usual speed, his speed becomes \( \frac{S}{2} \). The time taken to cover the same distance \( D \) at this speed is given by:
\[ \text{Time at half speed} = \frac{D}{\frac{S}{2}} = \frac{2D}{S} \]

### Step 3: Relate the times

According to the problem, when Raju travels at half speed, he is 24 minutes late. This means:
\[ \text{Time at half speed} = T + 24 \]

### Step 4: Set up the equation

Now we can set up the equation based on the time taken at half speed:
\[ \frac{2D}{S} = T + 24 \]

### Step 5: Substitute \( D \)

From our first equation \( D = S \times T \), we can substitute \( D \) in the equation:
\[ \frac{2(S \times T)}{S} = T + 24 \]

This simplifies to:
\[ 2T = T + 24 \]

### Step 6: Solve for \( T \)

Now, we can solve for \( T \):
\[ 2T - T = 24 \\ T = 24 \text{ minutes} \]

### Conclusion

Thus, the time taken by Raju to reach his office at his usual speed is **24 minutes**.

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