A man started 20 min late and traveling at a speed of ` 1 (1)/(2)` times of his usual speed reaches his office in times the time taken by the man to reaches his office in time the time taken by the man to reach his office at his usual speed is

Options

A. 40 min
B. 1 h 20 min
C. 1 h
D. 30 min

Answer

To solve the problem step by step, let's denote the following:

- Let the usual speed of the man be \( S \).
- Let the usual time taken to reach the office be \( T \).

### Step 1: Understand the relationship between speed, time, and distance
The distance to the office can be expressed as:
\[ \text{Distance} = \text{Speed} \times \text{Time} \]
Thus, the distance to the office is:
\[ D = S \times T \]

### Step 2: Determine the current speed of the man
The man is traveling at a speed of \( 1 \frac{1}{2} \) times his usual speed. This can be expressed as:
\[ \text{Current Speed} = \frac{3}{2} S \]

### Step 3: Calculate the time taken to reach the office when late
Since the man started 20 minutes late, let’s denote the time taken to reach the office at the current speed as \( T' \). According to the problem, he reaches the office in the same time he would have taken at his usual speed if he had not been late, which is \( T \).

### Step 4: Set up the equation for the time taken at current speed
Using the formula for distance, we can express the time taken at the current speed:
\[ D = \text{Current Speed} \times T' \]
Substituting the values we have:
\[ S \times T = \left(\frac{3}{2} S\right) \times (T - \frac{1}{3}) \quad \text{(since he is 20 minutes late, which is } \frac{1}{3} \text{ hours)} \]

### Step 5: Simplify the equation
Cancelling \( S \) from both sides (assuming \( S \neq 0 \)):
\[ T = \frac{3}{2} (T - \frac{1}{3}) \]
Now, distribute \( \frac{3}{2} \):
\[ T = \frac{3}{2} T - \frac{3}{6} \]
This simplifies to:
\[ T = \frac{3}{2} T - \frac{1}{2} \]

### Step 6: Rearranging the equation
Rearranging gives:
\[ T - \frac{3}{2} T = -\frac{1}{2} \]
\[ -\frac{1}{2} T = -\frac{1}{2} \]
Thus:
\[ T = 1 \text{ hour} \]

### Conclusion
The usual time taken by the man to reach his office at his usual speed is **60 minutes** or **1 hour**.