Understanding the Speed-Time Relationship

This problem involves calculating the usual time taken for a commute based on a change in speed and the resulting delay. We need to figure out the original time '$T$' when the man's speed was his usual speed '$S$', given that when he travels at $\frac{3}{4}$ of his usual speed, he arrives 4 minutes late.

Setting up the Equations

Let:

• '$S$' be the man's usual speed.
• '$T$' be the usual time (in minutes) he takes to reach the office.
• '$D$' be the distance to the office.

We know the fundamental relationship: Distance = Speed × Time. So, the distance to the office can be represented as:

$D = S \times T$

Now, consider the scenario where the man walks at three-fourth of his usual speed. His new speed is $\frac{3}{4}S$. Because of this slower speed, he reaches 4 minutes late. This means his new travel time is his usual time plus 4 minutes, which is $T + 4$ minutes.

The distance '$D$' remains the same. Therefore, we can write the equation for this new scenario:

$D = \left(\frac{3}{4}S\right) \times (T + 4)$

Calculating the Usual Time

Since the distance '$D$' is constant in both scenarios, we can set the two expressions for '$D$' equal to each other:

$S \times T = \left(\frac{3}{4}S\right) \times (T + 4)$

We can simplify this equation by canceling out the speed '$S$' from both sides (assuming $S \ne 0$):

$T = \frac{3}{4}(T + 4)$

Now, we solve for '$T$':

1. Multiply both sides by 4 to eliminate the fraction:

   $4 \times T = 4 \times \frac{3}{4}(T + 4)$

   $4T = 3(T + 4)$

2. Distribute the 3 on the right side:

   $4T = 3T + 12$

3. Subtract $3T$ from both sides to isolate '$T$':

   $4T - 3T = 12$

   $T = 12$

This means the usual time '$T$' the man takes to reach his office is 12 minutes.

Verification (Optional)

Let's check the result. If the usual time is 12 minutes, and he travels at $\frac{3}{4}$ speed:

• Usual Speed = $S$, Usual Time = 12 min. Distance = $12S$.
• New Speed = $\frac{3}{4}S$. New Time = $12 + 4 = 16$ min.
• Distance = New Speed × New Time = $(\frac{3}{4}S) \times 16 = \frac{48}{4}S = 12S$.

The distances match, confirming our calculation.

Conclusion

The man usually takes 12 minutes to reach his office.