A person saves 5 min in covering a certain distance by increasing his speed by `20%` . What is the time taken to cover the distance at his usual speed ?

Options

A. 25 min
B. 30 min
C. 45 min
D. 50 min

Answer

To solve the problem, we can follow these steps:

1. **Define Variables**:
- Let the distance be \( d \).
- Let the usual speed be \( v \).
- Let the usual time taken to cover the distance be \( t \).

2. **Relate Distance, Speed, and Time**:
- We know that \( d = v \times t \).

3. **Calculate New Speed**:
- When the speed is increased by 20%, the new speed becomes:
\[ \text{New Speed} = v + 0.2v = 1.2v \]

4. **Calculate Time at New Speed**:
- The time taken to cover the same distance \( d \) at the new speed is:
\[ \text{New Time} = \frac{d}{\text{New Speed}} = \frac{d}{1.2v} \]

5. **Express New Time in Terms of Usual Time**:
- Since \( d = vt \), we can substitute \( d \) in the new time equation:
\[ \text{New Time} = \frac{vt}{1.2v} = \frac{t}{1.2} \]

6. **Set Up the Equation**:
- According to the problem, the person saves 5 minutes by increasing his speed. Therefore, we can set up the equation:
\[ t - \frac{t}{1.2} = 5 \]

7. **Solve the Equation**:
- To solve for \( t \), first simplify the left side:
\[ t - \frac{t}{1.2} = t \left(1 - \frac{1}{1.2}\right) = t \left(\frac{1.2 - 1}{1.2}\right) = t \left(\frac{0.2}{1.2}\right) = \frac{t}{6} \]
- Now, we have:
\[ \frac{t}{6} = 5 \]
- Multiplying both sides by 6 gives:
\[ t = 30 \]

8. **Conclusion**:
- The time taken to cover the distance at his usual speed is \( 30 \) minutes.