If Deepesh had walked 20 km/h faster he would have saved 1 hour in the distance of 600 km.What is the usual speed of Dipesh?

Options

A. 100
B. 120
C. 150
D. none of these

Answer

To find the usual speed of Deepesh, we can set up the problem using the information given.

### Step-by-Step Solution:

1. **Define Variables:**
Let the usual speed of Deepesh be \( S \) km/h.

2. **Calculate Time at Usual Speed:**
The time taken to cover 600 km at the usual speed \( S \) is given by the formula:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} = \frac{600}{S} \]

3. **Calculate Time at Increased Speed:**
If Deepesh walked 20 km/h faster, his speed would be \( S + 20 \) km/h. The time taken to cover the same distance at this increased speed is:
\[ \text{Time} = \frac{600}{S + 20} \]

4. **Set Up the Equation:**
According to the problem, if Deepesh walked 20 km/h faster, he would save 1 hour. Therefore, we can set up the equation:
\[ \frac{600}{S} - \frac{600}{S + 20} = 1 \]

5. **Solve the Equation:**
To solve this equation, we first find a common denominator:
\[ \frac{600(S + 20) - 600S}{S(S + 20)} = 1 \]
Simplifying the numerator:
\[ \frac{600S + 12000 - 600S}{S(S + 20)} = 1 \]
This simplifies to:
\[ \frac{12000}{S(S + 20)} = 1 \]

6. **Cross Multiply:**
Cross multiplying gives us:
\[ 12000 = S(S + 20) \]
Expanding the right side:
\[ 12000 = S^2 + 20S \]

7. **Rearrange the Equation:**
Rearranging gives us a standard quadratic equation:
\[ S^2 + 20S - 12000 = 0 \]

8. **Use the Quadratic Formula:**
We can solve this quadratic equation using the quadratic formula:
\[ S = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1, b = 20, c = -12000 \):
\[ S = \frac{-20 \pm \sqrt{20^2 - 4 \cdot 1 \cdot (-12000)}}{2 \cdot 1} \]
\[ S = \frac{-20 \pm \sqrt{400 + 48000}}{2} \]
\[ S = \frac{-20 \pm \sqrt{48400}}{2} \]
\[ S = \frac{-20 \pm 220}{2} \]

9. **Calculate Possible Values:**
This gives us two possible solutions:
\[ S = \frac{200}{2} = 100 \quad \text{(valid speed)} \]
\[ S = \frac{-240}{2} = -120 \quad \text{(not valid)} \]

10. **Conclusion:**
Therefore, the usual speed of Deepesh is:
\[ \boxed{100 \text{ km/h}} \]