A passenger train takes two hours less for a journey of 300 km if its speed is increased by 5 km/h from its normal speed. The normal speed of the train is

Options

A. 35 km/h
B. 50 km/h
C. 25 km/h
D. 30 km/h

Answer

To find the normal speed of the train, we can follow these steps:

### Step 1: Define the Normal Speed
Let the normal speed of the train be \( x \) km/h.

### Step 2: Define the Increased Speed
If the speed is increased by 5 km/h, the new speed will be \( x + 5 \) km/h.

### Step 3: Write the Time Taken for Each Speed
The time taken to travel 300 km at normal speed \( x \) is given by:
\[ \text{Time at normal speed} = \frac{300}{x} \]
The time taken to travel 300 km at increased speed \( x + 5 \) is:
\[ \text{Time at increased speed} = \frac{300}{x + 5} \]

### Step 4: Set Up the Equation
According to the problem, the difference in time between the two speeds is 2 hours. Therefore, we can set up the equation:
\[ \frac{300}{x} - \frac{300}{x + 5} = 2 \]

### Step 5: Solve the Equation
To solve the equation, we first find a common denominator:
\[ \frac{300(x + 5) - 300x}{x(x + 5)} = 2 \]
This simplifies to:
\[ \frac{1500}{x(x + 5)} = 2 \]
Cross-multiplying gives:
\[ 1500 = 2x(x + 5) \]
Expanding the right side:
\[ 1500 = 2x^2 + 10x \]
Rearranging the equation:
\[ 2x^2 + 10x - 1500 = 0 \]
Dividing the entire equation by 2:
\[ x^2 + 5x - 750 = 0 \]

### Step 6: Factor or Use the Quadratic Formula
Now we can factor or use the quadratic formula to solve for \( x \). The quadratic formula is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1 \), \( b = 5 \), and \( c = -750 \):
\[ x = \frac{-5 \pm \sqrt{5^2 - 4 \cdot 1 \cdot (-750)}}{2 \cdot 1} \]
Calculating the discriminant:
\[ x = \frac{-5 \pm \sqrt{25 + 3000}}{2} \]
\[ x = \frac{-5 \pm \sqrt{3025}}{2} \]
\[ x = \frac{-5 \pm 55}{2} \]
Calculating the two possible values:
1. \( x = \frac{50}{2} = 25 \)
2. \( x = \frac{-60}{2} = -30 \) (not valid since speed cannot be negative)

### Conclusion
Thus, the normal speed of the train is:
\[ \boxed{25 \text{ km/h}} \]