The difference between the time taken by two buses to travel a distance of 350 km is 2 hours 20 minutes. If the difference between their speeds is 5 kmph, find the slower speed.

Options

A. 35 kmph
B. 30 kmph
C. 25 kmph
D. 20 kmph

Answer

To solve the problem step by step, we will follow these steps:

### Step 1: Define Variables
Let the speed of the faster bus be \( V_1 \) km/h and the speed of the slower bus be \( V_2 \) km/h.

### Step 2: Set Up the First Equation
According to the problem, the difference between their speeds is given as:
\[ V_1 - V_2 = 5 \quad \text{(Equation 1)} \]

### Step 3: Convert Time Difference
The difference in time taken by the two buses to travel 350 km is given as 2 hours and 20 minutes. We convert this into hours:
\[ 2 \text{ hours } 20 \text{ minutes} = 2 + \frac{20}{60} = 2 + \frac{1}{3} = \frac{7}{3} \text{ hours} \]

### Step 4: Set Up the Second Equation
The time taken by the faster bus to cover 350 km is:
\[ T_1 = \frac{350}{V_1} \]

The time taken by the slower bus to cover the same distance is:
\[ T_2 = \frac{350}{V_2} \]

According to the problem, the difference in time taken is:
\[ T_2 - T_1 = \frac{7}{3} \]
Substituting for \( T_1 \) and \( T_2 \):
\[ \frac{350}{V_2} - \frac{350}{V_1} = \frac{7}{3} \quad \text{(Equation 2)} \]

### Step 5: Simplify Equation 2
Multiply through by \( V_1 \times V_2 \) to eliminate the denominators:
\[ 350V_1 - 350V_2 = \frac{7}{3} V_1 V_2 \]

### Step 6: Rearranging the Equation
Rearranging gives:
\[ 350(V_1 - V_2) = \frac{7}{3} V_1 V_2 \]

### Step 7: Substitute Equation 1 into the Rearranged Equation
From Equation 1, we know \( V_1 - V_2 = 5 \):
\[ 350 \times 5 = \frac{7}{3} V_1 V_2 \]
\[ 1750 = \frac{7}{3} V_1 V_2 \]

### Step 8: Solve for \( V_1 V_2 \)
Multiply both sides by 3:
\[ 5250 = 7 V_1 V_2 \]
Now divide by 7:
\[ V_1 V_2 = 750 \quad \text{(Equation 3)} \]

### Step 9: Substitute \( V_1 \) in terms of \( V_2 \)
From Equation 1, we have:
\[ V_1 = V_2 + 5 \]
Substituting into Equation 3:
\[ (V_2 + 5)V_2 = 750 \]
\[ V_2^2 + 5V_2 - 750 = 0 \]

### Step 10: Solve the Quadratic Equation
Now we can solve the quadratic equation:
Using the quadratic formula \( V_2 = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
Here, \( a = 1, b = 5, c = -750 \):
\[ V_2 = \frac{-5 \pm \sqrt{5^2 - 4 \cdot 1 \cdot (-750)}}{2 \cdot 1} \]
\[ V_2 = \frac{-5 \pm \sqrt{25 + 3000}}{2} \]
\[ V_2 = \frac{-5 \pm \sqrt{3025}}{2} \]
\[ V_2 = \frac{-5 \pm 55}{2} \]

Calculating the two potential solutions:
1. \( V_2 = \frac{50}{2} = 25 \) (valid speed)
2. \( V_2 = \frac{-60}{2} = -30 \) (not valid)

### Conclusion
Thus, the slower speed \( V_2 \) is:
\[ V_2 = 25 \text{ km/h} \]