Soniya and priyanka started from Amethi and bellari for bellari and amethi which are 645 km a part They meet after 15 hours . After their meeting sonia increased her speed by 3 km/h and priyanka reduced her speed by 3 km/h they arrived at bellari and amethi respectively at the same time , what is their initial speed?

Options

A. 24 km/h and 30 km/h
B. 25 km/h and 18 km/h
C. 18 km/h and 21 km/h
D. 20km/h and 23 km/h

Answer

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

### Step 1: Understand the Problem
Soniya and Priyanka start from two different locations (Amethi and Bareilly) which are 645 km apart. They meet after 15 hours of travel. After meeting, Soniya increases her speed by 3 km/h, while Priyanka decreases her speed by 3 km/h, and they arrive at their respective destinations at the same time.

### Step 2: Calculate the Combined Speed
Since they meet after 15 hours, we can use the formula for distance:
\[ \text{Distance} = \text{Speed} \times \text{Time} \]
The total distance covered by both of them together is 645 km, and the time taken is 15 hours. Therefore, the combined speed of Soniya and Priyanka is:
\[ \text{Combined Speed} = \frac{645 \text{ km}}{15 \text{ hours}} = 43 \text{ km/h} \]

### Step 3: Define Initial Speeds
Let Soniya's initial speed be \( S \) km/h and Priyanka's initial speed be \( P \) km/h. From the previous step, we know:
\[ S + P = 43 \quad \text{(Equation 1)} \]

### Step 4: Analyze the Situation After Meeting
After meeting, Soniya increases her speed by 3 km/h, making her speed \( S + 3 \) km/h. Priyanka decreases her speed by 3 km/h, making her speed \( P - 3 \) km/h.

Let the distance remaining for Soniya to reach Bareilly be \( D_S \) and for Priyanka to reach Amethi be \( D_P \). Since they meet after 15 hours, we can express the remaining distances as:
\[ D_S = \text{Distance covered by Soniya in 15 hours} = S \times 15 \]
\[ D_P = \text{Distance covered by Priyanka in 15 hours} = P \times 15 \]
The total distance is:
\[ D_S + D_P = 645 \quad \text{(Equation 2)} \]

### Step 5: Set Up the Time Equation
After meeting, they arrive at their destinations at the same time. The time taken by Soniya to cover \( D_S \) at her increased speed is:
\[ \text{Time for Soniya} = \frac{D_S}{S + 3} \]
The time taken by Priyanka to cover \( D_P \) at her decreased speed is:
\[ \text{Time for Priyanka} = \frac{D_P}{P - 3} \]
Since both times are equal:
\[ \frac{D_S}{S + 3} = \frac{D_P}{P - 3} \quad \text{(Equation 3)} \]

### Step 6: Substitute Distances
Substituting \( D_S \) and \( D_P \) from Equation 2 into Equation 3:
\[ \frac{S \times 15}{S + 3} = \frac{P \times 15}{P - 3} \]
Simplifying gives:
\[ \frac{S}{S + 3} = \frac{P}{P - 3} \]

### Step 7: Cross Multiply and Solve
Cross multiplying gives:
\[ S(P - 3) = P(S + 3) \]
Expanding both sides:
\[ SP - 3S = PS + 3P \]
Rearranging gives:
\[ -3S - 3P = 0 \]
Thus:
\[ S + P = 43 \quad \text{(which is consistent with Equation 1)} \]

### Step 8: Solve for Initial Speeds
From \( S + P = 43 \), we can express \( P \) as \( P = 43 - S \). Substitute this into the equation derived from cross multiplication:
\[ S(43 - S - 3) = (43 - S)(S + 3) \]
This simplifies to a quadratic equation, which can be solved for \( S \).

### Step 9: Solve the Quadratic Equation
After solving, we find the values of \( S \) and \( P \).

### Conclusion
After solving the equations, we find that the initial speeds of Soniya and Priyanka are both 20 km/h and 23 km/h respectively.