Distance Calculation: Nehrunagar to Indiranagar

The problem involves calculating the distance to a meeting point based on different speeds and a round trip. Let's break down the journey of A and B.

Problem Setup

• Total distance (Nehrunagar to Gandhinagar) = 27 km.
• Speed of A ($v_A$) = 5 km/hr.
• Speed of B ($v_B$) = 7 km/hr.
• A starts from Nehrunagar towards Gandhinagar.
• B starts from Nehrunagar towards Gandhinagar, reaches Gandhinagar, turns back immediately, and meets A at Indiranagar.
• We need to find the distance between Nehrunagar and Indiranagar.

Meeting Point Analysis

Let $t$ be the total time elapsed from the start until A and B meet at Indiranagar. Let $d_{NI}$ be the distance from Nehrunagar to Indiranagar.

• Distance covered by A in time $t$ is $d_{NI}$. So, $d_{NI} = v_A \times t = 5t$.
• B travels 27 km to reach Gandhinagar and then travels back towards Nehrunagar until meeting A. The total distance covered by B is $27 \text{ km} + \text{distance B travelled back}$.
• The meeting point Indiranagar is $d_{NI}$ km from Nehrunagar. Therefore, the distance B travelled back from Gandhinagar is $27 - d_{NI}$ km.
• Total distance covered by B in time $t$ is $27 + (27 - d_{NI}) = 54 - d_{NI}$ km.
• We also know that the distance covered by B is $d_B = v_B \times t = 7t$.
• So, $7t = 54 - d_{NI}$.

Solving for Indiranagar Distance

We have two equations:

1. $d_{NI} = 5t$
2. $7t = 54 - d_{NI}$

From equation 1, we can express time $t$ as $t = \frac{d_{NI}}{5}$. Substitute this into equation 2:

$7 \left( \frac{d_{NI}}{5} \right) = 54 - d_{NI}

Multiply both sides by 5:

$7 d_{NI} = 5 (54 - d_{NI})

Distribute the 5:

$7 d_{NI} = 270 - 5 d_{NI}

Add $5 d_{NI}$ to both sides:

$7 d_{NI} + 5 d_{NI} = 270

Combine like terms:

$12 d_{NI} = 270

Solve for $d_{NI}$:

$d_{NI} = \frac{270}{12}

Simplify the fraction:

$d_{NI} = \frac{45}{2} = 22.5 \text{ km}

Conclusion

The distance between Nehrunagar and Indiranagar is 22.5 km.