To solve this problem, we need to determine the minimum flow on the road from A to B such that traffic keeps flowing smoothly in the entire network. Here's the step-by-step process:

Step 1: Analyze the Flow Network

We are given a traffic network where:

1. Flow rates on various roads are represented as variables $x_1, x_2, x_3, x_4, x_5, x_6$.
2. Flow conservation must hold at every intersection (nodes A and B). That is:
   • Flow into a node = Flow out of the node.

Step 2: Flow Conservation at Each Node

At Node A:

The incoming flow to node A is: $325 + x_3 = x_1 + x_4$ Thus, the equation becomes: 325 + x_3 = x_1 + x_4 \tag{1}

At Node B:

The incoming flow to node B is: $x_4 + 500 = x_3 + x_2$ Thus, the equation becomes: x_4 + 500 = x_3 + x_2 \tag{2}

Step 3: Minimum Flow from A to B

The road from A to B corresponds to $x_4$. To ensure the flow remains consistent throughout the network, $x_4$ must satisfy these constraints while maintaining flow conservation.

Step 4: Solve Equations

Combine equations (1) and (2) to solve for $x_4$:

1. Rearrange $325 + x_3 = x_1 + x_4$: x_4 = 325 + x_3 - x_1 \tag{3}
2. Rearrange $x_4 + 500 = x_3 + x_2$: x_4 = x_3 + x_2 - 500 \tag{4}
3. Equate equations (3) and (4): $325 + x_3 - x_1 = x_3 + x_2 - 500$ Simplify: x_2 - x_1 = 825 \tag{5}

Step 5: Use Known Flow Rates

From the diagram:

• $x_1 = 250$ (outflow from A to another road),
• $x_2 = 200$ (outflow from B to another road).

Substitute $x_1 = 250$ and $x_2 = 200$ into equation (5): $200 - 250 = 825$ Thus: