To solve this problem, we need to analyze the circuit and use Ohm's Law. The circuit has two branches with resistances, currents, and a voltage source $V_B$. We will calculate the resistance $R$ based on the given current and resistance values.

Step-by-Step Solution

1. Identify the known values:

   • The upper branch has a 1 A current through a 15 Ω resistor in series with an unknown resistor $R$.
   • The lower branch has a 2 A current through a 5 Ω resistor in series with the same resistor $R$.

2. Calculate the voltage drop in each branch: Since both branches are connected in parallel, the total voltage drop across each branch must be the same (i.e., across the 15 Ω and $R$ in the upper branch, and across the 5 Ω and $R$ in the lower branch).

3. Set up the equations:

   • Upper Branch: The voltage drop across the 15 Ω resistor and $R$ must equal $V_B$. $V_B = (1 \text{ A}) \times 15 \Omega + (1 \text{ A}) \times R = 15 + R$
   • Lower Branch: The voltage drop across the 5 Ω resistor and $R$ must also equal $V_B$. $V_B = (2 \text{ A}) \times 5 \Omega + (2 \text{ A}) \times R = 10 + 2R$

4. Equate the voltage expressions: Since both expressions represent $V_B$, we can set them equal to each other: $15 + R = 10 + 2R$

5. Solve for $R$: Rearrange the equation: $15 - 10 = 2R - R$ $5 = R$

Answer:

The value of $R$ is $5 \Omega$, which corresponds to option (c).

Would you like a more detailed explanation or have any questions?

Here are five related questions to further explore this topic:

1. How would you approach finding the equivalent resistance in parallel and series circuits?
2. What would happen to the current distribution if $R$ were a different value?
3. How would the circuit change if we added another resistor in one of the branches?
4. What formulas are essential for analyzing parallel circuits with multiple resistors?
5. How do you determine the total power consumption in this type of circuit?

Tip: Remember that in parallel circuits, the voltage across each branch remains the same, which is crucial for solving for unknown resistances.