To solve the problem, we need to analyze the circuit and apply Ohm's Law, which states that \( V = I \times R \), where \( V \) is the voltage, \( I \) is the current, and \( R \) is the resistance. ### Step-by-Step Explanation 1. **Understanding the Circuit**: - The circuit consists of a voltage source (which we will denote as \( V \)) and two resistors: the unknown resistance \( X \) and a variable resistor \( R \). The ammeter measures the current flowing through the circuit. 2. **Using the Given Data**: - When \( R = 8 \, \Omega \), the current \( I_1 = 5.0 \, A \). - When \( R = 5 \, \Omega \), the current \( I_2 = 7.0 \, A \). 3. **Applying Ohm's Law**: - For the first scenario (when \( R = 8 \, \Omega \)): \[ V = I_1 \times (R + X) = 5.0 \times (8 + X) \] - For the second scenario (when \( R = 5 \, \Omega \)): \[ V = I_2 \times (R + X) = 7.0 \times (5 + X) \] 4. **Setting the Equations Equal**: - Since the voltage \( V \) is the same in both cases, we can set the two equations equal to each other: \[ 5.0 \times (8 + X) = 7.0 \times (5 + X) \] 5. **Expanding Both Sides**: - Expanding the left side: \[ 40 + 5X \] - Expanding the right side: \[ 35 + 7X \] 6. **Setting Up the Equation**: - Now we have: \[ 40 + 5X = 35 + 7X \] 7. **Rearranging the Equation**: - To isolate \( X \), we can rearrange the equation: \[ 40 - 35 = 7X - 5X \] \[ 5 = 2X \] 8. **Solving for \( X \)**: - Dividing both sides by 2 gives: \[ X = \frac{5}{2} = 2.5 \, \Omega \] ### Conclusion The value of the unknown resistance \( X \) is **2.5Ω**, which corresponds to option **D**. ### Explanation of Other Options - **Option A (10.0Ω)**: This value is too high based on the current readings. If \( X \) were 10Ω, the current would be significantly lower than observed. - **Option B (7.5Ω)**: This value does not satisfy the equations derived from the current readings. It would lead to a voltage that does not match the conditions given. - **Option C (5.0Ω)**: Similar to option B, this value does not satisfy the conditions of the circuit based on the current readings. ### Revision Summary - Use Ohm's Law \( V = I \times R \) to analyze circuits. - Set up equations based on different conditions in the circuit. - Rearrange and solve for unknowns carefully. - Always check if the derived value satisfies the original conditions of the problem.