A: If the amplitude of a message signal is half of the amplitude of the carrier wave, then the index of modulation is 50%.
R: In amplitude modulation the amplitude of the carrier wave varies in accordance with the signal voltage of the message signal.

Answer

To solve the question, we need to analyze the assertion and reason provided, and then derive the modulation index based on the information given.

### Step-by-step Solution:

1. **Understanding the Assertion**:
The assertion states that if the amplitude of a message signal is half of the amplitude of the carrier wave, then the index of modulation is 50%.

2. **Defining the Modulation Index**:
The modulation index (μ) in amplitude modulation is defined by the formula:
\[ \mu = \frac{A_m}{A_c} \]
where \( A_m \) is the amplitude of the message signal and \( A_c \) is the amplitude of the carrier wave.

3. **Assigning Values**:
According to the assertion, the amplitude of the message signal \( A_m \) is half of the amplitude of the carrier wave \( A_c \). This can be expressed as:
\[ A_m = \frac{1}{2} A_c \]

4. **Calculating the Modulation Index**:
Substituting \( A_m \) in the modulation index formula:
\[ \mu = \frac{A_m}{A_c} = \frac{\frac{1}{2} A_c}{A_c} \]
Simplifying this gives:
\[ \mu = \frac{1}{2} = 0.5 \]

5. **Converting to Percentage**:
To express the modulation index as a percentage:
\[ \mu = 0.5 \times 100\% = 50\% \]
Thus, the assertion is true.

6. **Understanding the Reason**:
The reason states that in amplitude modulation, the amplitude of the carrier wave varies in accordance with the signal voltage of the message signal. This is indeed true as the carrier wave's amplitude is modulated by the message signal.

7. **Conclusion**:
Both the assertion and the reason are true. However, the reason does not provide a direct explanation for the assertion. Therefore, the correct answer is that both statements are true, but the reason is not the correct explanation for the assertion.

### Final Answer:
Both assertion and reason are true, but the reason is not the correct explanation for the assertion.

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