Assertion : The apparent weight of a body in an elevator moving with some downward acceleration is less than the actual weight of body.
Reason : The part of the weight is spent in producing downward acceleration, when body is in elevator.

Explanation

To solve the given question, we need to analyze both the assertion and the reason provided.

### Step-by-Step Solution:

1. **Understanding the Situation**:
- We have a body of mass \( m \) placed in an elevator (lift) that is accelerating downwards.
- The forces acting on the body are its weight \( W = mg \) acting downwards and the normal force \( N \) acting upwards.

2. **Applying Newton's Second Law**:
- According to Newton's second law, the net force acting on the body can be expressed as:
\[ F_{\text{net}} = ma \]
- Here, \( a \) is the downward acceleration of the elevator.

3. **Setting Up the Equation**:
- The net force acting on the body can also be expressed as the difference between the weight and the normal force:
\[ F_{\text{net}} = mg - N \]
- Equating the two expressions for net force gives:
\[ mg - N = ma \]

4. **Solving for Normal Force**:
- Rearranging the equation to solve for the normal force \( N \):
\[ N = mg - ma \]
- This shows that the normal force (which represents the apparent weight of the body) is less than the actual weight \( mg \) since \( ma \) is a positive quantity.

5. **Conclusion about Apparent Weight**:
- Since \( N = mg - ma \), it is clear that \( N < mg \). Therefore, the apparent weight of the body in the elevator moving with downward acceleration is indeed less than its actual weight.

6. **Analyzing the Reason**:
- The reason states that "part of the weight is spent in producing downward acceleration." This is correct because the force \( ma \) (which is the product of mass and downward acceleration) is effectively reducing the normal force acting on the body.

7. **Final Assessment**:
- Both the assertion and the reason are correct, and the reason correctly explains the assertion.

### Final Answer:
- **Assertion**: True
- **Reason**: True
- The reason is the correct explanation for the assertion.