The law of variable proportions is one of the most important, fundamental and unchallenged law of production. This law is also termed as return to a factor, as under it one factor is varied, while keeping all other factors fixed. With these variations in the quantity of one factor, keeping the quantity of other factors constant, the ratio of employment of the variable factor to that of the fixed factor keeps on changing. As we study the effects of variations in factor proportions under this law, this is called the law of variable proportions.

The law of variable proportion is a fundamental concept in economics that explains the relationship between inputs and outputs in production. It states that as the proportion of one input is varied while keeping the other inputs fixed, the marginal product of the variable input will eventually decline.

The correct answer to the question is option 'A'. Let's understand why this is the case:

**Explanation:**

**1. Law of Variable Proportion:**
The law of variable proportion, also known as the law of diminishing marginal returns, states that when one input is varied while keeping all other inputs fixed, there will come a point where the marginal product of the variable input will start to decline. In simple terms, it means that as you increase the quantity of one input, the additional output produced will eventually decrease.

**2. Only One Input is Variable and All Other Inputs are Fixed:**
Option 'A' states that only one input is variable and all other inputs are fixed. This is the correct condition for the law of variable proportion to be valid. When all factors of production except one are held constant, it allows us to isolate the impact of the variable input on the output.

**3. Fixed Inputs:**
The fixed inputs are those factors of production that are held constant during the production process. These inputs do not change in quantity regardless of the level of production. For example, in the context of a manufacturing firm, fixed inputs can include the size of the factory, machinery, and land.

**4. Variable Input:**
The variable input is the factor of production that can be varied in quantity during the production process. This input can be increased or decreased based on the desired level of output. For instance, in the manufacturing firm example, the variable input can be labor. By increasing or decreasing the number of workers, the firm can adjust its level of production.

**5. Marginal Product:**
The marginal product refers to the additional output that is produced when one additional unit of the variable input is added while keeping all other inputs fixed. Initially, the marginal product of the variable input may increase, indicating increasing returns to scale. However, as the variable input continues to increase, the marginal product will eventually start to decline.

**6. Conclusion:**
In conclusion, the law of variable proportion is valid when only one input is variable and all other inputs are fixed. This condition allows us to observe the impact of the variable input on the output, as the marginal product of the variable input declines. It is important to understand this concept in order to make informed production decisions and optimize resource allocation.

Answer of this question should be D as in law of variable proportion only one input is fixed and other input are variable