A farmer purchased several animals from a neighboring farmer: 6 animals costing `$100` each, 10 animals costing `$200` each, and k animals costing `$400` each, where k is a positive odd integer. If the median price for all the animals was `$200`, what is the greatest possible value of k?

Answer

To find the greatest possible value of \( k \) such that the median price of the animals is $200, we can follow these steps:

### Step 1: Determine the total number of animals
The total number of animals purchased by the farmer is given by:
\[ 6 \text{ (animals at $100)} + 10 \text{ (animals at $200)} + k \text{ (animals at $400)} = 16 + k \]

### Step 2: Identify the median position
Since the median is the middle value, we need to determine the position of the median in the ordered list of animal prices. For \( n = 16 + k \) (which is the total number of animals), the median position is given by:
\[ \text{Median position} = \frac{(16 + k) + 1}{2} = \frac{17 + k}{2} \]

### Step 3: Ensure the median is $200
To have $200 as the median, we need to ensure that the median position falls within the range of animals priced at $200. The animals are priced as follows:
- 6 animals at $100
- 10 animals at $200
- \( k \) animals at $400

The first 6 animals will be priced at $100, and the next 10 animals (positions 7 to 16) will be priced at $200. Therefore, for the median to be $200, the median position must be at or below 16 (the last position of the $200 animals).

### Step 4: Set up the inequality
To ensure that the median position is at most 16, we set up the inequality:
\[ \frac{17 + k}{2} \leq 16 \]

### Step 5: Solve the inequality
Multiplying both sides by 2 gives:
\[ 17 + k \leq 32 \]
Subtracting 17 from both sides results in:
\[ k \leq 15 \]

### Step 6: Ensure \( k \) is a positive odd integer
Since \( k \) must be a positive odd integer, the greatest possible value of \( k \) that satisfies \( k \leq 15 \) is:
\[ k = 15 \]

### Conclusion
Thus, the greatest possible value of \( k \) is:
\[ \boxed{15} \]