The sum of two numbers is 18. The greatest product of these two numbers can be :

Options

A. 17
B. 81
C. 80
D. can't be determined

Answer

To find the greatest product of two numbers whose sum is 18, we can follow these steps:

### Step 1: Define the two numbers
Let the two numbers be \( A \) and \( B \). According to the problem, we know that:
\[ A + B = 18 \]

### Step 2: Express one variable in terms of the other
From the equation \( A + B = 18 \), we can express \( B \) in terms of \( A \):
\[ B = 18 - A \]

### Step 3: Write the product in terms of one variable
The product \( P \) of the two numbers can be expressed as:
\[ P = A \times B = A \times (18 - A) \]
This simplifies to:
\[ P = 18A - A^2 \]

### Step 4: Find the maximum product
To find the maximum value of the product \( P \), we can complete the square or use calculus. However, a simpler method is to recognize that the product of two numbers with a fixed sum is maximized when the two numbers are equal.

If we set \( A = B \):
\[ A + A = 18 \implies 2A = 18 \implies A = 9 \]
Thus, \( B = 9 \) as well.

### Step 5: Calculate the maximum product
Now, we can calculate the product:
\[ P = A \times B = 9 \times 9 = 81 \]

### Conclusion
The greatest product of the two numbers is:
\[ \boxed{81} \]