Nails are sold in 8-ounces and 20-ounce boxes. If 50 boxes of nails were sold and the total weight of the nails sold was less than 600 ounces, what is the greatest possible number of 20-ounce boxes that could have been sold?

Options

A. `33`
B. `25`
C. `17`
D. `16`

Answer

To solve the problem step by step, we will define our variables, set up the inequalities based on the information provided, and then solve for the maximum number of 20-ounce boxes.

### Step 1: Define Variables
Let:
- \( x \) = number of 8-ounce boxes sold
- \( y \) = number of 20-ounce boxes sold

### Step 2: Set Up the Equations
From the problem, we know two things:
1. The total number of boxes sold is 50:
\[ x + y = 50 \]
2. The total weight of the nails sold is less than 600 ounces:
\[ 8x + 20y < 600 \]

### Step 3: Express \( x \) in Terms of \( y \)
From the first equation, we can express \( x \) in terms of \( y \):
\[ x = 50 - y \]

### Step 4: Substitute \( x \) in the Weight Inequality
Now, we substitute \( x \) in the weight inequality:
\[ 8(50 - y) + 20y < 600 \]
Expanding this gives:
\[ 400 - 8y + 20y < 600 \]
Combining like terms results in:
\[ 400 + 12y < 600 \]

### Step 5: Solve for \( y \)
Next, we isolate \( y \):
\[ 12y < 600 - 400 \]
\[ 12y < 200 \]
Dividing both sides by 12 gives:
\[ y < \frac{200}{12} \]
Simplifying this fraction:
\[ y < \frac{50}{3} \approx 16.67 \]

### Step 6: Determine the Greatest Integer Value for \( y \)
Since \( y \) must be an integer, the greatest possible value for \( y \) that satisfies this inequality is:
\[ y \leq 16 \]

### Conclusion
Thus, the greatest possible number of 20-ounce boxes that could have been sold is:
\[ \boxed{16} \]