The problem is asking us to determine how many small boxes and how many large boxes were purchased, given certain information. Let's break down the details from the problem:

• There are small and large boxes.
• Each small box contains 50 nails.
• Each large box contains 350 nails.
• The contractor bought a total of 13 boxes (both small and large combined).
• The total number of nails is 3350.

Let:

• $x$ be the number of small boxes.
• $y$ be the number of large boxes.

We can set up two equations based on the information provided:

1. The total number of boxes is 13: $x + y = 13$

2. The total number of nails is 3350: $50x + 350y = 3350$

Solving the system of equations:

1. From the first equation, solve for $y$: $y = 13 - x$

2. Substitute this into the second equation: $50x + 350(13 - x) = 3350$ Simplify: $50x + 4550 - 350x = 3350$ Combine like terms: $-300x + 4550 = 3350$ Subtract 4550 from both sides: $-300x = -1200$ Divide by -300: $x = 4$

3. Now that we know $x = 4$, substitute this back into the first equation: $4 + y = 13$ So: $y = 9$

Conclusion:

• The contractor bought 4 small boxes and 9 large boxes.

Would you like more details or have any questions?

Here are five related questions to explore:

1. What would happen if the contractor bought 5 small boxes instead of 4?
2. How many nails would the contractor have if they bought only large boxes?
3. Can you express the relationship between the number of boxes and total nails as a function?
4. How many nails are needed if the contractor bought 20 boxes in total with the same ratio of small to large boxes?
5. How would the problem change if each large box contained 400 nails instead?

Tip: For problems involving two variables, always look to set up a system of equations based on the given constraints.