Writing an Expression for the Number of Boxes of Nails

Oct 13, 2025 by miodeya.com 78 views

Introduction

Peter is at a lumber yard, and he's looking to buy some boards. However, there's a catch - for every 10 boards he buys, he gets 2 free boxes of nails. In this article, we'll explore how to write an expression for the number of boxes of nails Peter will get if he buys $n$ boards.

Understanding the Problem

Let's break down the problem step by step. Peter gets 2 free boxes of nails for every 10 boards he buys. This means that for every 10 boards, he gets 2 boxes of nails. To find the number of boxes of nails Peter will get if he buys $n$ boards, we need to determine how many sets of 10 boards he buys.

Writing an Expression for the Number of Boxes of Nails

To write an expression for the number of boxes of nails Peter will get, we need to divide the total number of boards he buys ( $n$ ) by 10. This will give us the number of sets of 10 boards he buys. Since each set of 10 boards comes with 2 boxes of nails, we can multiply the number of sets by 2 to get the total number of boxes of nails.

Let's write this mathematically:

Number of sets of 10 boards = $\frac{n}{10}$
Number of boxes of nails = $2 \times \frac{n}{10}$

Simplifying the expression, we get:

Number of boxes of nails = $\frac{n}{5}$

Interpretation

So, if Peter buys $n$ boards, he will get $\frac{n}{5}$ boxes of nails. This means that for every 5 boards he buys, he gets 1 box of nails.

Example

Let's say Peter buys 20 boards. How many boxes of nails will he get?

Using the expression we derived earlier, we get:

Number of boxes of nails = $\frac{20}{5}$
Number of boxes of nails = 4

So, Peter will get 4 boxes of nails if he buys 20 boards.

Conclusion

In this article, we explored how to write an expression for the number of boxes of nails Peter will get if he buys $n$ boards. We broke down the problem step by step and used mathematical expressions to determine the number of boxes of nails Peter will get. The expression we derived is $\frac{n}{5}$ , which means that for every 5 boards Peter buys, he gets 1 box of nails.

Additional Questions

What if Peter buys a fraction of a board? How would we calculate the number of boxes of nails he gets?
What if the lumber yard changes its policy and gives Peter 3 boxes of nails for every 10 boards he buys? How would we modify the expression to account for this change?

Answer to Additional Questions

If Peter buys a fraction of a board, we can use the same expression we derived earlier: $\frac{n}{5}$ . However, we need to be careful when dealing with fractions, as the expression may not give us a whole number of boxes of nails.
If the lumber yard changes its policy and gives Peter 3 boxes of nails for every 10 boards he buys, we can modify the expression as follows:
Number of sets of 10 boards = $\frac{n}{10}$
Number of boxes of nails = $3 \times \frac{n}{10}$

Simplifying the expression, we get:

Number of boxes of nails = $\frac{3n}{10}$
Free Nails at the Lumber Yard: A Q&A Session =====================================================

Introduction

In our previous article, we explored how to write an expression for the number of boxes of nails Peter will get if he buys $n$ boards. We broke down the problem step by step and used mathematical expressions to determine the number of boxes of nails Peter will get. In this article, we'll answer some frequently asked questions related to the problem.

Q&A Session

Q: What if Peter buys a fraction of a board? How would we calculate the number of boxes of nails he gets?

A: If Peter buys a fraction of a board, we can use the same expression we derived earlier: $\frac{n}{5}$ . However, we need to be careful when dealing with fractions, as the expression may not give us a whole number of boxes of nails.

Q: What if the lumber yard changes its policy and gives Peter 3 boxes of nails for every 10 boards he buys? How would we modify the expression to account for this change?

A: If the lumber yard changes its policy and gives Peter 3 boxes of nails for every 10 boards he buys, we can modify the expression as follows:

Number of sets of 10 boards = $\frac{n}{10}$
Number of boxes of nails = $3 \times \frac{n}{10}$

Simplifying the expression, we get:

Number of boxes of nails = $\frac{3n}{10}$

Q: How many boxes of nails will Peter get if he buys 25 boards?

A: Using the expression we derived earlier, we get:

Number of boxes of nails = $\frac{25}{5}$
Number of boxes of nails = 5

So, Peter will get 5 boxes of nails if he buys 25 boards.

Q: What if Peter buys 30 boards? How many boxes of nails will he get?

A: Using the expression we derived earlier, we get:

Number of boxes of nails = $\frac{30}{5}$
Number of boxes of nails = 6

So, Peter will get 6 boxes of nails if he buys 30 boards.

Q: Can we use the expression to calculate the number of boxes of nails Peter will get if he buys a negative number of boards?

A: No, we cannot use the expression to calculate the number of boxes of nails Peter will get if he buys a negative number of boards. The expression is only valid for non-negative values of $n$ .

Q: Can we use the expression to calculate the number of boxes of nails Peter will get if he buys a non-integer number of boards?

A: Yes, we can use the expression to calculate the number of boxes of nails Peter will get if he buys a non-integer number of boards. However, we need to be careful when dealing with fractions, as the expression may not give us a whole number of boxes of nails.

Conclusion

In this article, we answered some frequently asked questions related to the problem of calculating the number of boxes of nails Peter will get if he buys $n$ boards. We used mathematical expressions to determine the number of boxes of nails Peter will get and provided examples to illustrate the use of the expression.

Additional Questions

What if the lumber yard changes its policy and gives Peter 4 boxes of nails for every 10 boards he buys? How would we modify the expression to account for this change?
What if Peter buys 40 boards? How many boxes of nails will he get?
Can we use the expression to calculate the number of boxes of nails Peter will get if he buys a large number of boards?

Answer to Additional Questions

If the lumber yard changes its policy and gives Peter 4 boxes of nails for every 10 boards he buys, we can modify the expression as follows:
Number of sets of 10 boards = $\frac{n}{10}$
Number of boxes of nails = $4 \times \frac{n}{10}$

Simplifying the expression, we get:

Number of boxes of nails = $\frac{4n}{10}$
If Peter buys 40 boards, we can use the expression we derived earlier to calculate the number of boxes of nails he will get:
Number of boxes of nails = $\frac{40}{5}$
Number of boxes of nails = 8

So, Peter will get 8 boxes of nails if he buys 40 boards.

Yes, we can use the expression to calculate the number of boxes of nails Peter will get if he buys a large number of boards. However, we need to be careful when dealing with large numbers, as the expression may not give us a whole number of boxes of nails.