Question

Caleb is starting a business selling handmade necklaces. He has decided to invest an initial amount of $63 for advertising, and materials cost $8 for each necklace he makes. Caleb can sell his creations for $9 per necklace. Once he makes and sells a certain number of necklaces, he will break even, with identical expenses and sales. How many necklaces would that take? What would the total expenses and sales be then?


After selling
necklaces, Caleb would have total expenses and sales of $
.

Answer 1

To determine how many necklaces Calvin needs to sell to break even, we can start by defining the key components:

### Expenses:
1. **Initial Advertising Cost**: $63 (fixed cost)
2. **Cost per Necklace**: $8 (variable cost)

### Revenue:
- **Selling Price per Necklace**: $9

### Break-Even Point Calculation:
At the break-even point, total expenses will equal total sales.

#### Total Expenses:
Total expenses \( E \) can be expressed as:
\[
E = \text{initial cost} + \text{cost per necklace} \times \text{number of necklaces} = 63 + 8n
\]
where \( n \) is the number of necklaces.

#### Total Sales:
Total sales \( S \) can be expressed as:
\[
S = \text{selling price per necklace} \times \text{number of necklaces} = 9n
\]

#### Setting Expenses Equal to Sales for Break-Even:
To find the break-even point, we set total expenses equal to total sales:
\[
63 + 8n = 9n
\]

Now, solve for \( n \):
\[
63 = 9n - 8n
\]
\[
63 = n
\]

Thus, Caleb needs to sell **63 necklaces** to break even.

### Calculating Total Expenses and Sales:
Now we can calculate the total expenses and total sales when \( n = 63 \):

#### Total Expenses:
\[
E = 63 + 8 \times 63 = 63 + 504 = 567
\]

#### Total Sales:
\[
S = 9 \times 63 = 567
\]

### Conclusion:
After selling 63 necklaces, Caleb would have total expenses and sales of **$567**.

Final answers:
- Necklaces to break even: **63**
- Total expenses and sales: **$567**