Question

Suppose that you own a video store that has total costs of \(\$ 3,600\) per month. If you charge \(\$ 12\) for each DVD you sell, how many do you need to sell each month in order to break even? Explain how you arrived at your answer.

Short answer

Sell 300 DVDs to break even.

Step-by-step solution

Understand the Break-Even Point

The break-even point is when total revenue equals total costs. This means we do not make a loss or a profit. To find the break-even point, we need to determine how many DVDs we must sell so that the revenue from these sales equals the total costs each month.

Set Up the Equation

Let's denote the number of DVDs you need to sell as \( x \). The revenue from selling \( x \) DVDs at \\(12 each will be \( 12x \). The break-even condition requires that the total costs (\\)3600) are equal to the total revenue (\(12x\)). So the equation is: \( 12x = 3600 \).

Solve the Equation

Now, solve the equation \( 12x = 3600 \) to find the value of \( x \). Divide both sides of the equation by 12 to isolate \( x \):\[x = \frac{3600}{12}\]Calculate the division:\[x = 300\]

Interpret the Solution

The solution \( x = 300 \) means you need to sell 300 DVDs to break even. This number of sales will ensure your revenue equals the total monthly costs, resulting in no net loss or profit.

Key concepts

Cost Calculation

Understanding cost calculation is key for any business to manage its finances effectively. In this example, total costs include all expenses necessary to keep the video store running monthly. Costs can vary widely depending on the type of business, but in this case, the total monthly cost is given as \(\\(3,600\).
These costs could cover multiple things:

• Rent for the store space
• Employee wages
• Utility bills, such as electricity and water
• Inventory costs (e.g., purchasing new DVDs)

To calculate costs accurately, it's crucial to keep track of all recurring expenses. Doing so will help determine whether the revenue generated can cover these expenses and what sales volume is needed to break even. In our example, knowing the total cost of \(\\)3,600\) directly feeds into our calculations for achieving the break-even point.

Revenue Calculation

Revenue calculation involves determining the total amount of money a business brings in from selling goods or services. In the context of the video store, revenue comes from the sales of DVDs. Each DVD is sold at \(\\(12\).
For revenue to effectively support the business, it's calculated by multiplying the price of each DVD by the quantity sold. Mathematically, this is expressed as:
\[\text{Revenue} = 12x\]
Here, \(x\) is the number of DVDs sold.
Revenue needs to match or exceed costs to achieve the break-even point. As per the example, to find out how much you need to sell, you set the revenue equal to \(\\)3,600\) (the total cost), allowing us to find the break-even sales quantity.
This calculation provides necessary insight into how changes in sales volume or pricing may impact overall earnings.

Profit and Loss

Profit and loss are financial metrics that indicate whether a business is making money or losing money. The break-even point is a critical concept here, where profit is zero because revenue equals costs.
Before reaching this point, the business operates at a loss. After it, each additional sale contributes to profit.
To illustrate, using our example:

• If fewer than 300 DVDs are sold, expenses will exceed revenue, resulting in a loss.
• At exactly 300 DVDs, costs equal revenue, meaning no loss and no profit.
• Selling more than 300 DVDs will generate profit since revenue surpasses costs.

Managing profit and loss involves strategic planning to maintain favorable sales numbers and pricing. It's not just about reaching the break-even point but surpassing it to ensure business growth and sustainability.