Question

A doughnut shop sells a dozen doughnuts for \(\$ 3.95 .\) Beyond the fixed costs (rent, utilities, and insurance) of \(\$ 165\) per day, it costs \(\$ 1.45\) for enough materials (flour, sugar, and so on) and labor to produce a dozen doughnuts. The daily profit from doughnut sales varies between \(\$ 100\) and \(\$ 400\). Between what numbers of doughnuts (in dozens) do the daily sales vary?

Short answer

The daily sales vary between 106 dozens and 226 dozens of doughnuts.

Step-by-step solution

Calculate Net Income per Dozen

Start by calculating the net income generated from selling a dozen doughnuts. This can be done by subtracting the cost of a dozen doughnuts from the sale price of a dozen doughnuts. \[ netIncomeDozen = priceDozen - costDozen \] \[ netIncomeDozen = \$3.95 - \$1.45 = \$2.5 \] Each dozen doughnuts brings \$2.5 net income.

Calculate Minimum Number of Dozens

Next, calculate the minimum number of doughnuts dozens (numDozensMin) the shop needs to sell in order to make their minimum profit (\$100), taking into account the fixed costs (\$165). This can be calculated using the formula: \[ numDozensMin = (profitMin + fixedCosts) / netIncomeDozen \] Substituting the values: \[ numDozensMin = (\$100 + \$165) / \$2.5 \] \[ numDozensMin = 106 \]

Calculate Maximum Number of Dozens

Finally, calculate the maximum number of doughnuts dozens (numDozensMax) the shop can sell in order to make their maximum profit (\$400), taking into account the fixed costs (\$165). This can be calculated using the formula: \[ numDozensMax = (profitMax + fixedCosts) / netIncomeDozen \] Substituting the values: \[ numDozensMax = (\$400 + \$165) / \$2.5 \] \[ numDozensMax = 226 \]

Key concepts

Profit Calculation

To determine the profitability of your business, you first need to understand how profit is calculated. Profit is simply the amount of money left after all expenses have been deducted from the total revenue. In our scenario, to find the net income per dozen doughnuts, we subtract the cost per dozen from the selling price per dozen:
\[ \text{Net Income per Dozen} = \text{Selling Price} - \text{Cost Price} \]
Here, the selling price for a dozen doughnuts is \(\\(3.95\), and the cost price for producing a dozen doughnuts is \(\\)1.45\). By applying the formula:
\[ \\(3.95 - \\)1.45 = \\(2.50 \]
So for every dozen doughnuts sold, the shop earns a profit of \(\\)2.50\). This calculation is crucial as it tells you how much each sale contributes to your overall profitability.

Fixed Costs

Fixed costs are expenses that do not change with the level of goods produced or sold. They remain constant regardless of the business activity level. In the context of the doughnut shop, the fixed costs include rent, utilities, and insurance that sum up to \(\$165\) per day.

These costs are essential to consider because they exist even if no doughnuts are sold that day. No matter how many dozens of doughnuts the shop sells, these costs must be covered to simply keep the doors open. Understanding fixed costs helps in setting a baseline financial plan where the first aim is to cover these costs, and only then can net profit begin to accumulate.

Variable Costs

Variable costs, unlike fixed costs, change with the production level. These are the costs incurred each time a product is created. For the doughnut shop, the variable cost is \(\\(1.45\) for making a dozen doughnuts. This includes materials like flour and sugar, and labor costs.

The key to managing variable costs is understanding that they directly impact profit margins. As sales increase, so do variable costs. If every sold dozen generates \(\\)2.50\) of net income after covering the \(\$1.45\) cost, the variable cost becomes a crucial factor in deciding on pricing strategies and production levels. Proper management of these costs can enhance profitability.

Net Income Analysis

Net income analysis is a process where you evaluate the income after all business expenses. It helps in understanding the actual profitability and efficiency of a business operation. In our doughnut shop example, the shop aims for a daily net income ranging from \(\\(100\) to \(\\)400\).

To achieve the desired net income, you factor in fixed and variable costs and calculate how many dozens of doughnuts must be sold. First, determine the minimum sales needed:
\[ \text{Minimum Dozens} = \frac{\text{Minimum Profit} + \text{Fixed Costs}}{\text{Net Income per Dozen}} \]
\[ \text{Minimum Dozens} = \frac{\\(100 + \\)165}{\\(2.50} = 106 \]

Similarly, for maximum profit:
\[ \text{Maximum Dozens} = \frac{\text{Maximum Profit} + \text{Fixed Costs}}{\text{Net Income per Dozen}} \]
\[ \text{Maximum Dozens} = \frac{\\)400 + \\(165}{\\)2.50} = 226 \]
This analysis aids in setting realistic sales targets and budget planning, ensuring that the doughnut shop remains profitable within its operational capabilities.