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Printshop Expenses



You own a small printshop. Your monthly expenses include the following:



Fixed expenses:

  • Rent: $2,500
  • Insurance: $400
  • Equipment lease: $600



Variable expenses:

  • Paper cost per print job: $1.50
  • Ink cost per print job: $0.75
  • Other variable costs: $200



Construct an equation to model the total monthly expenses $E(q)$, where $q$ represents the print jobs completed in a month.



With the addition of a $0.50 per print job expense for binding materials, what is the updated equation to model the total monthly expenses $E(q)$ for the printshop?


A$E(q) = 2.75q + 3,700$
B$E(q) = 2.25q + 3,700$
C$E(q) = 2.75q + 3,800$
D$E(q) = 2.75q + 3,500$

Verified Answer :

  • Combine all fixed costs (\(\$2,500\), \(\$400\), \(\$600\), and \(\$200\)) to reach a total of \(\$3,700\), or \(3.7\) thousand dollars.
  • Sum the initial variable costs (\(\$1.50\) and \(\$0.75\)) with the new binding cost (\(\$0.50\)) to get a new per-job cost of \(\$2.75\).
  • Combine these components into the linear model \(E(q) = 2.75q + 3.7\).
  • Select the matching option, which is \(\boxed{a}\).

Explanation

  1. Identifying Monthly Expenses

First, we identify the monthly expenses for the printshop. The fixed expenses are Rent (\(\$2,500\)), Insurance (\(\$400\)), and Equipment lease (\(\$600\)). Summing these, we get:
\[\$2,500 + \$400 + \$600 = \$3,500.\]
Next, we identify the variable expenses. We have paper (\(\$1.50\) per job) and ink (\(\$0.75\) per job). The \"Other variable costs\" are \(\$200\). Based on the structure of the provided options, which have a constant term of \(3.7\), we interpret this \(\$200\) as an additional fixed monthly expense, bringing the total fixed monthly cost to \(\$3,500 + \$200 = \$3,700\), or \(\$3.7\) thousand.

  1. Calculating Updated Variable Expenses

Initially, the variable costs per print job were paper (\(\$1.50\)) and ink (\(\$0.75\)). With the addition of the new binding material cost of \(\$0.50\) per print job, the updated variable cost per print job, which we will call \(V_{total}\), is:
\[V_{total} = \$1.50 + \$0.75 + \$0.50 = \$2.75.\]
Thus, the variable expense per job is \(\$2.75\).

  1. Formulating the Total Monthly Expense Equation

Now we construct the final linear equation to represent the total monthly expenses \(E(q)\), where \(q\) is the number of print jobs completed in the month. The total expense is the sum of the total fixed costs and the variable cost multiplied by the number of jobs. Expressing all values in thousands of dollars, the fixed costs are \(3.7\) and the variable cost is \(2.75\):
\[E(q) = 2.75q + 3.7.\]
Comparing this result to the given options, we find that it corresponds to option (a).

Examples

This concept of modeling expenses as a linear equation is essential for any small business owner, such as an entrepreneur running a bakery or a service shop. By classifying costs into fixed (like rent, which you pay regardless of production) and variable (like ingredients or paper, which change depending on the number of customers), business owners can accurately project their monthly budget as \(E(q) = V \cdot q + F\). Understanding how these costs shift with new business requirements—such as adding a new product line—allows for better pricing strategies, break-even analysis, and overall financial health.

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