A retailer spends \[tex]$500 per month to keep its online shop active and updated. The store acquires shirts at a marginal cost of \$[/tex]5 per shirt. Each shirt sells for a marginal benefit of \$10 per shirt.

What is the minimum number of shirts the retailer needs to sell in order to pay for all its costs in a month?

[tex]\[
\begin{tabular}{|l|l|l|l|}
\hline
Quantity of shirts sold & Marginal cost & Total cost & Marginal benefit \\
\hline
0 & \[tex]$0 & \$[/tex]500 & \$0 \\
\hline
25 & \[tex]$125 & \$[/tex]625 & \$250 \\
\hline
50 & \[tex]$250 & \$[/tex]750 & \$500 \\
\hline
75 & \[tex]$375 & \$[/tex]875 & \$750 \\
\hline
100 & \[tex]$500 & \$[/tex]1,000 & \$1,000 \\
\hline
125 & \[tex]$625 & \$[/tex]1,125 & \$1,250 \\
\hline
\end{tabular}
\][/tex]

A. 25
B. 100
C. 125
D. 75



Answer :

To determine the minimum number of shirts the retailer needs to sell in order to cover all its monthly costs, we need to find the break-even point where total costs equal total revenue.

The retailer has a fixed monthly cost of \[tex]$500 for keeping the online shop active and updated, a marginal cost of \$[/tex]5 per shirt, and a marginal benefit (selling price) of \$10 per shirt.

To find out the break-even point, we must set the total costs equal to the total revenue.

Step-by-Step Solution:

1. Total Cost (C): This consists of the fixed monthly cost plus the variable cost (marginal cost per shirt times the number of shirts sold).

So, total cost \( C \) can be expressed as:
[tex]\[ C = 500 + 5 \times (\text{number of shirts}) \][/tex]

2. Total Revenue (R): This is the marginal benefit per shirt times the number of shirts sold.

Therefore, total revenue \( R \) can be expressed as:
[tex]\[ R = 10 \times (\text{number of shirts}) \][/tex]

3. Break-even Point: At the break-even point, total cost equals total revenue.

[tex]\[ 500 + 5 \times (\text{number of shirts}) = 10 \times (\text{number of shirts}) \][/tex]

4. Solve for the Number of Shirts:
[tex]\[ 500 + 5 \times (\text{number of shirts}) = 10 \times (\text{number of shirts}) \][/tex]
Move the term involving the number of shirts on the left-hand side to the right-hand side:
[tex]\[ 500 = 10 \times (\text{number of shirts}) - 5 \times (\text{number of shirts}) \][/tex]
Simplify:
[tex]\[ 500 = 5 \times (\text{number of shirts}) \][/tex]

Divide both sides of the equation by 5:
[tex]\[ \text{number of shirts} = \frac{500}{5} \][/tex]
[tex]\[ \text{number of shirts} = 100 \][/tex]

So, the retailer needs to sell 100 shirts in a month to break even and cover all its costs.

Answer: B. 100