Gina Chuez has considered starting her own custom greeting card business. With an initial start-up cost of $1500, she figures it will cost $0.45 to produce each card. In order to remain competitive with the larger greeting card companies, Gina must sell her cards for no more than $1.70 each. To make a profit, her income must exceed her costs. How many cards must she sell before making a profit?
The Correct Answer and Explanation is :
To determine how many cards Gina must sell to make a profit, we need to set up an equation for her costs and income.
Step 1: Define variables
- Let ( x ) be the number of cards Gina sells.
- The cost to produce each card is $0.45, so the total cost to produce ( x ) cards is ( 0.45x ).
- The initial start-up cost is $1500.
- The revenue she gets from selling each card is $1.70, so her total revenue from selling ( x ) cards is ( 1.70x ).
Step 2: Write the profit condition
Gina makes a profit when her income (revenue) exceeds her costs. This can be written as:
[
\text{Income} > \text{Costs}
]
The total income is ( 1.70x ), and the total cost is ( 1500 + 0.45x ) (the initial cost plus the production cost per card). So the inequality becomes:
[
1.70x > 1500 + 0.45x
]
Step 3: Solve the inequality
To solve for ( x ), subtract ( 0.45x ) from both sides of the inequality:
[
1.70x – 0.45x > 1500
]
[
1.25x > 1500
]
Now, divide both sides by 1.25 to isolate ( x ):
[
x > \frac{1500}{1.25}
]
[
x > 1200
]
Step 4: Conclusion
Gina must sell at least 1201 cards to make a profit. This is because she needs to sell more than 1200 cards, and since the number of cards sold must be a whole number, 1201 cards is the minimum number that will result in a profit.
Explanation:
The start-up cost of $1500 is a fixed cost, and each card produced costs $0.45 to make. To make a profit, the income from selling the cards must exceed both the start-up cost and the production cost. By solving the inequality, we found that selling 1201 cards will ensure that Gina’s income exceeds her total costs, which means she will make a profit.