Problem 40 - Algebra 2

Question

Business Your friend's mother plans to open a restaurant. The initial investment is \(\$ 90,000 .\) Weekly expenses will be about \(\$ 8200 .\) If the weekly income is about \(\$ 8900,\) in how many weeks will she get back her investment?

Step-by-Step Solution

Answer

It will take 10 weeks to recover the initial investment.

Step 1: Calculate Net Weekly Income

Subtract the weekly expenses from the weekly income to find the net weekly income. Net weekly income = weekly income - weekly expenses.

Step 2: Determine Total Investment Recovery Time

Divide the initial investment by the net weekly income to calculate the number of weeks required to recover the investment. Weeks to recover investment = initial investment / net weekly income.

Key Concepts

Net Weekly Income Calculation

Understanding how to calculate net weekly income is crucial for anyone running a business or managing personal finances. This figure is the amount of money that remains after all the weekly expenses have been paid.

To calculate net weekly income effectively, one would follow these steps:

• First, identify the total weekly income, which is the sum of all earnings in a week.
• Subtract all the weekly expenditures from this amount. These expenses can include salaries, utilities, rent, and supplies.
• The result of this subtraction will be the net weekly income.

In the context of the given exercise, the weekly income is \(\$8900\) and the weekly expenses are \(\$8200\). By subtracting the expenses from the income, \( 8900 - 8200 = 700\), we find that the net weekly income is \(\$700\) per week.

Break-Even Point

The break-even point is a key financial concept in business, as it represents the moment when total costs and total revenues are equal, meaning there is no net loss or gain.

For businesses, knowing the break-even point is essential for understanding when they will start to make a profit. To calculate this point, one must account for all costs—both fixed and variable—and the selling price of goods or services. The formula to find the break-even point in units is:

• Break-even point (units) = Fixed Costs / (Selling Price per Unit - Variable Cost per Unit).

The exercise provided does not directly ask for the break-even point but rather the investment recovery time which is related but distinct. While reaching the break-even point means the business is no longer losing money, recovering the full investment also takes into account the initial capital outlay. In this case, the initial investment is \(\$90,000\).

Algebraic Problem-Solving

Algebraic problem-solving is a systematic approach to finding the unknowns in mathematical equations and real-world scenarios. It often involves identifying variables, establishing relationships between them, and applying algebraic operations to solve for these unknowns.

Let's review the approach applied to the restaurant business scenario:

• After calculating the net weekly income as \(\$700\), we define the total time to recover investment as an unknown 'x'.
• We know that this 'x' times the net weekly income should equal the initial investment, leading us to set up the equation \(x \times 700 = 90,000\).
• By dividing both sides of the equation by 700, we solve for 'x': \(\frac{90,000}{700} = x\).
• This calculation tells us that 'x' is approximately 128.57, which indicates that it will take just under 129 weeks for your friend's mother to recover her \(\$90,000\) investment.

Clearly stating the problem, organizing the known information, and carefully performing algebraic operations are key steps in algebraic problem-solving.