Problem 77 - Elementary Algebra

Question

In order for a small business to break even on a project, it must have sales of $$\$ 21,000.$$ If the amount of sales was $$\$ 15,000,$$ how much money did this company fall short?

Step-by-Step Solution

Answer

Answer: The small business fell short by $6,000 in sales to break even on their project.

Step 1: Identify the break-even sales amount

The break-even sales amount for the project is $$\$ 21,000.$$ #Phase 2: Identifying the Actual Sales Amount#

Step 2: Identify the actual sales amount

The actual sales amount for the project is $$\$ 15,000.$$ #Phase 3: Calculating the Difference between Break-Even Sales Amount and Actual Sales Amount#

Step 3: Calculate the difference between break-even and actual sales amount

The difference between the break-even sales amount and actual sales amount can be calculated by subtracting the actual sales amount from the break-even sales amount: \(Difference = Break\_Even\_Sales - Actual\_Sales\) #Phase 4: Find out how much money the company fell short#

Step 4: Calculate the shortfall

Using the formula from the previous step, we get: \(Difference = \$ 21,000 - \$ 15,000 = \$ 6,000\) This means the company fell short by $$\$ 6,000.$$

Key Concepts

Break-even analysis

Break-even analysis is a crucial concept in financial and business planning. It helps businesses determine the point at which their revenues equal their costs, resulting in no profit or loss. This is known as the break-even point. Understanding this concept allows business owners to know how much they need to sell in order to cover their expenses and avoid losses.

To perform a break-even analysis, businesses must know their fixed costs, variable costs, and the revenues generated from sales. Fixed costs are expenses that do not change regardless of the business activity level, such as rent and salaries. Variable costs change with the level of production or sales, such as the cost of raw materials.

To calculate the break-even point in sales terms, the formula used is:

• Break-even Quantity = Fixed Costs / (Revenue per Unit - Variable Cost per Unit)

In our example, the business needed sales of $21,000 to cover its costs, marking the break-even point.

Financial problem-solving

Financial problem-solving involves identifying financial challenges and determining solutions. In business, it is critical for maintaining profitability and ensuring sound financial management. It includes analyzing sales, costs, profits, and any financial discrepancies that may arise.

Effective financial problem-solving requires:

• Accurate identification of financial goals and shortfalls
• Comprehensive understanding of the business finances
• Strategic planning to optimize resources
• Use of mathematical calculations to clarify the financial situation

In the given exercise, the problem-solving process involved recognizing that the company needed to reach $21,000 in sales to break even but only managed to sell $15,000. By calculating the shortfall, the business was able to identify their financial challenge and plan for future adjustments.

Subtraction in mathematics

Subtraction in mathematics is one of the fundamental arithmetic operations. It involves taking one number away from another and is written as a minus sign (-) between the numbers. Subtraction is essential for determining differences, such as in financial calculations where one needs to find how much is lacking or in surplus.

In our exercise, subtraction was used to find the difference between the break-even sales amount and the actual sales amount. By subtracting the actual sales ($15,000) from the break-even amount ($21,000), the shortfall was calculated:

• Formula: Difference = Break-Even Sales - Actual Sales
• Calculation: $21,000 - $15,000 = $6,000

Thus, subtraction not only denotes a mathematical process but also supports financial analysis, helping businesses understand variances and resource deficiencies.