Question

Danielle owns a business that is in financial trouble. Her accountant informs her that her net worth is 5267 dollar Her net worth should be at least 2500 dollar by the end of the month; otherwise, she will be forced to close her business. What is the minimum profit she must clear to remain in business?

Short answer

Answer: Danielle must make a minimum profit of 2767 dollars less to remain in business.

Step-by-step solution

Identify Given Values

Danielle's current net worth = 5267 dollars Desired net worth at the end of the month = 2500 dollars

Write the Equation

Let P be the profit she must make. Net Worth at the end of the month = Current Net Worth + Profit Desired Net Worth = Current Net Worth + P

Solve for Profit (P)

Now we can plug in the given values into the equation. 2500 = 5267 + P

Isolate P

To solve for P, subtract 5267 from both sides of the equation: P = 2500 - 5267

Calculate the Minimum Profit

Now, we can find the minimum profit by calculating the value of P: P = -2767 So, Danielle must make a minimum profit of 2767 dollars (by the negative sign) less to remain in business. Note that any additional profit over this amount would assist in bettering her financial situation, but she must achieve at least this reduction in profit to avoid closing her business.

Key concepts

Algebraic Expressions

An algebraic expression is a mathematical phrase that can contain numbers, variables, and arithmetic operations. Variables, represented by letters such as P for profit in the given exercise, play a crucial role because they stand for unknown values that we aim to find. By combining these elements, algebraic expressions can model real-world situations, like financial calculations in a business context.

For Danielle’s problem, the expression is the net worth at the end of the month, formulated as her current net worth plus the desired profit, or symbolically, 5267 + P. Understanding how to manipulate these expressions is essential, as it allows us to derive meaningful information, such as the minimum profit needed to maintain business operations.

Isolate Variables

To isolate a variable means to solve an equation such that the variable is alone on one side, thus finding its value. In our exercise, the goal was to isolate the profit variable, P. This is done through inverse operations - in this case, by subtracting Danielle’s current net worth from both sides of the equation.

Mastering the skill of isolating variables is fundamental in solving any algebra problem and is especially useful in financial mathematics where variables often represent monetary values. The step-by-step process ensures that the balance of the equation is maintained and the true value of the variable is discovered.

Net Worth Calculation

Net worth is a key financial concept representing the value of what is owned minus what is owed. Calculating net worth involves summing all assets and then subtracting any liabilities.

In the context of the provided problem, Danielle’s net worth calculation is simplified; it is assumed to be her business’s current net worth (assets minus liabilities) plus any profit (or additional assets) she will make within the month. This concept is crucial for both personal and business financial planning because it provides a snapshot of financial health at a given time and helps in making informed decisions.

Financial Mathematics

Financial mathematics is a branch of applied mathematics concerned with financial markets and economic data. It utilizes mathematical models to analyze and predict financial outcomes. This discipline is not only about complex market theories but also addresses everyday issues like figuring out profit thresholds to avoid business closure — which Danielle faces in our exercise.

By applying algebraic expressions and the ability to isolate variables within equations, we can solve real-life financial problems. In learning such mathematics, students gain valuable skills that enable them to apply logical and analytical thinking to manage personal and business finances effectively.