Problem 25 - Precalculus Mathematics for Calculus

Question

An executive in an engineering firm earns a monthly salary plus a Christmas bonus of \(\$ 8500 .\) If she earns a total of \(\$ 97,300\) per year, what is her monthly salary?

Step-by-Step Solution

Answer

The monthly salary is \( \$7,400 \).

Step 1: Identify the Known Values

We know the total annual earning, including the Christmas bonus, is \( \\(97,300 \). The bonus amount is \( \\)8,500 \).

Step 2: Subtract the Christmas Bonus

To find the total annual salary excluding the Christmas bonus, subtract the bonus from the total earnings. So, \( \\(97,300 - \\)8,500 = \$88,800 \).

Step 3: Calculate the Monthly Salary

To find the monthly salary, divide the annual salary (excluding the bonus) by 12 months. So, \( \frac{\\(88,800}{12} = \\)7,400 \).

Key Concepts

Annual Salary Calculation

Calculating an annual salary is a fundamental concept in understanding one's financial earnings over a year. It takes into account the total amount of money an individual earns before any deductions, such as taxes or bonuses, which may be included in a contract or agreement.

To determine the annual salary based on the given problem, consider the following steps:

• First, note the total annual earnings, which include any bonuses. In this case, it was given as \(\\(97,300\).
• Identify any bonuses that are already included in this figure, like the Christmas bonus of \(\\)8,500\).
• Subtract the known bonus from the total annual earnings to find the pure annual salary. This gives us the base salary excluding extra bonuses.

This step helps in understanding what part of the earnings comes from regular monthly income and what part is supplemental. In financial contexts, knowing how much you make regularly versus what comes from occasional bonuses is crucial for budgeting and financial planning.

Monthly Salary Division

Dividing the annual salary to determine a monthly paycheck is essential for monthly budget planning and understanding cash flow. Once the annual salary has been calculated without any additional bonuses, the next step is to break it down into manageable portions that reflect how often most employees receive their pay.

For this problem, you follow these steps:

• Take the annual salary calculated after removing the bonus, \(\\(88,800\), and divide it by 12 months, reflecting the number of months in a year.
• Perform the division to find the monthly salary: \(\frac{\\)88,800}{12} = \$7,400\).

This method allows you to see exactly how much income you receive each month from your regular salary. It's crucial to differentiate between this steady income and occasional bonus payouts when planning expenses. Understanding this helps in setting realistic budgets and financial goals.

Financial Problem-Solving Steps

Solving financial problems often involves several systematic steps that ensure all relevant factors are considered. This process includes identifying known values, performing necessary calculations, and critically analyzing the results.

To tackle the problem given:

• Start by clearly identifying what is known—total annual earnings and the specific bonus amount.
• Next, proceed with the calculations necessary to derive unknowns from knowns. Here, the subtraction of the bonus from the total was crucial.
• Finally, divide the simplified annual figure by 12 to find the recurring amount, reinforcing the regular monthly pay structure.

These steps offer a general framework that can be applied to similar financial problems. They provide a reliable method to break down complex problems into simple, manageable parts. This structured approach not only enhances problem-solving efficiency but also minimizes errors in financial computations.