Problem 78
Question
Let \(S(x)\) represent the weekly salary of a salesperson, where \(x\) is the weekly dollar amount of sales generated. If the salesperson pays \(15 \%\) of her salary in federal taxes, express her after-tax salary in terms of \(S(x)\) Assume there are no other deductions to her salary.
Step-by-Step Solution
Verified Answer
The after-tax salary of the salesperson in terms of \( S(x) \) is \( 0.85 * S(x) \).
1Step 1: Understand the problem
The weekly salary of the salesperson is represented by \( S(x) \). The salesperson pays 15% of this salary in federal taxes. So, the first step is to calculate the amount of tax that the salesperson pays.
2Step 2: Calculate the Tax
To calculate the income tax paid, multiply the income by the tax rate. Here, the income is \( S(x) \) and the tax rate is 15% or 0.15. So, the tax paid is \( 0.15 * S(x) \).
3Step 3: Calculate the After-Tax Salary
Subtract the tax amount from the initial salary to get the after-tax salary. The after-tax salary is therefore \( S(x) - 0.15 * S(x) \). Simplify it further to get the after-tax salary: \( 0.85 * S(x) \).
Key Concepts
Income TaxMathematical ModelingPercentages
Income Tax
Income tax is a crucial part of any salary calculation. It refers to the percentage of income an individual or business pays to the government. The primary goal is to fund public services, like education, healthcare, and infrastructure.
For a salesperson with a weekly salary represented by a function, such as \( S(x) \), understanding income tax begins with identifying the portion of this income that goes to taxes. In the problem above, the salesperson pays 15% of her salary in federal taxes. This means that for every dollar earned, 15 cents will be paid as tax.
In real-world applications, income tax rates can vary based on the total income, filing status, or local laws. However, in mathematical problems, we usually operate with a fixed percentage, simplifying the process of calculating taxes.
For a salesperson with a weekly salary represented by a function, such as \( S(x) \), understanding income tax begins with identifying the portion of this income that goes to taxes. In the problem above, the salesperson pays 15% of her salary in federal taxes. This means that for every dollar earned, 15 cents will be paid as tax.
In real-world applications, income tax rates can vary based on the total income, filing status, or local laws. However, in mathematical problems, we usually operate with a fixed percentage, simplifying the process of calculating taxes.
Mathematical Modeling
Mathematical modeling is a technique used to represent real-world scenarios in a mathematical format. It helps in understanding, analyzing, and solving various problems by converting them into mathematical terms.
In this exercise, the weekly salary \( S(x) \) is a mathematical model of the salesperson's earnings based on sales generated. By using this model, we can easily compute how taxes affect the take-home pay.
This approach allows individuals to predict outcomes based on different variables. In this scenario, knowing \( S(x) \) and the tax rate helps us calculate the net salary after taxes. Mathematical models are powerful because they provide a structured approach to solve problems logically and efficiently.
In this exercise, the weekly salary \( S(x) \) is a mathematical model of the salesperson's earnings based on sales generated. By using this model, we can easily compute how taxes affect the take-home pay.
This approach allows individuals to predict outcomes based on different variables. In this scenario, knowing \( S(x) \) and the tax rate helps us calculate the net salary after taxes. Mathematical models are powerful because they provide a structured approach to solve problems logically and efficiently.
Percentages
Percentages are a common way to express proportions and comparisons among numbers. They represent parts per hundred and are often used in finance to describe things like interest rates, discounts, or taxes.
When dealing with income tax, percentages simplify the calculation of tax obligations. For example, if the tax rate is 15%, this means that for every $100 of income, $15 is paid as tax.
When dealing with income tax, percentages simplify the calculation of tax obligations. For example, if the tax rate is 15%, this means that for every $100 of income, $15 is paid as tax.
- To compute taxes, multiply the income by the tax rate (expressed as a decimal).
- A 15% tax rate is the same as multiplying by 0.15.
- For an after-tax salary, subtract the tax from the original income.
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