Problem 38 - Elementary Algebra Within Reach

Question

The state income tax on a gross income of \(I\) dollars in Pennsylvania is \(3.07 \%\). Write an algebraic expression that represents the total amount of income tax.

Step-by-Step Solution

Answer

The algebraic expression that represents the total amount of income tax in Pennsylvania is \(T = I*0.0307\)

Step 1: Understand the problem

The main task is to express the amount of state income tax as an algebraic expression. Given that the gross income is defined as \(I\) dollars and the tax rate in Pennsylvania is $3.07 \%$, the algebraic expression will relate \(I\) and the tax rate.

Step 2: Convert percentage to decimal

Percentages can be converted to decimals by moving the decimal point two places to the left. Therefore, the tax rate in decimal form is \(0.0307\).

Step 3: Formulate the algebraic expression

The total income tax amount is calculated as a percentage of gross income. This relationship is expressed algebraically by multiplying the gross income \(I\) with the tax rate in decimal form. So, the total amount of income tax \(T\) can be expressed as \(T = I*0.0307\).

Key Concepts

Income Tax Calculations

Income tax calculations involve understanding how taxes are derived from a person's or entity's gross income. In various states, a specific income tax rate is applied to calculate the total tax payable. In our example, we’re considering the state of Pennsylvania, which imposes a fixed rate income tax on individuals. Here’s how it works:

• The state’s income tax rate is applied to the total gross income an individual earns.
• To find the tax amount, you multiply the gross income by the applicable tax rate.
• This process results in the total amount that must be paid as income tax to the state.

Let's take an example: if a person earns $50,000 annually, you would multiply this income by the tax rate (in decimal form). This helps calculate the total tax liability. Understanding this process can help in planning and financial decision-making.

Decimal Conversion

Decimal conversion is a fundamental mathematical skill that involves translating numbers from one form to another, such as percentages to decimals. This is crucial for various calculations, especially in finance and taxes. Here's how you can convert percentages into decimals:

• Move the decimal point two spots to the left.
• This movement changes a percentage into its equivalent decimal form.
• For example, 3.07% becomes 0.0307 when converted into a decimal.

Conversion from percentages to decimals makes calculations easier, as it simplifies the multiplication and division processes involved in tax and interest calculations. Decimal conversion is essential because most mathematical formulas used in finance employ decimals rather than percentages.

Percentage to Decimal

Understanding how to convert percentage values to decimals is essential for anyone dealing with mathematics, finance, or data analysis. It’s a simple yet invaluable skill in algebra and everyday life. When converting a percentage to a decimal, the basic rule is straightforward:

• Drop the percentage sign (%) and move the decimal point two places to the left.
• This change is because percentages are essentially fractions out of 100.
• For example, converting 4.5% to a decimal results in 0.045.

This simple conversion is practical in real scenarios such as calculating interest rates, discount rates, and more. When using percentages in mathematical equations, converting them to decimals allows for easier and more precise calculations.