Problem 25 - Prealgebra

Question

The amount of sales tax paid on a purchase is proportional to the price of the item. If the sales tax rate is \(6.25 \%\), find the equation if \(p\) is the price of the item and \(t\) is the sales tax. \(\mathbf{F} \quad p=0.0625 t\) \(\mathbf{H} t=6.25 p\) \(\mathbf{G} p=6.25 t\) \(\mathbf{J} \quad t=0.0625 p\)

Step-by-Step Solution

Answer

The correct equation is \(t = 0.0625p\) (Option J).

Step 1: Understand the Proportionality

The sales tax is proportional to the price of the item, meaning the sales tax is a fixed percentage of the item's price. This percentage is given as a rate of \(6.25\%\). We need to express this relationship in the form of an equation where \(t\) (sales tax) is a function of \(p\) (price of the item).

Step 2: Convert the Percentage to a Decimal

To use the percentage in our equation, we must convert \(6.25\%\) to a decimal. The conversion is done by dividing the percentage by 100. Therefore, \(6.25\% = \frac{6.25}{100} = 0.0625\).

Step 3: Form the Equation

Since the sales tax \(t\) is \(6.25\%\) of the item's price \(p\), we can write the relationship as \(t = 0.0625p\). This equation shows that \(t\) is directly proportional to \(p\) with a proportionality constant of \(0.0625\).

Key Concepts

Proportionality

Proportionality is a fundamental mathematical concept that expresses a specific type of relationship between two quantities. When we say that the sales tax is proportional to the price of an item, it means that the sales tax maintains a constant ratio or percentage of the item's price, irrespective of how high or low that price is. In practical terms, if you double the price of the item, the sales tax will also double.

• For instance, if the price of an item is $100 with a proportional tax rate of 6.25%, the tax amount will be $6.25.
• If the price increases to $200, the tax will be $12.50, which is proportionally the same percentage (6.25%).

Understanding proportionality helps in forming equations that accurately represent the relationship between different quantities, enabling us to predict and calculate values accurately. It simplifies our understanding of how changing one variable affects the other without requiring complex calculations each time.

Percentage to Decimal Conversion

Percentage to decimal conversion is crucial in mathematical equations involving rates, such as sales tax calculations. A percentage provides an easy-to-understand format but isn't directly usable in calculations without conversion.
To convert a percentage to a decimal, divide it by 100. This is because a percentage is a way of expressing a number as a fraction of 100.

• For example, converting 6.25%:
  Start by dividing 6.25 by 100, which gives us 0.0625.
• Another instance: 50% would convert to 0.50 since 50 divided by 100 equals 0.50.

Once in decimal form, the number becomes usable in equations and formulas. This step is vital for maintaining consistency and accuracy in mathematical operations, ensuring that all calculations are performed correctly.

Equation Formation

Forming equations is the backbone of solving many mathematical problems, as it sets up the relationship between different variables in a structured and solvable manner. In our sales tax example, forming an equation enables us to express the relationship between the price of an item and the sales tax.
To form the equation:

• Identify the variables: For this scenario, let
  x = price of the item (p)
  y = sales tax amount (t)
• Determine the relationship: The tax rate is 0.0625. Thus, the equation becomes:
  \[ t = 0.0625p \]
• This equation tells us that the sales tax (t) is 6.25% of the price (p).

The equation is simple yet effective in calculating the sales tax for any given item price, removing guesswork and empowering accurate financial decisions.Equations provide clarity, allowing every component of the problem to be understood and utilized effectively.