Problem 7 - Prealgebra

Question

Tax Rate Suppose the purchase price of a dining room set is $$ 450 .\( If the sales tax is \)22.50, what is the sales tax rate?

Step-by-Step Solution

Answer

The sales tax rate is 5\%.

Step 1: Identify Given Values

We are given the purchase price of the dining room set, which is $450, and the sales tax amount, which is $22.50. We need to find the tax rate from this information.

Step 2: Understanding the Relationship

The sales tax amount is determined by multiplying the purchase price by the tax rate. In equation form, this can be represented as: \( \text{Sales Tax} = \text{Purchase Price} \times \text{Tax Rate} \).

Step 3: Setting Up the Equation

Substitute the known values into the equation: \( 22.50 = 450 \times \text{Tax Rate} \).

Step 4: Solving for Tax Rate

To find the tax rate, divide the sales tax by the purchase price: \( \text{Tax Rate} = \frac{22.50}{450} \).

Step 5: Calculate Tax Rate

Perform the division: \( \text{Tax Rate} = \frac{22.50}{450} = 0.05 \).

Step 6: Expressing as a Percentage

Convert the decimal tax rate to a percentage by multiplying by 100: \( 0.05 \times 100 = 5\% \). The sales tax rate is 5\%.

Key Concepts

Percentage Calculation

Understanding percentage calculations is crucial for determining how much of a value changes compared to its original amount. In this context, percentages help us express the sales tax as a part of the overall purchase price. To convert a fraction or ratio into a percentage, you multiply it by 100. This is because percentage literally means 'per hundred'.
For example, if we found a tax rate of 0.05 (as a decimal), we convert it to a percentage by multiplying by 100, getting 5%. So, the basic steps for calculating a percentage are:

• Identify the ratio you need to convert to a percentage.
• Multiply the ratio by 100 to get the percentage.

This simple calculation is an essential skill in finance, allowing you to quickly convert between decimals and percentages, which is frequently needed when dealing with taxes and discounts.

Algebraic Equations

Algebraic equations are mathematical statements where two expressions are set equal to each other. They can be used to solve for unknown values, such as the tax rate in the problem we have. The power of algebra lies in its ability to provide a systematic way to solve problems involving unknowns.
In this problem, we used the equation:\[22.50 = 450 \times \text{Tax Rate}\]Here, the sales tax amount is equated to the product of the purchase price and the tax rate. Algebraic manipulation allows us to isolate 'Tax Rate' by dividing both sides by 450:\[\text{Tax Rate} = \frac{22.50}{450}\]These steps showcase how algebra helps break down a problem into manageable parts, making it easier to find solutions.

Problem Solving Steps

Problem-solving is an essential skill that involves a strategic approach to finding solutions. When dealing with mathematical problems such as calculating a sales tax rate, it helps to have a clear plan:

• **Identify the given values:** Know what information is provided. Here, the purchase price is $450, and the sales tax is $22.50.
• **Understand relationships:** Connect the given values through a known formula, such as the sales tax formula.
• **Set up the equation:** Substitute the known values into your algebraic equation.
• **Solve for the unknown:** Manipulate the equation to isolate the variable you need, in this case, the tax rate.
• **Calculate and convert:** Perform the necessary calculations, and convert your answer into the desired format, such as a percentage.

By following these problem-solving steps meticulously, you ensure a methodical approach that reduces errors and provides clarity.