Problem 20 - Prealgebra

Question

Cigarette Tax In an effort to encourage people to quit smoking, many states place a high tax on a pack of cigarettes. Nine states place a tax of $$ 2.00\( or more on a pack of cigarettes, with New Jersey being the highest at \) 2.575\( per pack. If this is \)39 \%$ of the cost of a pack of cigarettes in New Jersey, how much does a single pack cost?

Step-by-Step Solution

Answer

A pack costs approximately $6.60 in New Jersey.

Step 1: Identify the Given Information

We are given that the tax is $2.575, which represents 39% of the total cost of a pack of cigarettes in New Jersey.

Step 2: Set Up the Equation

Let the total cost of a pack of cigarettes be represented by \( x \). According to the problem, 39% of \( x \) is $2.575. This can be expressed as an equation: \( 0.39x = 2.575 \).

Step 3: Solve for x

To find \( x \), divide both sides of the equation by 0.39 to isolate \( x \): \[ x = \frac{2.575}{0.39} \]

Step 4: Calculate the Total Cost

Perform the division to find \( x \): \[ x \approx 6.6025 \] Thus, the total cost of a pack of cigarettes in New Jersey is approximately $6.60.

Key Concepts

Understanding Percentages

Percentages are a way of expressing a number as a fraction of 100. This is particularly useful for comparing relative sizes of various quantities. When working with percentages, remember that:

• 100% represents the whole of something.
• A percentage like 39% means 39 out of every 100 units of the whole.

In the exercise, the tax of $2.575 is described as 39% of the total cost of a pack of cigarettes. So, we are essentially saying that $2.575 is 39 parts out of the 100 parts that make up the total cost. To find the whole (i.e., the total cost), you need to determine what 100% of the cost is based on the known part. This is achieved by setting up an equation that relates the percentage to the total.

Building and Solving Equations

Equations are mathematical statements that assert the equality of two expressions. They are vital tools for expressing relationships and solving problems like the one given.

• First, translate the problem into a mathematical statement. Here, it’s about finding the total cost when 39% of it is $2.575.
• To write this as an equation, denote the total cost by a variable, for example, \( x \).

The problem tells us \( 0.39x = 2.575 \). This equation links the percentage to the known amount, allowing us to solve for the unknown \( x \).
To solve this equation, divide both sides by 0.39. This isolates \( x \) on one side and gives:\[x = \frac{2.575}{0.39}\]Carrying out this division provides the value of \( x \), which is the total cost of a pack of cigarettes.

Problem Solving Techniques

Problems in mathematics often require a systematic approach to solve. This exercise demonstrates several key problem-solving techniques:

• Identify known and unknown quantities: First, we identify the tax amount ($2.575) and understand it represents 39% of the total cost.
• Translate words into math: Create an equation using the percentage information and set it equal to the known part, establishing a relationship.
• Solve the equation: Use algebraic methods to isolate the variable representing the unknown part, in this case, the total cost.
• Verify your results: Check your solution by substituting it back into the context of the problem. Here, multiply the solution by the percentage to see if it matches the given tax amount.

Each step builds on the previous one to navigate from the question to the solution efficiently. Always ensure your interpretations are consistent with the problem statement to avoid errors.