Problem 54 - Prealgebra

Question

In \(1990,\) Congress passed a law, which took effect on January \(1,1991,\) requiring an additional tax of \(10 \%\) on a portion of the purchase price of certain luxury items. (In 1996 the law was amended so that this tax on luxury automobiles expired in 2003.) For expensive cars, it was paid on the part of the purchase price that exceeded 30,000 dollars. For example, if you purchased a Jaguar XJ-S for 53,000 dollars , you would pay a luxury tax of \(10 \%\) of 23,000 dollars , because the purchase price, 53,000 dollars , is 23,000 dollars above 30,000 dollars Suppose you bought a car in \(1991 .\) One of the cars you were interested in had a sticker price of $$ 35,500,\( while another had a sticker price of $$ 28,500\). If you expected to pay full price for either car, how much did you save if you bought the less expensive car?

Step-by-Step Solution

Answer

By choosing the less expensive car, you save $7,550.

Step 1: Understanding Luxury Tax Calculation

For cars purchased in 1991, the luxury tax is applicable on the amount exceeding $30,000 of the car's sticker price. This tax is calculated at a rate of 10% on the excess amount.

Step 2: Calculating Excess Amount for Both Cars

For a car with a sticker price of $35,500, the excess amount over $30,000 is $35,500 - $30,000 = $5,500. For a car priced at $28,500, since it is below $30,000, there is no excess to calculate.

Step 3: Calculating Luxury Tax for the More Expensive Car

The luxury tax on the car priced at \(35,500 is 10% of the excess amount \)5,500. Calculate: \( 0.10 \times 5,500 = 550 \) dollars.

Step 4: Comparing Costs

The more expensive car costs $35,500 plus a luxury tax of $550, totaling $36,050. The less expensive car costs $28,500 with no additional tax. Thus, the difference in total cost is $36,050 - $28,500.

Step 5: Final Savings Calculation

By buying the less expensive car, you would save $36,050 - $28,500 = $7,550 in total. This includes both the lower base price and the absence of the luxury tax.

Key Concepts

Luxury Tax

A luxury tax is an additional cost imposed on certain high-end items, specifically to items perceived as non-essential. In the context of our problem, this tax was relevant to cars over $30,000 in value. The tax rate set by Congress was 10%, applied just to the amount over this threshold. For instance, if a car had a sticker price of $35,500, the tax would be applied to the $5,500 excess over $30,000. Luxury taxes serve as a way for the government to earn revenue while simultaneously discouraging excessive spending on luxury goods. They make it more expensive to purchase high-end products, thus potentially reducing demand. This specific tax was in effect from 1991 till 2003, at which point the regulation changed.

Basic Arithmetic

Basic arithmetic is foundational for solving our problem, especially operations like addition and subtraction. These operations help determine the excess price portion of a luxury car and consequently the applicable tax. Let’s break it down:

• Subtraction is used to find the excess portion of the car’s price over $30,000. For the more expensive car, we subtract $30,000 from $35,500 to find an excess of $5,500.
• Addition comes into play when calculating the total cost of the car, which is the sum of its sticker price and the luxury tax. Here, for example, the total cost is $35,500 (car price) + $550 (luxury tax) = $36,050.

These simple arithmetic operations allow us to navigate through such financial problems with ease and efficiency.

Problem Solving

Problem solving in this context involves a strategic approach to decide which car is the smarter purchase. Here, it prompts us to consider both the sticker price and additional taxes.
First, calculate the tax for the more expensive car. Then, add it to the purchase price to find its total cost. Next, compare this with the cost of the less expensive car, which involves no extra tax. Problem solving requires a clear method to decide the smartest financial move. In this case, it's about finding total costs, making comparisons, and understanding the benefit of avoiding the luxury tax by choosing the less expensive car. This method helps highlight how additional costs can alter your final decision and allow you to maximize savings.

Percentage Calculation

Percentage calculation is key when dealing with taxes like our 10% luxury tax. The task involves calculating a percentage of a given numerical value, often referred to as the 'rate'. Follow this simple formula:\[\text{Tax Amount} = \left(\frac{\text{Percentage}}{100}\right) \times \text{Excess Amount}\]For example, the excess amount in our exercise was \(5,500. The luxury tax rate was 10%, so we calculate the tax as follows:\[0.10 \times 5,500 = 550 \]This means you pay \)550 as the luxury tax on a car priced $35,500. Understanding how to compute percentages is crucial in numerous financial contexts like sales, discounts, and taxes. In our example, it's a critical step for determining additional costs linked to purchasing a luxury vehicle.