Problem 50 - 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 If you purchased a Mercedes \(300 \mathrm{E}\) for $$ 43,500\( on January \)20,1991,\( in California, where the sales tax rate was \)6 \%,$ how much more would you pay in sales tax than luxury tax?

Step-by-Step Solution

Answer

You pay $1,260 more in sales tax than luxury tax.

Step 1: Identify the Luxury Tax Amount

First, we calculate the luxury tax for the car. The purchase price is \(43,500, and the luxury tax applies to the portion above \)30,000. Calculate this excess amount as \( 43,500 - 30,000 = 13,500 \). The luxury tax is 10% of this excess, so the luxury tax amount is \( 0.10 \times 13,500 = 1,350 \).

Step 2: Calculate Sales Tax Amount

Next, we calculate the sales tax for the car. The total purchase price is $43,500, and the sales tax rate is 6%. Therefore, the sales tax is \( 0.06 \times 43,500 = 2,610 \).

Step 3: Compare Sales Tax to Luxury Tax

Finally, compare the amounts of sales tax and luxury tax. The sales tax paid is \(2,610, and the luxury tax paid is \)1,350. To find out how much more is paid in sales tax than luxury tax, calculate \( 2,610 - 1,350 = 1,260 \).

Key Concepts

Luxury Tax

Luxury tax is an additional charge imposed on certain high-end or expensive items. Governments often introduce luxury taxes to generate more revenue from those able to afford luxury items. In the context of automobiles, like in the exercise, it was applied to the portion of the purchase price that exceeded a specified threshold, in this case, $30,000.
For calculating luxury tax:

• Determine the amount by which the item's price exceeds the threshold.
• Apply the tax percentage to this excess amount.

For example, if a car costs $43,500, the luxury tax is calculated on $13,500, which is the amount over $30,000. We then find 10% of $13,500, leading to a luxury tax of $1,350.

Sales Tax Calculation

Sales tax is a standard percentage added to the purchase price of goods and services.
Purchasers pay it and businesses collect it for the government. The rate depends on the location where the purchase occurs. In this case, California has a 6% sales tax.
To calculate sales tax for a given product:

• Determine the total purchase price.
• Multiply the total price by the sales tax rate.

Using our exercise, a car priced at \(43,500 with a 6% sales tax rate would incur a sales tax of:\[0.06 \times 43,500 = 2,610\]Therefore, buyers would pay an additional \)2,610 for the sales tax.

Percentage Operations

Percentage operations are used to find parts of a whole, often seen in calculations involving taxes, discounts, or interest rates. Understanding percentages is crucial for figuring out how much of an item is being taxed or discounted.
To work with percentages, follow these steps:

• Convert the percentage to a decimal for calculations by dividing by 100.
• Multiply this decimal by the number you're finding the percentage of.

For instance, finding 10% of $13,500 involves:\[10\% = \frac{10}{100} = 0.10\]Then calculate:\[0.10 \times 13,500 = 1,350\]This operation tells us how much the luxury tax costs based on the excess price.

Step-by-Step Math Solutions

Breaking down a problem into step-by-step math solutions is essential for understanding and solving complex problems. This approach makes it easier to grasp each part of a problem and see how they connect.
Here's how to solve our sample problem:

• Step 1: Calculate the luxury tax on the excess purchase price above $30,000.
• Step 2: Compute the sales tax on the total purchase price.
• Step 3: Compare the two results to find the difference in tax amounts.

Each stage involves specific calculations using percentage operations to find the tax amounts. This systematic approach aids in verifying each part of your solution and ensuring accuracy.