Problem 53 - 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 \(1991 .\) How much did you save on a car with a sticker price of $$ 31,500,\( if you persuaded the car dealer to reduce the price to $$ 29,900 ?\)

Step-by-Step Solution

Answer

You saved $150 by persuading the car dealer to reduce the price.

Step 1: Determine the taxable amount before reduction

Initially, the car's sticker price is \(31,500. The taxable portion is the amount exceeding \)30,000. Calculate this by subtracting \(30,000 from \)31,500. This gives us:\[ 31,500 - 30,000 = 1,500 \].

Step 2: Calculate the luxury tax before reduction

The luxury tax is 10% of the taxable amount. Use the taxable amount from Step 1 to calculate the tax:\[ 0.10 \times 1,500 = 150 \].

Step 3: Determine the taxable amount after reduction

After convincing the dealer to lower the price to \(29,900, check if there is any taxable amount. Since \)29,900 is less than $30,000, the taxable amount is:\[ 29,900 - 30,000 = -100 \].Negative taxable amount means there is no tax due.

Step 4: Calculate the tax after price reduction

Since the reduced price is below $30,000, the luxury tax on this amount is $0.

Step 5: Calculate the savings

The savings from the price reduction are simply the original tax minus the tax after negotiation:\[ 150 - 0 = 150 \].

Key Concepts

Understanding Tax Calculations

Tax calculations play a crucial role in determining the final cost of purchases, especially when it comes to higher-priced goods like luxury items. Here's how it works:

• Taxes are typically calculated based on a percentage of the amount over a certain threshold. For luxury goods, this can mean additional costs.
• In the given exercise, a tax of 10% applied only to the amount exceeding $30,000. This kind of tax is designed to target high-value purchases.
• Understanding the threshold helps buyers estimate if they'll owe any tax and how much it could be.

Knowing how tax calculations work can empower you to make better financial decisions, such as negotiating lower prices to avoid or reduce taxes.

What is Luxury Tax?

Luxury tax is a surcharge levied on items that are considered non-essential or high-end goods. This tax is different from regular sales tax in a few ways:

• It's imposed specifically on goods that surpass a defined price level. For example, our exercise used a $30,000 mark as a starting point.
• Luxury taxes are intended to generate additional revenue from individuals who are likely able to afford it, essentially taxing the wealthy at higher rates.
• They can encourage manufacturers and buyers to produce and purchase more cost-effective products.

This type of tax can influence purchasing behavior by making luxury items more expensive, thus potentially reducing demand.

Effective Step-by-Step Solutions

Breaking down a problem into steps can greatly simplify the process of understanding and solving it. Let's use a step-by-step approach similar to our exercise:

• Step 1: Identify the portion of the purchase subject to tax. Subtract the threshold ($30,000) from the item's price.
• Step 2: Calculate the tax due on that amount by applying the tax rate (10%).
• Step 3: If the price changes, reassess the taxable amount using the new price.
• Step 4: Recalculate the tax with the revised price, checking if it falls below the threshold, making it tax-free.
• Step 5: Determine savings by comparing taxes before and after any price negotiation.

This methodical approach takes complex tax scenarios and translates them into manageable steps. It's a valuable skill when handling real-world financial decisions.