Problem 75 - Elementary Algebra

Question

The regular price of a 14 -piece cookware set is \(\$ 728\). It is on sale at a discount of \(35 \%\). The sales tax is \(6.5 \%\). Find the final cost of the cookware including tax. Round to the nearest hundredth.

Step-by-Step Solution

Answer

The final cost is \$503.96\.

Step 1: Calculate the discount amount

First, find the amount of the discount by multiplying the regular price by the discount percentage. The regular price is \$728\, and the discount is \35 \%\. \[\text{Discount Amount} = 728 \times 0.35 = 254.8\]

Step 2: Subtract the discount from the regular price

Next, subtract the discount amount from the regular price to find the sale price: \[\text{Sale Price} = 728 - 254.8 = 473.2\]

Step 3: Calculate the sales tax amount

Now calculate the sales tax by multiplying the sale price by the sales tax percentage, which is \6.5 \%\. \[\text{Sales Tax Amount} = 473.2 \times 0.065 = 30.758\]

Step 4: Add the sales tax to the sale price

Finally, add the sales tax amount to the sale price to find the final cost of the cookware set: \[\text{Final Cost} = 473.2 + 30.758 = 503.96\]

Key Concepts

Discount Calculations

Discounts are common in many sales situations. They reduce the regular price of an item, making it more attractive to buyers. Understanding how to calculate a discount is crucial for making informed purchasing decisions. Here's how you can do it:

First, convert the discount percentage into a decimal. For instance, a 35% discount becomes 0.35. This conversion is done by dividing the percentage by 100.

Next, multiply the regular price by this decimal to find the discount amount. In our example, the regular price of the cookware set is \(728, and the discount percentage is 35%. So, we do:

\[\text{Discount Amount} = 728 \times 0.35 = 254.8\]

This calculation tells us that the discount amount is \)254.80. Understanding this step helps you to see how much money you save from the original price.

Finally, subtract the discount amount from the regular price to find the sale price:

\[\text{Sale Price} = 728 - 254.8 = 473.2\]Understanding discount calculations allows you to evaluate the true value of sales and promotions.

Sales Tax Calculation

Sales tax is an additional amount added to the sale price of goods and services, set by government regulations. Knowing how to compute sales tax is essential to determine the total cost of an item.

First, convert the sales tax percentage into a decimal form. For example, a sales tax of 6.5% becomes 0.065. This is done by dividing the percentage by 100.

Then, multiply the sale price by the sales tax percentage to find the sales tax amount. Using our example, the sale price of the cookware set is \(473.2, and the sales tax percentage is 6.5%. So, we do:\[\text{Sales Tax Amount} = 473.2 \times 0.065 = 30.758\]

This calculation gives us the sales tax amount, which is \)30.76. Calculating sales tax helps you to understand how much extra you need to pay over the listed sale price.

Adding this tax to the sale price gives you the total amount you need to pay. This is critical for budgeting and financial planning.

Final Cost Determination

The final cost is the total amount you pay, which includes both the sale price and the sales tax. Understanding this helps you to be prepared for the total expenditure.

To find the final cost, simply add the sales tax amount to the sale price. Using our example, the sale price is \(473.2, and the sales tax amount is \)30.76. Therefore:

\[\text{Final Cost} = 473.2 + 30.758 = 503.96\]

This means the final cost of the cookware set, after accounting for the discount and adding the sales tax, is $503.96.

Knowing how to determine the final cost is essential when making purchases, as it gives you a complete understanding of how much you are actually spending. This is especially useful for managing personal finances and ensuring you do not overspend.