Problem 48 - Algebra 2

Question

For Exercises \(48-50,\) use the following information. Liluye wants to buy a pair of inline skates that are on sale for 30\(\%\) off the original price of \(\$ 149 .\) The sales tax is 5.75\(\% .\) Express the price of the inline skates after the discount and the price of the inline skates after the sales tax using function notation. Let \(x\) represent the price of the inline skates \(p(x)\) represent the price after the 30\(\%\) discount, and \(s(x)\) represent the price after the sales tax.

Step-by-Step Solution

Answer

The price after the 30% discount is $104.30, and after the sales tax, it is $110.30.

Step 1: Define the Discount Function

First, we need to calculate the discount on the inline skates. The original price of the skates is $149. They are on sale at a 30% discount. In function notation, to express the discounted price, let \( p(x) \) be the function that gives the price after the discount on a pair of skates. The discount portion is 30% of \( x \), so we have: \[ p(x) = x - 0.30x = 0.70x \]Substitute the original price into the function: \[ p(149) = 0.70 \times 149 \]

Step 2: Calculate the Discounted Price

Calculate the result from the previous expression: \[ p(149) = 0.70 \times 149 = 104.30 \]Therefore, the price of the inline skates after applying the 30% discount is \$104.30.

Step 3: Define the Sales Tax Function

Next, we must include sales tax in our calculations. The sales tax rate is 5.75%. With the discounted price \( p(x) \), we want to express the price after tax as \( s(x) \). The tax added is 5.75% of the price of the skates after the discount. Thus, our function becomes: \[ s(x) = x + 0.0575x = 1.0575x \]Substitute the discounted price into this function: \[ s(104.30) = 1.0575 \times 104.30 \]

Step 4: Calculate the Final Price with Sales Tax

Calculate the final price as follows: \[ s(104.30) = 1.0575 \times 104.30 = 110.30 \]Therefore, the price of the inline skates after the discount and sales tax is \$110.30.

Key Concepts

Discount Calculation

To determine the cost of the inline skates after a discount, we start with the original price, which is \(149. Discounts in percentage decrease the original price by a certain rate. In this case, the discount rate is 30%.
Function notation effectively helps us express these mathematical changes. Here, the function will show how much the skates cost after applying the discount.
- **Original Price:** \)149- **Discount Percentage:** 30%- **Discounted Amount:** 30% of \(149, which is calculated as:- Formula: \[ p(x) = x - 0.30x = 0.70x \]- Calculation: \[ p(149) = 0.70 \times 149 \]When you multiply 0.70 by 149, the result is \)104.30, which is the price after the discount has been applied.

Sales Tax Calculation

Once we've determined the price of the inline skates after the discount, we then add sales tax. Sales tax is a percentage added on top of the discounted price. This means we're increasing the total cost by the tax percentage.
Using function notation, we define a function that adds the sales tax of 5.75% to the already discounted price.- **Discounted Price:** \(104.30- **Sales Tax Percentage:** 5.75%- **Tax Calculation:** - Formula: \[ s(x) = x + 0.0575x = 1.0575x \]- Calculation: \[ s(104.30) = 1.0575 \times 104.30 \]This multiplication results in approximately \)110.30, which becomes the final price after incorporating the sales tax.

Step-by-Step Solution

When solving problems involving discounts and sales tax, following a clear step-by-step approach aids in understanding and accuracy.
1. **Identify Given Values and Percentages**: Always start with identifying what you know—here, the original price, discount rate, and tax rate. - Original Price = \(149 - Discount = 30% - Tax = 5.75% 2. **Calculate the Discounted Price**: Use the discount percentage to calculate the deduction from the original price using function notation. - Discount Calculation: \[ p(x) = x - 0.30x = 0.70x \] which simplifies to \)104.30 3. **Calculate Total with Sales Tax**: Apply the sales tax to the new price post-discount using another function notation. - Tax Calculation: \[ s(x) = x + 0.0575x = 1.0575x \] which equals about $110.30 By using function notation at each stage, you systematically convert percentages to decimals, apply these to the original or current price, and ensure clarity in each calculation.