Problem 10 - Algebra 2

Question

For Exercises \(10-13,\) use the following information. Mai-Lin is shopping for computer software. She finds a CD-ROM that costs \(\$ 49.99,\) but is on sale at a 25\(\%\) discount. She also has a \(\$ 5\) coupon she can use. Express the price of the CD after the discount and the price of the CD after the coupon. Let \(x\) represent the price of the CD, \(p(x)\) represent the price after the 25\(\%\) discount, and \(c(x)\) represent the price after the coupon.

Step-by-Step Solution

Answer

The final price of the CD after the discount and coupon is \(c(x) = 32.49\).

Step 1: Determine the Discount Amount

We start by calculating the discount amount from the original price of the CD. Given that the CD-ROM costs \$49.99 and is on sale with a 25\% discount, the discount amount can be found as follows:\[\text{Discount} = 0.25 \times 49.99\]

Step 2: Calculate the Price After Discount

Next, we subtract the discount amount from the original price to find the price after the discount. Let \( p(x) \) be this price:\[p(x) = 49.99 - \text{Discount} = 49.99 - (0.25 \times 49.99)\]

Step 3: Apply the Coupon

After calculating the price after the discount, we then apply the \$5 coupon. This gives us the final price of the CD-ROM. Let \( c(x) \) represent this final price:\[c(x) = p(x) - 5\]

Step 4: Express the Final Result

Substitute the value from Step 2 into the expression for \( c(x) \) to find the final price:\[c(x) = (49.99 - (0.25 \times 49.99)) - 5\]

Key Concepts

Discount Calculation

Discount calculation is an essential skill when shopping, as it helps you understand how much you're saving on a purchase. In this exercise, Mai-Lin spots a CD-ROM priced at \( \\(49.99 \) with a 25\% discount. To find out the discount value, you multiply the original price by the discount percentage in decimal form.

Here's how you can calculate it:

• Convert the percentage discount to a decimal by dividing by 100. So, 25\% becomes 0.25.
• Multiply the original price by this decimal: \( 0.25 \times 49.99 \).
• This gives you the discount amount, which is \( \\)12.50 \).

Remember, calculating discounts using algebraic expressions can make shopping much more straightforward and helps you manage your expenses better.

Price Reduction

Price reduction involves reducing the original price by a specific amount or percentage, which benefits shoppers like Mai-Lin looking to save money. After calculating the discount amount, you need to figure out the new price of the CD after the discount has been applied.

To achieve this, you subtract the discount amount from the original price:

• The formula becomes \( p(x) = \text{Original Price} - \text{Discount} \).
• Substitute the numbers: \( p(x) = 49.99 - 12.50 \).
• This results in a new reduced price of \( \$37.49 \).

This method of price reduction through a simple subtraction in algebra helps you find the new cost easily, emphasizing the importance of understanding budget management.

Step-by-Step Solution

Understanding the step-by-step solution can simplify complex buying decisions and enhance your overall shopping experience. Through each stage, you're guided to the final discounted and reduced price in a structured manner.

Here's how these steps come together:

• **Step 1:** Calculate the discount using \( 0.25 \times 49.99 \), which gives a discount of \( \\(12.50 \).
• **Step 2:** Subtract the discount from the original price, resulting in \( 49.99 - 12.50 = 37.49 \).
• **Step 3:** Finally, apply the coupon by further subtracting \( \\)5 \) from the discounted price: \( 37.49 - 5 = 32.49 \).

Following these simple steps ensures you don’t miss out on any savings, and can skillfully find the final price, in this case, concluding with \( c(x) = \$32.49 \). Such step-by-step problem-solving is a handy tool, not just in mathematics, but in everyday shopping scenarios.