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Chapter 2: Problem 13

APPLICATION A CD is on sale for \(15 \%\) off its normal price of \(\$ 13.95\). What is its sale price? Write a direct variation equation to solve this problem.

Short Answer

Expert verified
The sale price is $11.86.

Step by step solution

01

Understand the Problem

We need to find the sale price of a CD that is on sale for 15% off its normal price of $13.95. We'll solve this by creating an equation based on direct variation to find the discount amount and subtract it from the original price.
02

Set Up the Equation for Discount

A direct variation equation can be used to find the discount amount. The equation is given by \( y = kx \), where \( y \) is the discount, \( k \) is the percentage of the discount as a decimal, and \( x \) is the original price. Here, \( k = 0.15 \) and \( x = 13.95 \). The equation is: \( y = 0.15 \times 13.95 \).
03

Calculate the Discount Amount

Substitute the given values into the equation to find \( y \): \( y = 0.15 \times 13.95 = 2.0925 \). Thus, the discount amount is $2.09 when rounded to the nearest cent.
04

Calculate the Sale Price

Subtract the discount from the original price to find the sale price. The sale price is: \( 13.95 - 2.09 = 11.86 \). Thus, the sale price of the CD is $11.86.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Discount Calculation
Calculating a discount is an essential skill in math, especially when shopping or budgeting. When an item is marked as being on discount, it means you're paying less than the original price. Let's dive into how you can calculate the discount amount using percentages.
To find out how much you save during a discount:
  • Identify the percentage of the discount. In our example, it's 15%.
  • Convert this percentage into a decimal by dividing by 100. For 15%, this lessens to 0.15.
  • Use the direct variation equation: \( y = kx \), where \( y \) is the discount amount, \( k \) is the decimal form of the percentage, and \( x \) is the original price. Here, \( x \) is \(13.95, so we calculate \( y = 0.15 \times 13.95 \).
  • Solve it to find the discount amount. Here, it equals roughly \)2.09.
Breaking it down like this makes the calculation straightforward and ensures you know exactly how much you're saving.
Sale Price Determination
After calculating the discount amount, the next step is to figure out the sale price of the item. This is crucial because it tells you the amount you must pay at checkout.
To determine the sale price:
  • Start with the original price of the item. In our case, the CD is priced at \(13.95.
  • Subtract the discount amount you calculated earlier. If the discount is \)2.09, subtract it from $13.95.
  • The difference between the original price and the discount gives the sale price: \( 13.95 - 2.09 = 11.86 \).
This method ensures that you have accounted for the discount perfectly and know exactly what you're expected to pay. Understanding this process helps in managing finances smartly, making sure you're always aware of any cost reductions.
Percentages in Math
Percentages represent parts per hundred and are a fundamental concept in mathematics used across various applications such as discounts, interest rates, and statistics. Understanding percentages is key to solving real-life problems like determining sales prices and discounts.
Here's how percentages operate in math language:
  • They allow representation of proportions and comparisons easily. For example, 15% means 15 out of every 100.
  • Converting between percentages, decimals, and fractions is frequently required. For instance, 15% = 0.15 = \(\frac{15}{100}\).
  • In discount calculation, always transform the percentage figure into a decimal before using it in equations, such as \(0.15\) for 15%.
  • In practical scenarios, multiplying the decimal percentage by the price gives the discount or interest amount directly.
Mastering percentages simplifies understanding everyday financial decisions and comparative analyses, thus proving to be extremely beneficial across multiple situations.

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