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Chapter 9: Problem 31

An item is on sale for \(40 \%\) off its original price. If it is then marked down an additional \(60 \%\), does this mean the item is free? Discuss why or why not.

Short Answer

Expert verified
The item is not free after discounts; it costs 24% of the original price.

Step by step solution

01

Calculate the First Discount

The first discount reduces the item's price by 40%. If the original price is \( x \), the reduction is \( 0.4x \). Therefore, the sale price after the first discount is \( x - 0.4x = 0.6x \).
02

Apply the Second Discount

The second discount is 60% off the sale price obtained in Step 1. The reduction is \( 0.6 \times 0.6x = 0.36x \). Therefore, the final price after the second discount is \( 0.6x - 0.36x = 0.24x \).
03

Determine if the Item is Free

The final price is \( 0.24x \), which is not zero. The discounts do not make the item free because the final price is 24% of the original price, not zero.

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Key Concepts

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

Understanding Percentage Discount
When we talk about a percentage discount, we refer to reducing the item's price by a specified percentage of its original value. For example, a 40% discount means that the item is being sold for 40% less than its original price. This can be calculated by multiplying the original price by the percentage (expressed as a decimal) to find the discount amount.
For an item originally priced at \( x \), a 40% discount is calculated as follows:
  • Take 40% of \( x \), which is \( 0.4x \).
  • Subtract this amount from \( x \) to get the new price: \( x - 0.4x = 0.6x \).
This process effectively reduces the price by the percentage stated. It's essential to remember that percentage discounts are always relative to the price they are applied to.
Grasping Price Reduction
Price reduction is the outcome of applying a discount. It's important to note that the reduction is applied to the value at the time of discount, not to the original price.
In the problem example, after a 40% initial discount, we have an item priced at 60% of its original value. Additional discounts, like the 60% in the example, are applied to this already-reduced price.
  • After the first discount, the price becomes \( 0.6x \).
  • Then the 60% discount is calculated on this new price \( 0.6 \times 0.6x = 0.36x \).
  • The final price is \( 0.6x - 0.36x = 0.24x \).
This is why combining discounts doesn't necessarily mean the price reaches zero. It depends on how each discount percentage is applied successively.
Applying Mathematical Reasoning
When solving percentage discounts, mathematical reasoning is key to understanding how multiple discounts interact. Each discount is applied to the current price, not the original. This cascading effect creates a compounded reduction rather than a straightforward subtraction.
If we take the step-by-step approach:
  • First, determine the remaining percentage of the price after each discount.
  • Recognize that multiplying through these remaining percentages reveals the final fraction of the price remaining.
  • In this example, the remaining percentage after both discounts is \( 0.6 \times 0.6 = 0.36 \), and finally \( 0.36 \times x = 0.24x \).
This kind of reasoning ensures the solution is logical and accurately reflects real-world scenarios, leaving no room for misunderstanding about whether an item is free or not. Mathematically, this clarifies that two consecutive discounts do not sum up directly but act upon whatever price remains at each stage.

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