Problem 52 - Algebra for College Students

Question

Is a \(10 \%\) discount followed by a \(30 \%\) discount the same as a \(30 \%\) discount followed by a \(10 \%\) discount? Justify your answer.

Step-by-Step Solution

Answer

Yes, both discount orders result in the same final price.

Step 1: Understand the problem

We need to determine whether applying a 10% discount followed by a 30% discount results in the same price as a 30% discount followed by a 10% discount. This involves calculating the final price for each scenario starting from the same original price.

Step 2: Define the original price

Let's assume the original price of an item is \( P \). This will be the starting point for our calculations in both scenarios.

Step 3: Calculate the final price for 10% followed by 30% discount

First, apply a 10% discount on \( P \). This reduces the price to \( 0.9P \). Next, apply a 30% discount on the new price, \( 0.9P \), reducing it to \( 0.7 \times 0.9P = 0.63P \).

Step 4: Calculate the final price for 30% followed by 10% discount

Apply a 30% discount on \( P \) first. The new price is \( 0.7P \). Now apply a 10% discount on this price, which results in \( 0.9 \times 0.7P = 0.63P \).

Step 5: Compare results from both scenarios

Both discount sequences (10% followed by 30%, and 30% followed by 10%) result in the same final price of \( 0.63P \). Thus, the order of applying these discounts does not matter.

Key Concepts

Percentage discount calculations

When dealing with percentage discount calculations, it's important to understand how discounts affect the price of an item differently based on the percentage applied. A discount is essentially a reduction on the price, which means you're paying less than the original amount.

Let's break down the concept: if you have an original price, say represented by the variable \( P \), applying a discount of \( 10\% \) means you're effectively reducing the price to \( 90\% \) of its original value. Mathematically, this means multiplying the original price by \( 0.9 \).

If you then apply another discount – say \( 30\% \), you're reducing the new price further to \( 70\% \) of the value after the \( 10\% \) discount. Thus, instead of calculating the final price after each discount separately, you compound the discounts by direct multiplication.

• A \( 10\% \) discount leaves \( 0.9 \) of the price.
• A \( 30\% \) discount leaves \( 0.7 \) of the previous price.

Understanding these principles helps in knowing how your total savings will be calculated.

Step-by-step solution

When solving problems like this, using a step-by-step solution approach helps clarify each part of the process. Let's go through the steps we summarized from the problem.

First, set an original price as a variable \( P \). This helps us understand how the discounts apply rather than needing an explicit numeric amount.

• Calculate \( 10\% \) discount then \( 30\% \): Start with reducing \( P \) to \( 0.9P \), then apply a \( 30\% \) discount on this new amount, giving us \( 0.63P \).
• Next, calculate \( 30\% \) discount then \( 10\% \): Begin by reducing \( P \) to \( 0.7P \), then apply another \( 10\% \) discount on it, resulting in an equivalent \( 0.63P \).

The step-by-step dissection allows you to see how each operation leads to the final price and clarifies that the order of operations in this scenario does not affect the result.

Mathematical justification

The mathematical justification for this problem lies in understanding the properties of multiplication. In arithmetic, when we perform operations such as discounts in percentage form, we are actually multiplying the values. It's important to know that multiplication is commutative.

This means that the order of the numbers you're multiplying doesn't change the result. Hence, multiplying \( 0.9 \) and \( 0.7 \) in any sequence still results in the same reduction to \( 0.63 \) of the original price \( P \).

Let’s break it down further:

• The expression \( 0.9 \times 0.7 \) represents compound multiplication.
• The concern here is about the position of multiplication (order), just like \( a \times b = b \times a \).
• This property provides the justification that it doesn’t matter whether you apply the 10% discount first or the 30% discount first; you'll end up with the same final price.

This mathematical property eliminates uncertainty in calculating consecutive discounts by emphasizing that the order doesn't affect the cumulative result.