Problem 23 - Prealgebra

Question

FIND THE ERROR The price of bottled water at the concession stand was doubled. Scott says this is a \(50 \%\) increase, but David says it's a \(100 \%\) increase. Who is correct? Explain your reasoning.

Step-by-Step Solution

Answer

David is correct; it's a 100% increase.

Step 1: Define the Initial Situation

Start by assuming the initial price of bottled water is \( P \).

Step 2: Calculate the New Price

Since the price is doubled, the new price is given as \( 2P \).

Step 3: Calculate the Percentage Increase

The percentage increase can be calculated using the formula: \[ \text{Percentage Increase} = \left( \frac{\text{New Price} - \text{Old Price}}{\text{Old Price}} \right) \times 100\% \]Substituting into the formula gives: \[ \left( \frac{2P - P}{P} \right) \times 100\% = 100\% \]

Step 4: Evaluate the Claim

Scott's claim of a 50% increase is incorrect, while David's claim of a 100% increase is correct as shown by the calculation.

Key Concepts

Error Analysis

When analyzing errors like Scott's in this exercise, it's important to understand where the misunderstanding occurred. Error analysis helps in identifying and correcting misconceptions we may have in problem-solving.
Scott thought that doubling the price resulted in a 50% increase. This implies that he might have confused the concept of 'doubling' with 'increasing by half'. Here, Scott equated the doubling effect to adding another 50% of the original, which is incorrect. It shows the importance of:

• Careful reading of the problem statement
• Double-checking calculations and assumptions
• Comparing different approaches or interpretations to see which is valid

Therefore, in mathematics, verifying each step and understanding basic operations can help in avoiding such errors.

Mathematical Reasoning

Mathematical reasoning is the process of using logic to solve problems step by step.
In this exercise, David used reasoning to calculate the correct percentage increase of 100%. This shows how important it is to follow a structured approach.First, defining the initial situation by assuming the original price helps establish a reference point:

• Let the initial price be denoted by a variable, such as \( P \)
• Calculate the new situation. Here, since the price was doubled: new price = \(2P\)
• Apply the formula for percentage increase or any other relevant formula to arrive at a logical result

Employing such logical reasoning avoids reliance on intuition and helps ensure accuracy.

Prealgebra Concepts

Prealgebra serves as a foundation for understanding basic mathematical concepts, like percentage increases. In this case, we use simple calculations that form the basis of algebraic thinking.Key prealgebra concepts useful in this type of problem include:

• Understanding variables: Representing unknowns or initial values with symbols such as \( P \)
• Grasping basic operations: Knowing how to multiply, subtract, and rearrange terms
• Using formulas: Applying the percentage increase formula with appropriate substitutions

These fundamentals set the stage for more advanced topics in mathematics, ensuring that students develop strong analytical and logical reasoning skills.