Problem 16 - Prealgebra

Question

Solve each problem using the percent equation. 49.5 is what percent of \(132 ?\)

Step-by-Step Solution

Answer

49.5 is 37.5% of 132.

Step 1: Understand the Percent Equation

We use the percent equation: \( \text{part} = \text{percent} \times \text{whole} \). In this problem, the part is 49.5 and the whole is 132. We need to find the percent.

Step 2: Set Up the Equation

Set up the equation for this problem using the percent equation: \( 49.5 = x \times 132 \), where \( x \) is the percent expressed as a decimal.

Step 3: Solve for the Percent (as a Decimal)

Rearrange the equation to solve for \( x \) by dividing both sides by 132: \( x = \frac{49.5}{132} \).

Step 4: Calculate the Decimal

Perform the division: \( x = \frac{49.5}{132} \approx 0.375 \).

Step 5: Convert the Decimal to a Percent

Convert the decimal to a percent by multiplying by 100: \( 0.375 \times 100 = 37.5 \).

Key Concepts

Understanding Percent Calculations

Percent calculations are a vital part of mathematics, especially when dealing with comparisons and proportions. A percent represents a number out of 100. In the context of the above problem, we are trying to determine what 49.5 is in terms of a percentage of the whole, which is 132. To solve this, we use the percent equation:

• \( \text{part} = \text{percent} \times \text{whole} \)

In the given problem, "part" equals 49.5, and "whole" is 132. Our task is to determine the "percent." This involves setting up and solving an equation, which requires some basic arithmetic operations like division and multiplication.Practicing these calculations improves one's ability to think proportionally and understand relationships between numbers.

Approaching Problems with Mathematical Problem-Solving

Effective problem-solving in mathematics grows from understanding how to apply key concepts to different types of problems. Here, using the percent equation is our main tool for finding the percentage value. To solve the problem, we follow a step-by-step approach:

• Set up the equation using the given values. Here, that's setting the equation as \( 49.5 = x \times 132 \).
• Rearrange to solve for the unknown, \( x \), by isolating it. In this context, divide both sides by 132: \( x = \frac{49.5}{132} \).
• Finally, convert the decimal value to a percentage by multiplying by 100.

By breaking down the problem into smaller, manageable steps, you can solve even complicated mathematical scenarios more effectively.

Foundational Prealgebra Concepts at Work

Prealgebra provides the groundwork for understanding basic algebraic equations and mathematical concepts, such as percentages. In this exercise, key prealgebra skills are tested: setting up equations and simple arithmetic operations.

• Setting up a clear equation from a word problem demonstrates how algebra translates real-world situations into mathematical representations.
• Rearranging the equation involves understanding inverse operations—such as division being the inverse of multiplication.
• Handling decimals and converting them to percentages shows the practical application of fractions and proportions.

Mastering these concepts will make transitioning to more complex algebra and higher mathematical reasoning much more seamless. Focus on building a solid understanding here, and you'll be well prepared for more advanced studies.