Problem 27 - Introductory Algebra for College Students

Question

Use the percent formula, \(A=P B: A\) is \(P\) percent of \(B,\) to solve. What is \(3 \%\) of \(200 ?\)

Step-by-Step Solution

Answer

The result is 6.

Step 1: Converting Percentage to Decimal

Firstly, it is essential to convert the percentage into a decimal number. To do so, divide the percentage by 100. So, \(3\%\) will turn into \(0.03\) when divided by 100.

Step 2: Applying the Percent Formula

Substitute \(P\) with \(0.03\) and \(B\) with \(200\) in the percent formula \(A=P \cdot B\), which becomes \(A = 0.03 \cdot 200\).

Step 3: Solving the Equation

Finally, multiply \(0.03\) by \(200\) to get the final answer.

Key Concepts

Converting Percentages to Decimals

Understanding how to convert percentages to decimals is an essential skill in math and is particularly useful in various real-world applications. A percentage represents a part per hundred, which directly translates to the concept of decimals. For example, to convert a percentage into a decimal, simply divide the percentage by 100. This shifts the decimal two places to the left.

Take the percentage 3%. You divide 3 by 100, which is essentially moving the decimal two places to the left. So, 3% becomes 0.03. It's like reducing something down to its essence—a smaller, cleaner number that is easier to work with, especially in calculations.

• 25% becomes 0.25
• 60% becomes 0.60 or 0.6 (since trailing zeros after the decimal do not change the value)
• 100% becomes 1 (because 100 divided by 100 equals 1)

Keeping these conversions in mind helps streamline the process of working with percentages in algebraic problems.

Solving Percent Problems

Solving percent problems can often intimidate students, but it doesn't have to be complicated. Once you've converted your percentage to a decimal, you're halfway there. The basic percent formula, which is useful for finding out what one value is as a percentage of another, is: \( A = P \cdot B \) where \( A \) is the result, \( P \) is the percentage converted to decimal, and \( B \) is the other value you're comparing it to.

Let's tackle the example of finding 3% of 200. With the percentage converted to a decimal (0.03 in this case), you simply multiply this decimal by 200 to find your answer. The multiplication step is straightforward—just apply basic arithmetic or a calculator:

\[ A = 0.03 \times 200 = 6 \]
This result shows that 3% of 200 is 6. Following this process, you can solve a wide variety of percent problems with ease. Whether you're determining discounts, calculating interest, or just figuring out grades, the percent formula is an invaluable tool.

Basic Algebra

At its heart, basic algebra is about using letters to represent numbers and understanding how to manipulate these symbols to solve equations. It forms the foundation for advanced mathematical problem-solving. In our percent problem, we use a basic algebraic equation to represent the relationship between the total amount, the percentage of the amount we're interested in, and the part of the amount that percentage represents.

In the exercise, \( P \) and \( B \) are known, and we solve for \( A \) using multiplication. The formula \( A = P \cdot B \) can be considered a linear equation, which is the simplest form of an algebraic equation.

• If \( P \) is unknown but \( A \) and \( B \) are known, divide \( A \) by \( B \) to find \( P \) (the percentage).
• If \( B \) is unknown but \( A \) and \( P \) are known, divide \( A \) by \( P \) to find \( B \) (the total).

Through basic algebra, not only are you solving for unknowns but also developing logical thinking and problem-solving skills that apply well beyond math class.