Problem 38 - Introductory and Intermediate Algebra for College Students 4th

Question

Use the percent formula, \(A=P B: A\) is \(P\) percent of \(B,\) to solve Exercises \(27-42\) What percent of 7.5 is \(0.6 ?\)

Step-by-Step Solution

Answer

P = 8%

Step 1: Understand the problem

Identify the values for 'A' and 'B'. In this problem, 'A' is 0.6, and 'B' is 7.5.

Step 2: Substitute A and B in the formula

Substitute 0.6 for 'A' and 7.5 for 'B' in the percent formula, \(P = A / B\). This results in \(P = 0.6 / 7.5\).

Step 3: Calculate the value of P

Solve \(P = 0.6 / 7.5\) to get 'P', which is the percentage value. The solution gives a decimal value.

Step 4: Convert to percentage

The decimal value calculated in Step 3 can be converted to a percentage by multiplying it by 100. This will give us the final answer in terms of percentage.

Key Concepts

Algebra

Algebra is a fascinating area of mathematics dealing with symbols and the rules for manipulating these symbols. In essence, algebra allows us to solve equations and understand relationships between different variables. In the context of our problem, we are using the language of algebra to explore the relationship between a part and a whole, as represented by percentages. Understanding the percent formula, which is expressed as \(A = P \cdot B\), requires recognizing each component’s role in this equation. We often need to rearrange the formula to solve for the missing variable. In our scenario, finding what percent \(P\) of \(B\) is equivalent to \(A\) means we need to solve the equation \(P = \frac{A}{B}\), an expression that exemplifies algebra's power in problem-solving. When we substitute known values into this equation, algebra becomes our tool for calculating unknown quantities. The use of variables \(A\) and \(B\) allows us to solve this problem and many similar ones methodically.

Problem-Solving

In this context, problem-solving involves understanding how to approach a question methodically using established mathematical principles. Firstly, it is important to clearly identify the elements of the problem, which are often already hinted at in the question. For instance, in our problem, we identify \(A\) as 0.6 and \(B\) as 7.5 from the text.Understanding how these parts interplay is crucial. We then use these numbers and apply them into the relevant formula, aiming to isolate the unknown percentage \(P\). **Substitution** is a key technique here, where we replace abstract variables with concrete values from our problem statement: \(P = \frac{0.6}{7.5}\).Finally, solving the equation requires performing the division to determine \(P\), which often results in a decimal. This reinforces the importance of accurate calculations. Adjusting your approach with each step based on new information obtained from previous steps of solving an equation is a critical part of the problem-solving process.

Percentage Calculation

Percentage calculation provides a means to express proportions in an intuitive manner. It involves converting a fraction or decimal to a percentage, making it easier to comprehend comparisons or changes between values on a standard scale. In our solution, after solving the equation \(P = \frac{0.6}{7.5}\), we reach a decimal result. However, percentages are most conveniently understood when presented as whole numbers. To convert the found decimal into a percentage, we multiply by 100—transforming the decimal into a familiar percentage form. For instance, if the result of our division is 0.08, multiplying by 100 converts it to 8, hence 0.6 is 8 percent of 7.5. This final conversion step is what lends clarity to the data, allowing us a more digestible overview of the relationship between quantities.