Problem 34 - 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 \(27-42\) \(32 \%\) of what number is \(51.2 ?\)

Step-by-Step Solution

Answer

The base number is 160

Step 1: Identify A, P, and B

We are given that 32% of a certain number is 51.2. So, A is 51.2, and P is 32. We need to find the base number B.

Step 2: Convert percentage to decimal

We need to convert the percentage to decimal for the calculation. To convert percentage to decimal, divide the percentage by 100. So, P becomes \(P = 32 / 100 = 0.32\)

Step 3: Apply the Formula

Substitute A and P in the formula B = A / P. So, \(B = 51.2 / 0.32\)

Step 4: Compute the result

Now, calculate B to get the base number. B = 51.2 / 0.32 gives B = 160

Key Concepts

Algebra

Algebra is a branch of mathematics that uses symbols and letters to represent numbers and quantities in equations and formulas. When we encounter a mathematical problem, such as determining a base number from a percentage, algebra is incredibly useful for setting up equations.

To solve these types of problems, we use known quantities (like the percentage and a part of the whole) to find an unknown quantity (in this case, the base number). In the exercise, we had to find the base number which a given percentage of it equaled another number. This is essentially setting up an equation with one unknown variable, conventionally represented as "B."

• Known elements: percentage = 32%, result number = 51.2
• Unknown: base number (B)

The algebraic equation based on the percent formula is: \( A = P \times B \), where we rearrange it to solve for the unknown \( B \), resulting in \( B = \frac{A}{P} \).

Algebra strips away the complexity of words and allows us to focus on the numbers and operations needed to derive a solution.

Percentage Conversion

Percentage conversion is a critical step when working with problems involving percentages. Percentages are a way to express a number as a fraction of 100, which can sometimes make calculations cumbersome. Converting percentages to decimals simplifies the process since decimals are well-suited for multiplication and division.

In this problem, converting 32% into a decimal is crucial as we needed to perform an arithmetic operation (division) involving this percentage. To convert a percentage into a decimal, you divide by 100, essentially moving the decimal point two places to the left. Here's the conversion process that was applied:

• Take the percentage: 32%
• Divide by 100: \( 32 \div 100 = 0.32 \)

Once in decimal form, it becomes a straightforward number that is much easier to work with in formulas or equations. The reason we convert to decimals is because mathematical operations can be directly applied to these forms without additional modifications.

Mathematical Problem Solving

Mathematical problem solving is the process of finding solutions to problems or equations using a systematic approach. It involves understanding the problem, devising a plan, carrying out that plan, and then reviewing the solution for accuracy.

For this exercise:

• **Understanding the problem:** Recognize what you are given and what you need to find. We knew 32% of a number equals 51.2, and we needed the base number.
• **Devising a plan:** Use the percent formula \( A = P \times B \) and rearrange it to solve for B, which gives us \( B = \frac{A}{P} \).
• **Carrying out the plan:** Convert the percentage to a decimal (32% becomes 0.32) and perform the division: \( B = \frac{51.2}{0.32} \).
• **Reviewing the solution:** Check the result to ensure everything adds up correctly to yield the correct base number, which is 160.

Mathematical problem solving is essential not only because it helps solve mathematical equations but also because it enhances logical reasoning and critical thinking skills, which are valuable in all areas of life.