Problem 34 - Introductory Algebra for College Students

Question

Use the percent formula, \(A=P B: A\) is \(P\) percent of \(B,\) to solve. \(32 \%\) of what number is \(51.2 ?\)

Step-by-Step Solution

Answer

32% of the number 160 equals 51.2

Step 1: Write Down What You Know

We have that \(A = 51.2\) and \(P = 32\%\). Now, remember that percentages need to be converted into decimal form. So, turns out that \(P = 0.32\).

Step 2: Rearrange Formula to Solve for B

The formula we have is \(A=P \cdot B\). We want to solve for \(B\), so we need to rearrange the equation. To do this, we'll divide each side by \(P\), which gives us the rearranged formula: \(B = \frac{A}{P}\).

Step 3: Substitute and Solve

Now we plug in the known values for \(A\) and \(P\): \(B = \frac{51.2}{0.32}\). When you do this division, you get \(B=160\). So, 32% of 160 is 51.2.

Key Concepts

Algebraic Problem Solving

Algebraic problem solving is a methodical approach to untangle mathematical puzzles that involve unknown variables. In the context of the given exercise, algebraic problem solving starts with the identification of what we know and what we wish to find out. In this case, we are given a percent value and a part of a whole, and we aim to find that “whole” number.

Key to solving such algebraic problems is understanding and manipulating formulas. The percent formula, which states that a percent (\( P \times B \)) of a base (\( B \)) is equal to the amount (\( A \)), is crucial here. The process involves reordering this formula and plugging in the known values to solve for the unknown, a technique broadly applicable in many algebraic scenarios. By doing so methodically, we turn the abstract into the concrete, moving step by step – from rewriting the formula to substituting values – until we reach a solution.

Converting Percentages to Decimals

Understanding how to convert percentages to decimals is essential for properly handling many mathematical and real-world problems. This conversion is central to solving the provided exercise. A percentage represents a part per hundred and hence to convert it to a decimal form, we divide the percentage by 100.

For example, a conversion of 32% involves moving the decimal point two places to the left (since percent means per hundred), resulting in the decimal 0.32. Regardless of the complexity of the algebra at hand, mastering this conversion step is a fundamental aspect of dealing with percentages and is widely applicable, from calculating taxes to interpreting data in various fields.

Solving for a Variable

Solving for a variable is like finding the missing piece of a puzzle. In algebra, it means figuring out the value of an unknown that makes an equation true. In our exercise, the variable 'B' represents the base number that we are trying to find.

The process of finding 'B' has a few steps. First, we arrange the percent formula to isolate 'B' on one side: \( B = \frac{A}{P} \). Next, we substitute known values into this new equation. Here, \( A = 51.2 \) and \( P = 0.32 \), giving us \( B = 160 \). Such a method is vital for solving numerous algebraic equations, and it helps us to define one quantity in terms of others, offering a clear path to the solution.