Problem 17 - Prealgebra

Question

Solve each problem using the percent equation. 37.5 is what percent of \(30 ?\)

Step-by-Step Solution

Answer

37.5 is 125% of 30.

Step 1: Understand the Percent Equation

The percent equation is usually expressed as: \( \text{part} = \text{percent} \times \text{whole} \). In this problem, our "part" is \(37.5\), and our "whole" is \(30\). We need to find the "percent".

Step 2: Set Up the Equation

Based on the equation \( \text{part} = \text{percent} \times \text{whole} \), plug in the values: \(37.5 = \text{percent} \times 30\). Let \(x\) represent the percent in decimal form. So the equation becomes \(37.5 = x \times 30\).

Step 3: Solve for the Percent in Decimal Form

To find \(x\), divide both sides of the equation by \(30\):\[ x = \frac{37.5}{30} \]Calculate the division to find \(x\).

Step 4: Convert Decimal to Percent

After calculating \(x = \frac{37.5}{30}\), you get \(x = 1.25\). To convert \(1.25\) to a percent, multiply by \(100\) and add the percent sign. Thus, \(1.25 \times 100 = 125\%\).

Key Concepts

Solving Equations

When solving equations, we want to find the value of a variable that makes the equation true. In our example, the problem is to find out what percent 37.5 is of 30. To do this, we use the percent equation formula:

• Part = Percent × Whole

In this equation, 37.5 is the "part" and 30 is the "whole." We want to find the "percent," represented by the variable \( x \). To solve for \( x \), set up the equation as \( 37.5 = x \times 30 \). This is a simple linear equation. The next step is to isolate \( x \) by performing mathematical operations. By dividing both sides by 30, we simplify the equation to \( x = \frac{37.5}{30} \). Solving equations often involves reversing operations. Here, division helps us "undo" the multiplication by 30.

Percent Calculation

Percentages are a way to express numbers as parts of 100. They are incredibly handy when you need to compare parts of a whole or see changes in quantities. In the context of our problem, solving for \( x \) using \( x = \frac{37.5}{30} \) gives a decimal result, \( x = 1.25 \). To convert this to a percent, multiply the decimal by 100. This step transforms the decimal to a percentage: \( 1.25 \times 100 = 125\% \). Essentially, we're saying that 37.5 is 125% of 30.

• Use the percent equation to relate the part, percent, and whole.
• Use division to move from an equation to a decimal.
• Multiply by 100 to convert from a decimal to a percent.

Always remember that anytime you convert a decimal to a percentage, you're expressing how many parts of 100 you're working with.

Mathematical Operations

To navigate through percent problems effectively, understanding mathematical operations is crucial. These operations are the tools we use to manipulate equations and solve problems. Let's delve into some key operations:

• Multiplication: Used in the percent equation to relate the parts and the whole.
• Division: This operation is used to isolate and solve for the variable \( x \) by "undoing" multiplication.
• Conversion: Convert the result into a percent by multiplying by 100.

Operations like addition and subtraction are not involved here, but they are essential in other contexts. The main takeaway is that multiplication and division often go hand in hand when solving percent problems. Being comfortable with these operations allows you to solve percent equations neatly and efficiently. The process of converting a decimal to a percentage is a straightforward exercise in multiplication and is a vital skill for dealing with any percentage-related problem.