Problem 5 - Algebra A Combined Function

Question

The number 45 is \(25 \%\) of what number?

Step-by-Step Solution

Answer

The number is 180.

Step 1: Understand the Problem

We need to find a number such that 25% of this number equals 45. We can set up an equation to represent this relationship.

Step 2: Set Up the Equation

Let \( x \) be the number we are trying to find. According to the problem, 25% of \( x \) is equal to 45. Therefore, we can write the equation as \( 0.25x = 45 \).

Step 3: Solve the Equation

To solve for \( x \), we need to isolate it on one side of the equation. We can do this by dividing both sides of the equation by 0.25: \[ x = \frac{45}{0.25} \]

Step 4: Calculate the Solution

Perform the division: \[ x = \frac{45}{0.25} = 180 \]. Hence, the number is 180.

Key Concepts

Understanding Algebraic Equations

Algebraic equations are mathematical statements that show the equality of two expressions using variables. They form the backbone of many mathematical concepts and problem-solving scenarios.
To grasp algebraic equations better, consider them as a balance scale. Each side of the equation must be equal. Here, our equation is expressed as: \( 0.25x = 45 \).
The equation uses a variable, \( x \), which represents an unknown number we want to find. In simpler terms, algebraic equations provide a structured way of representing relationships between numbers using operations and variables.
In everyday problems like percentage calculations, setting up an algebraic equation helps to translate the problem from words to a mathematical expression that can be solved step by step.

Mastering Isolating Variables

Isolating variables is a key step in solving algebraic equations. It involves rearranging the equation so that the variable is by itself on one side of the equation.
In the context of our exercise, we start with the equation \( 0.25x = 45 \). The aim is to solve for \( x \), the variable representing the unknown number.
To isolate \( x \), we perform operations that inverse the effect around it. Since \( x \) is multiplied by 0.25, we divide both sides of the equation by 0.25 to counter this effect:

• Divide both sides by 0.25: \( \frac{0.25x}{0.25} = \frac{45}{0.25} \)
• The result simplifies to \( x = 180 \)

Through these steps, we successfully isolated \( x \). This means we found the value that satisfies the initial equation. Isolating variables often requires using basic operations such as addition, subtraction, multiplication, or division.
The goal is always to "free" the variable, making it the subject of the equation.

Efficient Problem-Solving Steps

Solving mathematical problems, like the given percentage problem, can seem daunting at first. However, using a structured approach makes it manageable. Here are the essential problem-solving steps:

• Understand the problem: Grasp what you're being asked. In our case, find a number where 25% equals 45.
• Set up the equation: Translate the problem into mathematical language. We wrote it as \( 0.25x = 45 \).
• Solve for the variable: Perform operations to isolate the variable, calculating as required to find the solution. Here, \( x = 180 \).
• Review and validate: Verify the solution by substituting it back into the original setup. Is 25% of 180 actually 45? Definitely! Thus, the steps yield the correct answer.

This process not only ensures accuracy but enhances problem-solving skills critical to tackling more complex mathematical challenges effectively. By following these steps, you build a reliable framework for working through any mathematical problem.