Problem 44 - Prealgebra

Question

Write a basic percent problem, the solution to which can be found by solving the equation \(75=0.46 \cdot n\)

Step-by-Step Solution

Answer

The value of \(n\) is approximately 163.04.

Step 1: Understand the Problem

The equation given is \(75 = 0.46 \cdot n\). We need to find the value of \(n\) such that it satisfies this equation. The equation can be read as "75 is 46% of \(n\)."

Step 2: Isolate the Variable

To solve for \(n\), isolate it by dividing both sides of the equation by 0.46. This gives us \(n = \frac{75}{0.46}\).

Step 3: Calculate the Solution

Compute the division \(\frac{75}{0.46}\) using a calculator or long division. When dividing, this gives us \(n \approx 163.04\).

Step 4: Verify Your Solution

Check if the solution is correct by multiplying \(n\) with 0.46. If \(75 \approx 0.46 \times 163.04\), then our solution is verified.

Key Concepts

Understanding Basic Algebra

The foundation of solving percent problems begins with basic algebra. Algebra involves using symbols, usually letters, to represent unknown values. These symbols allow us to create equations, which are mathematical statements showing the relationship between different numbers and variables.

In the exercise given, the variable is represented by the letter \(n\). This represents the unknown quantity we want to find. The formula shown, \(75 = 0.46 \cdot n\), depicts a relationship where 75 is 46% of the value of \(n\).

Basic algebra helps us rearrange and manipulate these symbols and numbers to isolate our variable, which is the key step in solving equations.

Mastering Equation Solving

Equation solving is a crucial part of basic algebra, especially in percent problems. To solve an equation means to find the value of the variable that makes the equation true.

In our problem, the equation \(75 = 0.46 \cdot n\) shows that 75 is equal to 46% of \(n\). To find \(n\), we need to 'undo' the multiplication by dividing both sides of the equation by 0.46. This results in \(n = \frac{75}{0.46}\).

By performing this division, we isolate \(n\) and find its value. Dividing each side equally is a fundamental principle in equation solving that helps us maintain balance and find the correct answer.

The Role of Mathematical Reasoning

Mathematical reasoning is the logical process of concluding based on evidence and relationships in math problems. It helps in verifying whether the solution we've found is correct and consistent.

After calculating \(n\), or \(n \approx 163.04\), mathematical reasoning prompts us to check our work. We multiply the found \(n\) by 0.46 to ensure that it indeed approximates 75. This step is crucial because it allows us to verify that our calculations have been executed correctly.

This reasoning ensures not only accuracy but also that we've understood both the problem and the solution. It develops strong problem-solving skills, encouraging a deeper understanding of percent problems and mathematics as a whole.