Solving for a Number: Understanding and Applying Algebraic Expressions

Mathematics is a universal language that helps us understand and solve the world around us. One common problem in algebra involves determining a specific number based on given mathematical expressions. In this article, we delve into the process of solving the expression 'one half of one third of a number equals the difference between that number and 15.' We will break down the problem step-by-step, ensuring clarity and accessibility.

Problem Statement and Initial Setup

Given: 1/2 × 1/3 × x x - 15

Here, x represents the unknown number whose value we need to determine. By setting up the equation, we can solve for x using basic algebraic principles. This problem tests our understanding of fractions, division, and how to manipulate algebraic expressions.

Step-by-Step Solution

Step 1: Simplify the Left Side of the Equation

Let us start by simplifying the left side of the equation. The expression 1/2 × 1/3 can be simplified as follows:

[frac{1}{2} times frac{1}{3} frac{1}{6}]

Therefore, the equation becomes:

[frac{x}{6} x - 15]

Step 2: Eliminate the Fraction

To eliminate the fraction, we can multiply both sides of the equation by 6:

[x 6(x - 15)]

Expanding the right side of the equation we get:

[x 6x - 90]

Step 3: Isolate x

To isolate x, we need to move all terms involving x to one side of the equation:

[x - 6x -90]

Simplifying further:

[-5x -90]

Dividing both sides of the equation by -5:

[x frac{-90}{-5} 18]

Verification

Let us verify our solution by substituting x 18 back into the original equation:

1/2 × 1/3 × 18 18 - 15

Simplifying the left side:

[frac{1}{2} times frac{1}{3} times 18 6]

And the right side:

[18 - 15 3]

Thus, the equation holds true, confirming that our solution x 18 is correct.

Additional Context and Applications

Understanding how to solve algebraic equations like these is crucial for a wide range of applications, from basic arithmetic to advanced mathematical models. In real-world scenarios, such calculations can be found in financial planning, engineering, and data analysis. By practicing similar problems, one can improve their algebraic proficiency and problem-solving skills.

For instance, in financial planning, one may need to calculate future values or interest, where such algebraic expressions play a key role. In engineering, these principles help in designing and modeling systems that require precise calculations for components.

This article aims to provide a clear, step-by-step guide to solving algebraic expressions, making it accessible to students, professionals, and anyone interested in improving their mathematical skills.

Key Points to Remember:

Maintain the balance of the equation when performing operations on both sides. Understand and apply basic algebraic rules such as distribution and simplification. Verify your solution to ensure accuracy.