Solving X = 160 + 20% Of X A Step-by-Step Guide

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Introduction

In the realm of mathematical problem-solving, encountering equations that require careful manipulation and understanding of percentages is a common occurrence. One such equation is x = 160 + 20% of x, which might appear perplexing at first glance. However, by systematically breaking down the equation and applying fundamental algebraic principles, we can unravel its solution and gain a deeper understanding of the underlying concepts. This comprehensive guide will walk you through the step-by-step process of solving this equation, providing clear explanations and insights along the way. We will also delve into the significance of percentages in mathematical equations and explore real-world applications where such equations might arise.

Decoding the Equation: x = 160 + 20% of x

At its core, the equation x = 160 + 20% of x represents a relationship between a variable, x, and a constant, 160, with an additional component that involves a percentage of the variable itself. To effectively tackle this equation, we need to dissect its components and understand their individual roles.

The left-hand side of the equation, x, represents the unknown value we are aiming to determine. It is the variable that we need to isolate to find its numerical value.

The right-hand side of the equation consists of two terms: 160 and 20% of x. The first term, 160, is a constant value that remains unchanged throughout the equation. The second term, 20% of x, introduces the concept of percentage. A percentage is simply a way of expressing a number as a fraction of 100. In this case, 20% represents 20 out of every 100, which can be written as a fraction (20/100) or a decimal (0.20). The phrase "of x" indicates that we need to multiply 20% (or 0.20) by the variable x.

Therefore, the term 20% of x can be mathematically expressed as 0.20x. This term represents a portion of the unknown value x, specifically 20% of it.

Now, let's rewrite the equation using this understanding:

x = 160 + 0.20x

This revised equation provides a clearer representation of the relationship between x, the constant 160, and 20% of x. The next step involves isolating the variable x to determine its value.

Step-by-Step Solution: Isolating the Variable x

To solve for x in the equation x = 160 + 0.20x, we need to isolate x on one side of the equation. This involves rearranging the terms so that all terms containing x are on one side and the constant terms are on the other side.

The first step is to subtract 0.20x from both sides of the equation. This will eliminate the term containing x from the right-hand side:

x - 0.20x = 160 + 0.20x - 0.20x

Simplifying both sides, we get:

0.80x = 160

Now, we have an equation where the variable x is multiplied by a constant (0.80). To isolate x, we need to divide both sides of the equation by 0.80:

0.80x / 0.80 = 160 / 0.80

Performing the division, we obtain:

x = 200

Therefore, the solution to the equation x = 160 + 20% of x is x = 200. This means that the value of x that satisfies the equation is 200.

Verifying the Solution: Plugging x = 200 Back into the Equation

To ensure the accuracy of our solution, we can substitute the value of x we found (x = 200) back into the original equation and verify that it holds true. This process is called verification and is an essential step in problem-solving.

Let's substitute x = 200 into the original equation:

200 = 160 + 20% of 200

Now, we need to calculate 20% of 200. As we discussed earlier, 20% can be expressed as 0.20. So, 20% of 200 is:

0. 20 * 200 = 40

Substituting this value back into the equation, we get:

200 = 160 + 40

Simplifying the right-hand side, we have:

200 = 200

Since both sides of the equation are equal, our solution x = 200 is verified. This confirms that the value we found for x satisfies the original equation.

The Significance of Percentages in Mathematical Equations

Percentages play a crucial role in various mathematical equations and real-world applications. They provide a standardized way of expressing proportions and ratios, making it easier to compare and analyze different quantities. In the equation x = 160 + 20% of x, the percentage component (20% of x) introduces a proportional relationship between the variable x and a portion of itself.

Understanding percentages is essential for solving a wide range of problems, including those related to:

  • Finance: Calculating interest rates, discounts, and investment returns.
  • Statistics: Analyzing data and interpreting statistical results.
  • Everyday life: Calculating tips, sales taxes, and price reductions.

Mastering the concepts of percentages and their applications in mathematical equations is a valuable skill that can enhance problem-solving abilities and provide a deeper understanding of the world around us.

Real-World Applications: Where Equations Like This Might Arise

Equations similar to x = 160 + 20% of x can arise in various real-world scenarios. Let's explore a few examples:

  • Sales Commissions: Imagine a salesperson who earns a base salary of $160 per week plus a 20% commission on their sales. If they want to earn a total of x dollars in a week, the equation x = 160 + 20% of x can be used to determine the amount of sales they need to generate.
  • Price Increases: Suppose an item's price increases by 20% after a certain period, and the new price is $160 more than the original price. The equation x = 160 + 20% of x can be used to find the original price of the item.
  • Mixture Problems: Consider a solution that is made up of a certain amount of a substance plus 20% of that same amount. If the total amount of the solution is represented by x and the constant amount is 160, the equation x = 160 + 20% of x can be used to determine the amount of the substance in the solution.

These are just a few examples of how equations involving percentages can be applied in real-world situations. By understanding the underlying principles of these equations, we can effectively solve problems in various fields.

Conclusion: Mastering Equations with Percentages

In this comprehensive guide, we have explored the equation x = 160 + 20% of x, providing a step-by-step solution and delving into the significance of percentages in mathematical equations. We have also examined real-world applications where such equations might arise.

By understanding the concepts presented in this guide, you can confidently tackle similar equations involving percentages and apply your knowledge to solve real-world problems. Mastering these skills will not only enhance your mathematical abilities but also provide you with a valuable tool for analyzing and understanding the world around you.

Remember, the key to success in mathematics lies in practice and perseverance. So, continue exploring different types of equations and challenging yourself to solve them. With dedication and a solid understanding of the fundamentals, you can unlock the power of mathematics and achieve your goals.