Solving For R In 28% Of R = 49 A Step By Step Guide
Introduction
In this article, we will delve into the process of solving a common mathematical problem: finding the value of a variable when given a percentage of it. Specifically, we'll address the equation "28% of r is 49". This type of problem frequently appears in various real-life scenarios, from calculating discounts to determining proportions in data analysis. To effectively solve this, we need to understand how to convert percentages into decimals and apply basic algebraic principles. This article will guide you through the steps, ensuring you grasp the underlying concepts and can confidently tackle similar problems in the future. Whether you're a student learning algebra or someone looking to brush up on your math skills, this guide will provide a clear and comprehensive explanation.
Converting the Percentage to a Decimal
The initial step in solving the equation "28% of r is 49" involves transforming the percentage into its decimal equivalent. Percentages are essentially fractions with a denominator of 100, meaning that 28% is the same as 28 out of 100. To convert this percentage into a decimal, we divide 28 by 100. This division results in 0.28, which is the decimal form of 28%. Understanding this conversion is crucial because it allows us to work with percentages in a more mathematically manageable form. By expressing 28% as 0.28, we can rewrite the original problem as a straightforward algebraic equation. This step is not just a mathematical procedure; it's a foundational concept in understanding how percentages relate to decimals and whole numbers, a relationship that is vital in various mathematical applications and everyday calculations.
Setting Up the Algebraic Equation
Having converted the percentage into its decimal equivalent, the next crucial step is to translate the problem statement "28% of r is 49" into an algebraic equation. This transformation is key to solving the problem using mathematical methods. The phrase "28% of r" can be mathematically expressed as 0.28 multiplied by r, which is written as 0.28r. The word "is" in the context of the problem signifies equality, so we equate 0.28r to 49. This process yields the equation 0.28r = 49. This equation is a simple yet powerful representation of the original problem, making it easier to manipulate and solve. By setting up the equation correctly, we lay the groundwork for finding the value of 'r'. This step highlights the importance of understanding how to translate word problems into mathematical expressions, a skill that is essential in algebra and beyond.
Solving for 'r'
With the algebraic equation 0.28r = 49 established, the next step is to isolate 'r' and solve for its value. This is a fundamental process in algebra that involves using inverse operations to undo the operations performed on the variable. In this case, 'r' is being multiplied by 0.28. To isolate 'r', we need to perform the inverse operation, which is division. We divide both sides of the equation by 0.28. This maintains the balance of the equation, ensuring that the equality remains true. Performing this division, we get r = 49 / 0.28. This step is a practical application of algebraic principles and demonstrates how equations can be manipulated to reveal the value of an unknown variable. The ability to solve for 'r' in this manner is a crucial skill in mathematics, applicable in various contexts, from basic algebra to more complex mathematical problems.
Calculating the Value of 'r'
After setting up the division r = 49 / 0.28, the next step is to perform the calculation to find the numerical value of 'r'. This involves dividing 49 by 0.28. When you perform this division, the result is 175. Therefore, the value of 'r' that satisfies the equation 0.28r = 49 is 175. This calculation is a straightforward arithmetic operation, but it's a critical step in solving the problem. It demonstrates how mathematical operations are used to arrive at a final answer. This process of calculating 'r' not only provides the solution to the problem but also reinforces the understanding of the relationship between the percentage, the variable, and the final value. By finding that r = 175, we complete the mathematical process and answer the question posed in the original problem.
Verifying the Solution
Once we've calculated the value of 'r' as 175, it's important to verify this solution to ensure its accuracy. This step is a crucial part of the problem-solving process in mathematics. To verify the solution, we substitute the value of 'r' back into the original equation, which was 0.28r = 49. By replacing 'r' with 175, the equation becomes 0.28 * 175. Performing this multiplication, we find that 0.28 multiplied by 175 equals 49. This result confirms that our calculated value of 'r', which is 175, is correct because it satisfies the original equation. Verification is not just a procedural step; it's a way to build confidence in your answer and ensure that no errors were made during the problem-solving process. This practice reinforces the importance of accuracy and attention to detail in mathematics.
Real-World Applications
Understanding how to solve equations like "28% of r is 49" has numerous real-world applications. These types of calculations are not just confined to the classroom; they are frequently used in everyday situations and professional settings. For example, in finance, you might use this type of calculation to determine the original price of an item after a discount, or to calculate interest rates on loans. In retail, understanding percentages is crucial for pricing strategies, sales promotions, and calculating profit margins. In data analysis, percentages are used to interpret data, identify trends, and make informed decisions. Moreover, in personal finance, these calculations are essential for budgeting, saving, and investment planning. The ability to work with percentages and solve for unknowns is a valuable skill that empowers individuals to make informed decisions in various aspects of life. By mastering these concepts, you gain a practical tool that can be applied in a wide range of contexts, making it a valuable asset in both personal and professional endeavors.
Conclusion
In conclusion, solving the problem "28% of r is 49" involves a series of steps that are fundamental to mathematical problem-solving. We began by converting the percentage to a decimal, which allowed us to translate the problem into an algebraic equation. We then set up the equation 0.28r = 49, which provided a clear representation of the problem. By dividing both sides of the equation by 0.28, we isolated 'r' and found its value to be 175. We verified this solution by substituting it back into the original equation, confirming its accuracy. This process not only solves the specific problem but also reinforces key mathematical concepts such as percentages, decimals, algebraic equations, and the importance of verification. The ability to solve these types of problems is crucial in various real-world applications, from finance and retail to data analysis and personal finance. By understanding and mastering these concepts, individuals can confidently tackle mathematical challenges and make informed decisions in their daily lives.
