Solving Percentage Problems What Number Is 120% Of 90
In the realm of mathematics, percentage problems often present themselves as intriguing puzzles. One such puzzle is the question: 120% of what number is 90? This question challenges us to think beyond the direct calculation of percentages and delve into the inverse process – determining the original number when a percentage of it is known. In this comprehensive exploration, we will embark on a step-by-step journey to unravel this mathematical enigma, equipping you with the tools and understanding to confidently tackle similar percentage problems.
Unveiling the Essence of Percentages
To effectively solve the problem at hand, we must first grasp the fundamental concept of percentages. A percentage, derived from the Latin "per centum," signifies "out of one hundred." It represents a proportion or ratio expressed as a fraction of 100. For instance, 50% signifies 50 out of every 100, or equivalently, one-half. Similarly, 120% represents 120 out of every 100, indicating a value greater than the original whole.
Understanding this core concept is crucial, as it forms the bedrock for solving percentage-related problems. When we say "120% of a number," we are essentially referring to 1.20 times that number. This conversion from percentage to a decimal or fraction is a key step in deciphering the puzzle.
Translating the Words into an Equation
The next step in our journey is to translate the word problem into a mathematical equation. This translation allows us to represent the problem in a concise and symbolic form, making it easier to manipulate and solve. Let's break down the given statement: "120% of what number is 90?"
We can represent the unknown number with a variable, say "x." The phrase "120% of" implies multiplication, and "is" signifies equality. Thus, we can rewrite the statement as an equation:
120% * x = 90
Now, to work with the percentage in an equation, we convert it to a decimal by dividing by 100:
- 20 * x = 90
This equation succinctly captures the essence of the problem, paving the way for us to isolate the unknown variable and determine its value.
Solving for the Unknown: A Step-by-Step Approach
With our equation in place, we now embark on the final stage – solving for the unknown variable, "x." The equation 1.20 * x = 90 presents a straightforward algebraic task. To isolate "x," we need to undo the multiplication by 1.20. This is achieved by dividing both sides of the equation by 1.20:
x = 90 / 1.20
Performing this division, we arrive at the solution:
x = 75
Therefore, the number for which 120% equals 90 is 75. This completes our journey, successfully unraveling the mathematical puzzle.
Verification: Ensuring Accuracy
As a crucial step in problem-solving, we must always verify our solution. This ensures that our answer aligns with the original problem statement and that no errors have crept into our calculations. To verify our answer, we substitute x = 75 back into the original equation:
120% * 75 = 90
Converting 120% to a decimal, we have:
- 20 * 75 = 90
Performing the multiplication, we get:
90 = 90
This confirms that our solution, x = 75, is indeed correct. The left-hand side of the equation equals the right-hand side, validating our answer.
Beyond the Specific: Generalizing the Approach
While we have successfully solved the specific problem of finding the number for which 120% equals 90, the real power lies in understanding the underlying approach. This approach can be generalized to tackle a wide range of percentage problems. Let's encapsulate the key steps:
- Understand the concept of percentages: Grasp that a percentage represents a proportion out of 100.
- Translate the word problem into an equation: Represent the unknown with a variable, and use mathematical symbols to express the relationships described in the problem.
- Solve for the unknown: Employ algebraic techniques to isolate the variable and determine its value.
- Verify the solution: Substitute the answer back into the original equation to ensure accuracy.
By mastering these steps, you can confidently navigate the realm of percentage problems, transforming them from daunting challenges into solvable puzzles.
Real-World Applications: Percentages in Action
Percentages are not confined to the realm of textbooks and mathematics classrooms. They permeate our daily lives, playing a crucial role in various real-world scenarios. From calculating discounts and sales tax to understanding financial investments and interpreting statistical data, percentages are an indispensable tool.
Consider a scenario where a store offers a 20% discount on a product. To determine the discounted price, you would calculate 20% of the original price and subtract it from the original price. Similarly, when calculating sales tax, you would determine the percentage of tax applicable and add it to the purchase price.
In the realm of finance, percentages are used to represent interest rates, investment returns, and inflation rates. Understanding these percentages is crucial for making informed financial decisions. Moreover, percentages are widely used in statistics to represent proportions and ratios, enabling us to analyze and interpret data effectively.
By recognizing the ubiquitous nature of percentages, we appreciate the importance of mastering the concepts and techniques for solving percentage-related problems.
Elevating Your Percentage Prowess: Practice and Exploration
As with any mathematical skill, proficiency in percentage problem-solving is honed through practice and exploration. The more you engage with different types of percentage problems, the more adept you become at identifying the underlying relationships and applying the appropriate techniques.
Seek out a variety of practice problems, ranging from simple calculations to more complex scenarios. Challenge yourself to translate word problems into equations, and meticulously work through the steps to arrive at the solution. Don't shy away from seeking assistance when needed, whether from textbooks, online resources, or instructors.
Furthermore, explore the diverse applications of percentages in real-world contexts. This will not only solidify your understanding but also enhance your appreciation for the practical significance of this mathematical concept. By actively practicing and exploring, you can elevate your percentage prowess and unlock a world of mathematical possibilities.
Conclusion: Embracing the Power of Percentages
In conclusion, the question "120% of what number is 90?" serves as a gateway to understanding the intricacies of percentage problems. By unraveling this puzzle, we have not only determined the specific answer but also gained a deeper appreciation for the underlying concepts and techniques. The ability to translate word problems into equations, solve for unknowns, and verify solutions empowers us to tackle a wide range of percentage challenges.
Moreover, we have recognized the pervasive role of percentages in our daily lives, from financial calculations to statistical analysis. This underscores the importance of mastering percentage problem-solving skills, enabling us to navigate the world with greater mathematical confidence.
So, embrace the power of percentages, continue to practice and explore, and unlock the mathematical potential that lies within.
