Solving Percentage Problems What Number Is 90 When Increased By 20 Percent

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Understanding percentages is a fundamental skill in mathematics, with applications spanning from everyday calculations like figuring out discounts to more complex financial analysis. This article delves into a specific percentage problem: "120% of what number is 90?" We will break down the problem, explore different methods to solve it, and discuss the underlying concepts that make percentage calculations so versatile. Whether you're a student grappling with math homework or simply someone looking to brush up on your percentage skills, this guide will provide you with a clear and comprehensive understanding of how to tackle this type of problem.

Decoding the Percentage Puzzle: 120% of What Number is 90?

At first glance, the question might seem a bit perplexing: 120% of what number is 90? Percentages, after all, are typically associated with fractions of a whole, so how can we have 120% of something? The key to understanding this lies in recognizing that percentages greater than 100% represent values larger than the original whole. In this case, we're dealing with a number that, when increased by 20% of itself, results in 90. This is a classic percentage problem that can be solved using various approaches, each offering a slightly different perspective on the underlying mathematical principles. Let's explore these methods in detail.

Method 1: The Algebraic Approach

One of the most straightforward ways to solve percentage problems is by using algebra. This method involves translating the word problem into a mathematical equation, which we can then solve for the unknown variable. Let's break down the steps:

  1. Define the variable: Let 'x' represent the unknown number we're trying to find. This is a crucial first step in any algebraic problem.

  2. Translate the words into an equation: The phrase "120% of what number" can be written as 1.20 * x (since 120% is equivalent to 1.20 in decimal form). The word "is" translates to the equals sign (=), and the number 90 remains as it is. Therefore, the equation becomes:

    1. 20 * x = 90

  3. Solve for x: To isolate 'x', we need to divide both sides of the equation by 1.20:

    x = 90 / 1.20

    x = 75

Therefore, 120% of 75 is 90. This algebraic method provides a clear and concise way to solve the problem, demonstrating the power of translating word problems into mathematical equations. It's a versatile approach that can be applied to a wide range of percentage calculations.

Method 2: The Proportion Method

Another effective technique for solving percentage problems is the proportion method. This approach relies on the principle that a percentage represents a ratio, and we can set up a proportion to relate the given information to the unknown value. Here's how it works:

  1. Set up the proportion: We know that 120% corresponds to 90, and we want to find the number that represents 100% (the whole). We can set up the following proportion:

    120 / 100 = 90 / x

    This proportion states that the ratio of 120 to 100 is equal to the ratio of 90 to the unknown number 'x'.

  2. Cross-multiply: To solve the proportion, we cross-multiply:

    120 * x = 90 * 100

    120x = 9000

  3. Solve for x: Divide both sides of the equation by 120:

    x = 9000 / 120

    x = 75

Again, we arrive at the same answer: 120% of 75 is 90. The proportion method offers a visual way to understand the relationship between percentages and ratios, making it a valuable tool for solving percentage problems.

Method 3: The Unitary Method

The unitary method involves finding the value of one unit (in this case, 1%) and then using that value to calculate the desired quantity. This method can be particularly helpful for building a conceptual understanding of percentages.

  1. Find the value of 1%: We know that 120% corresponds to 90. To find the value of 1%, we divide 90 by 120:

    1% = 90 / 120

    1% = 0.75

  2. Find the value of 100%: Since 100% represents the whole, we multiply the value of 1% by 100:

    100% = 0.75 * 100

    100% = 75

Thus, 120% of 75 is 90. The unitary method emphasizes the fundamental relationship between percentages and the whole, providing a step-by-step approach to solving percentage problems.

The Significance of Percentages Beyond the Classroom

Understanding percentages is not just an academic exercise; it's a crucial life skill. Percentages are ubiquitous in our daily lives, appearing in various contexts:

  • Finance: Interest rates on loans, investment returns, and inflation rates are all expressed as percentages. A solid grasp of percentages is essential for making informed financial decisions.
  • Retail: Discounts, sales tax, and markups are often presented as percentages. Knowing how to calculate these percentages can help you save money and make smart purchasing choices.
  • Statistics: Percentages are used to represent proportions and rates in statistical data, allowing for easy comparison and analysis of different groups or trends.
  • Health: Nutritional information on food labels often includes percentages of daily recommended values. Understanding these percentages can help you make healthier dietary choices.
  • Everyday Life: From tipping at restaurants to calculating the percentage of battery life remaining on your phone, percentages are a constant presence in our daily routines.

Mastering percentage calculations empowers you to navigate these situations with confidence and make informed decisions. The ability to quickly and accurately calculate percentages is a valuable asset in both personal and professional settings.

Real-World Examples of Percentage Calculations

To further illustrate the practical applications of percentage calculations, let's consider a few real-world examples:

  1. Sales Discount: Imagine a store is offering a 30% discount on a shirt that originally costs $40. To calculate the discount amount, you would multiply the original price by the discount percentage (40 * 0.30 = $12). The sale price would then be the original price minus the discount amount (40 - 12 = $28). This simple percentage calculation can help you determine the actual cost of an item on sale.
  2. Calculating Tips: When dining at a restaurant, it's customary to leave a tip for the server. A typical tip percentage is 15-20% of the pre-tax bill. To calculate a 18% tip on a bill of $60, you would multiply the bill amount by the tip percentage (60 * 0.18 = $10.80). This allows you to easily determine the appropriate amount to tip.
  3. Interest on Savings: If you have money in a savings account that earns 2% annual interest, you can calculate the amount of interest you'll earn in a year by multiplying your account balance by the interest rate. For example, if you have $1000 in the account, you'll earn $20 in interest (1000 * 0.02 = $20). Understanding how interest is calculated can help you make informed decisions about your savings and investments.
  4. Exam Scores: Exam scores are often expressed as percentages. If you score 85 out of 100 on a test, your percentage score is 85%. This allows for easy comparison of performance across different tests with varying total points.

These examples demonstrate the wide range of situations where percentage calculations are essential. By mastering these skills, you can confidently navigate everyday financial transactions, make informed decisions, and better understand the world around you.

Common Pitfalls and How to Avoid Them

While percentage calculations are relatively straightforward, there are a few common pitfalls that can lead to errors. Being aware of these potential issues and knowing how to avoid them can significantly improve your accuracy.

  1. Misinterpreting the Base: One of the most common mistakes is applying the percentage to the wrong base. It's crucial to identify the quantity that represents 100% in the problem. For example, if a price is increased by 20% and then decreased by 20%, the final price will not be the same as the original price because the second percentage is calculated on a different base (the increased price). Always carefully consider what the percentage is being applied to.
  2. Converting Percentages to Decimals: When performing calculations, it's essential to convert percentages to decimals or fractions. A common mistake is forgetting to divide the percentage by 100. For example, 25% should be converted to 0.25 or 1/4 before being used in a calculation.
  3. Confusing Percentage Increase and Decrease: It's important to distinguish between percentage increase and percentage decrease. A percentage increase is calculated by dividing the amount of increase by the original value, while a percentage decrease is calculated by dividing the amount of decrease by the original value. Using the wrong base can lead to significant errors.
  4. Rounding Errors: Rounding intermediate results can introduce inaccuracies in your final answer. It's generally best to carry out calculations with as many decimal places as possible and only round the final answer to the desired level of precision.
  5. Not Double-Checking Your Work: A simple way to catch errors is to double-check your calculations. If possible, use a different method to solve the problem and see if you arrive at the same answer. This can help you identify any mistakes you may have made.

By being mindful of these common pitfalls and practicing careful calculation techniques, you can minimize errors and ensure the accuracy of your percentage calculations.

Conclusion: Mastering Percentages for Success

In conclusion, the problem "120% of what number is 90?" provides a valuable opportunity to explore the concepts and techniques involved in percentage calculations. We've examined three different methods – the algebraic approach, the proportion method, and the unitary method – each offering a unique perspective on solving this type of problem. By understanding these methods, you can choose the approach that best suits your learning style and the specific problem at hand.

More importantly, we've emphasized the broader significance of percentages in everyday life. From financial planning to retail shopping to understanding statistical data, percentages play a crucial role in our ability to make informed decisions and navigate the world around us. By mastering percentage calculations, you empower yourself with a valuable skill that will serve you well in both personal and professional endeavors. So, continue to practice, explore real-world examples, and confidently tackle any percentage problem that comes your way. The world of percentages is vast and varied, and the more you understand it, the better equipped you'll be to succeed.