Solving The Equation N + (1/5)n = 24 A Step-by-Step Guide

This article delves into solving a mathematical problem where a number, when added to one-fifth of itself, equals 24. We will explore the equation that models this problem, n + (1/5)n = 24, and meticulously walk through the steps to determine the value of n. This exploration is not only a practical exercise in algebra but also a demonstration of how mathematical equations can represent and solve real-world scenarios.

Understanding the Problem

In mathematical problem-solving, understanding the problem is the crucial first step. The question states: "When a number is added to 1/5 of itself, the result is 24." This sentence encapsulates a mathematical relationship that can be translated into an equation. The key here is to identify the unknown, which in this case is "a number," which we represent with the variable n. The phrase "1/5 of itself" translates to (1/5) n, and "added to" signifies addition. Finally, "the result is 24" indicates that the sum equals 24. By carefully dissecting the problem statement, we can transform it into the algebraic equation n + (1/5)n = 24. This equation is the cornerstone of our solution, and a clear understanding of its origins is paramount for solving it effectively.

Setting up the Equation

Setting up the equation correctly is paramount to finding the correct solution. The problem describes a scenario where a number (n) is added to one-fifth of itself ((1/5)n), resulting in 24. This translates directly into the algebraic equation n + (1/5)n = 24. The left side of the equation, n + (1/5)n, represents the sum of the number and its one-fifth, while the right side, 24, represents the total. This equation is a linear equation in one variable, which means it can be solved using basic algebraic principles. The variable n represents the unknown number we are trying to find. The coefficient of n on the left side is effectively 1 + (1/5), which simplifies to 6/5. Therefore, the equation can be rewritten as (6/5)*n = 24. This form makes it clearer how to isolate n and find its value. Accurate equation setup is a critical skill in mathematics, as it forms the foundation for the entire solution process.

Solving the Equation: Step-by-Step

Solving the equation n + (1/5)n = 24 involves a series of algebraic steps designed to isolate the variable n. First, we need to combine like terms on the left side of the equation. The term n can be thought of as (5/5)n. Adding this to (1/5)n gives us (6/5)n. So, the equation becomes (6/5)n = 24. To isolate n, we need to undo the multiplication by (6/5). We can do this by multiplying both sides of the equation by the reciprocal of (6/5), which is (5/6). This gives us (5/6) * (6/5) * n = 24 * (5/6)*. On the left side, (5/6) * (6/5) simplifies to 1, leaving us with just n. On the right side, 24 * (5/6) can be simplified by first dividing 24 by 6, which gives us 4, and then multiplying by 5, resulting in 20. Therefore, the equation simplifies to n = 20. This step-by-step process demonstrates how to methodically solve for n using basic algebraic manipulations. Each step is crucial to maintaining the balance of the equation and arriving at the correct solution.

Verification of the Solution

Verification of the solution is a critical step in problem-solving to ensure accuracy. We found that n = 20. To verify this solution, we substitute 20 back into the original equation, n + (1/5)n = 24. Replacing n with 20, we get 20 + (1/5)20 = 24. Now, we simplify the left side of the equation. (1/5)20 equals 4, so the equation becomes 20 + 4 = 24. Adding 20 and 4, we get 24. Therefore, the equation becomes 24 = 24*, which is a true statement. This confirms that our solution, n = 20, is correct. Verification not only provides confidence in the answer but also helps identify any errors made during the solving process. It reinforces the importance of meticulousness and attention to detail in mathematics.

Conclusion

In conclusion, by systematically translating the word problem into an algebraic equation and employing step-by-step solving techniques, we have successfully determined the value of n in the equation n + (1/5)n = 24. We found that n = 20, and we verified this solution by substituting it back into the original equation. This exercise highlights the power of algebra in representing and solving real-world problems. The ability to translate verbal descriptions into mathematical expressions is a fundamental skill in mathematics and has broad applications across various fields. This problem serves as a valuable example of how algebraic principles can be applied to find unknown quantities, emphasizing the importance of understanding the problem, setting up the equation correctly, solving it methodically, and verifying the solution.

Understanding the equation n + (1/5)n = 24 requires a clear grasp of its components and how they relate to the original problem statement. This equation is an algebraic representation of the statement: "When a number is added to 1/5 of itself, the result is 24." The variable n represents the unknown number we are trying to find. The term (1/5)n represents one-fifth of the number n. The addition symbol (+) indicates that we are adding the number n to its one-fifth. The equals sign (=) signifies that the sum of n and (1/5)n is equivalent to 24. This equation is a linear equation in one variable, which means it involves only one unknown (n) and the highest power of n is 1. Linear equations are fundamental in algebra and have a wide range of applications. To effectively solve this equation, we need to combine like terms, isolate the variable n, and perform algebraic operations on both sides to maintain equality. A thorough understanding of each component of the equation is crucial for successfully solving the problem and interpreting the result.

Breaking Down the Components

To break down the components of the equation n + (1/5)n = 24, let's examine each term individually. The variable n is the unknown number we are trying to determine. It represents the whole quantity in question. The term (1/5)n signifies one-fifth of the number n. This can be thought of as dividing n into five equal parts and considering one of those parts. The plus sign (+) indicates the operation of addition, meaning we are combining n and (1/5)n. The equals sign (=) asserts the equivalence between the left side of the equation, which is the sum of n and (1/5)n, and the right side, which is 24. The number 24 is the result or total obtained when n and (1/5)n are added together. By understanding each component, we can see how the equation accurately reflects the problem statement. The equation n + (1/5)n = 24 is a concise and precise way to express the relationship between the number, its one-fifth, and their sum. This detailed breakdown is essential for grasping the mathematical meaning of the equation and preparing for its solution.

Why this Equation Models the Problem

This equation models the problem effectively because it accurately translates the verbal statement into a mathematical expression. The phrase "a number" is represented by the variable n. The phrase "1/5 of itself" is represented by the term (1/5)n. The phrase "added to" corresponds to the addition operation (+). The phrase "the result is 24" is represented by the equals sign (=) and the number 24. Therefore, the equation n + (1/5)n = 24 is a direct translation of the problem statement. It captures the essential mathematical relationship described in the problem. The equation states that when the number n is added to one-fifth of itself, the total is 24. This is precisely what the problem statement describes. The equation is a powerful tool for solving the problem because it allows us to use algebraic techniques to find the value of the unknown number n. By understanding how each part of the equation corresponds to the problem statement, we gain confidence in its accuracy and its ability to lead us to the correct solution.

Alternative Ways to Represent the Problem

While the equation n + (1/5)n = 24 is a standard way to represent the problem, there are alternative ways to represent the problem that can offer different perspectives. One alternative is to express the left side of the equation as a single term. Since n is equivalent to (5/5)n, we can rewrite the equation as (5/5)n + (1/5)n = 24. Combining the like terms on the left side gives us (6/5)n = 24. This form of the equation highlights that 6/5 of the number n is equal to 24. Another way to represent the problem is using a visual model, such as a bar diagram. We can represent the number n as a bar divided into five equal parts. One of these parts represents (1/5)n. The sum of the whole bar (n) and one part (1/5)n is equal to 24. This visual representation can help in understanding the relationship between n and (1/5)n. Yet another approach is to express the problem in words, emphasizing different aspects of the relationship. For example, we could say, "Six-fifths of a number is 24. What is the number?" These alternative representations can provide additional insights into the problem and help reinforce understanding of the underlying mathematical concepts.

Solving for n in the equation n + (1/5)n = 24 requires a systematic approach using algebraic principles. This detailed walkthrough will provide a step-by-step explanation of the solution process. The first step is to simplify the equation by combining like terms. The terms n and (1/5)n are like terms because they both contain the variable n. To combine them, we need to express n as a fraction with a denominator of 5. We can rewrite n as (5/5)n. Now, the equation becomes (5/5)n + (1/5)n = 24. Adding the like terms on the left side gives us (6/5)n = 24. The next step is to isolate the variable n. To do this, we need to undo the multiplication by (6/5). We can achieve this by multiplying both sides of the equation by the reciprocal of (6/5), which is (5/6). Multiplying both sides by (5/6) gives us (5/6) * (6/5)n = 24 * (5/6). On the left side, (5/6) * (6/5) simplifies to 1, leaving us with just n. On the right side, we need to calculate 24 * (5/6)*. We can simplify this by dividing 24 by 6, which gives us 4, and then multiplying by 5, resulting in 20. Therefore, the equation simplifies to n = 20. This step-by-step walkthrough demonstrates the logical progression of algebraic manipulations needed to solve for n. Each step builds upon the previous one, leading to the final solution.

Step 1: Combining Like Terms

Step 1: Combining like terms in the equation n + (1/5)n = 24 is a crucial step in simplifying the equation. Like terms are terms that have the same variable raised to the same power. In this equation, n and (1/5)n are like terms because they both contain the variable n raised to the power of 1. To combine these terms, we need to express them with a common denominator. The term n can be thought of as 1n, which can be rewritten as (5/5)n. Now we have (5/5)n + (1/5)n = 24. Adding the coefficients of the like terms, we have (5/5 + 1/5) * n = 24*. This simplifies to (6/5)n = 24. Combining like terms streamlines the equation, making it easier to isolate the variable n. This step demonstrates the importance of understanding fractions and their role in algebraic manipulations. By combining like terms, we reduce the complexity of the equation and move closer to finding the value of n. This initial simplification is a key element in solving linear equations.

Step 2: Isolating the Variable n

Step 2: Isolating the variable n is the core of solving the equation (6/5)n = 24. Isolating a variable means getting it by itself on one side of the equation. In this case, n is being multiplied by the fraction (6/5). To undo this multiplication, we need to perform the inverse operation, which is division. However, instead of dividing by a fraction, it is often easier to multiply by the reciprocal of the fraction. The reciprocal of (6/5) is (5/6). To maintain the balance of the equation, we must multiply both sides by (5/6). This gives us (5/6) * (6/5)n = 24 * (5/6). On the left side, (5/6) * (6/5) simplifies to 1, leaving us with 1n, which is just n. On the right side, we need to calculate 24 * (5/6)*. This can be simplified by first dividing 24 by 6, which gives us 4, and then multiplying by 5, resulting in 20. Therefore, the equation simplifies to n = 20. This step demonstrates the fundamental algebraic principle of performing the same operation on both sides of an equation to maintain equality. By multiplying both sides by the reciprocal, we successfully isolated the variable n and determined its value.

Step 3: Calculating the Value of n

Step 3: Calculating the value of n in the equation n = 24 * (5/6) involves performing the multiplication on the right side. We have n = 24 * (5/6). To simplify this, we can first divide 24 by the denominator 6, which gives us 4. So, the equation becomes n = 4 * 5. Multiplying 4 by 5 gives us 20. Therefore, n = 20. This calculation reveals the numerical value of the unknown variable n. This step is a straightforward application of arithmetic operations, but it is crucial for arriving at the final solution. The result, n = 20, represents the number that, when added to one-fifth of itself, equals 24. This calculated value can now be verified by substituting it back into the original equation to ensure its correctness. The accurate calculation of n is the culmination of the algebraic steps and provides the answer to the problem.

Verifying the solution n = 20 is an essential step in the problem-solving process. It ensures that the calculated value is correct and satisfies the original equation, n + (1/5)n = 24. To verify the solution, we substitute n = 20 back into the original equation. This gives us 20 + (1/5) * 20 = 24. Next, we simplify the left side of the equation. (1/5) * 20 equals 4, so the equation becomes 20 + 4 = 24. Adding 20 and 4 gives us 24. Therefore, the equation simplifies to 24 = 24, which is a true statement. This confirms that our solution, n = 20, is correct. Verification provides confidence in the answer and demonstrates that the algebraic manipulations were performed accurately. It also highlights the importance of checking solutions to avoid errors and ensure a thorough understanding of the problem. This step is a crucial part of the mathematical problem-solving process.

Substituting n = 20 into the Original Equation

Substituting n = 20 into the original equation n + (1/5)n = 24 is the first step in verifying our solution. We replace every instance of n in the equation with the value 20. This gives us the equation 20 + (1/5) * 20 = 24. This substitution transforms the algebraic equation into an arithmetic expression, which we can then simplify to check for equality. Accurate substitution is critical for verification; any error in this step will lead to an incorrect conclusion. By carefully replacing n with 20, we set the stage for the subsequent simplification and comparison that will confirm or deny the validity of our solution. This process underscores the connection between algebraic manipulation and arithmetic evaluation in problem-solving.

Simplifying the Equation with n = 20

Simplifying the equation with n = 20 involves performing the arithmetic operations in the substituted equation 20 + (1/5) * 20 = 24. Following the order of operations, we first calculate (1/5) * 20. This is equivalent to dividing 20 by 5, which results in 4. The equation now becomes 20 + 4 = 24. Next, we add 20 and 4, which gives us 24. The equation simplifies to 24 = 24. This simplification process reduces the complexity of the equation and allows us to clearly see if the left side is equal to the right side. Accurate simplification is crucial for the verification process; any error in arithmetic will lead to an incorrect conclusion. By methodically performing each operation, we arrive at a straightforward comparison that confirms whether our solution satisfies the original equation. This step highlights the importance of arithmetic skills in the context of algebraic problem-solving.

Confirming the Solution

Confirming the solution is the final step in the verification process. After substituting n = 20 into the original equation and simplifying, we arrived at the statement 24 = 24. This statement is a true equality, meaning that the left side of the equation is indeed equal to the right side. This confirms that n = 20 is the correct solution to the equation n + (1/5)n = 24. If the simplified equation had resulted in an inequality (e.g., 24 = 25), it would indicate that our solution was incorrect and that we needed to re-examine our steps. The confirmation step provides a definitive answer to whether our calculated value of n satisfies the original problem. It is a crucial part of the problem-solving process, as it ensures accuracy and builds confidence in the result. This final check underscores the importance of meticulousness and thoroughness in mathematical problem-solving.

Solving equations like n + (1/5)n = 24 might seem like a purely mathematical exercise, but the underlying principles have practical applications in various real-world scenarios. These types of equations, which involve a variable and a fraction of that variable, often arise in situations involving proportions, percentages, and financial calculations. For instance, consider a scenario where a store offers a discount. If an item is priced at n dollars and there's a 20% discount, the discounted price can be represented as n - 0.20n. If the discounted price is known, we can set up an equation to solve for the original price n. Similarly, in financial calculations, understanding compound interest involves equations that relate the principal amount, interest rate, and time period. These equations often involve adding a fraction of the principal to itself, mirroring the structure of n + (1/5)n. Moreover, in fields like engineering and physics, problems involving ratios and proportions frequently require solving equations of this form. The ability to translate real-world scenarios into mathematical equations and solve them is a valuable skill that extends beyond the classroom and into various professional and everyday contexts.

Real-World Scenarios

Real-world scenarios that can be modeled by equations similar to n + (1/5)n = 24 are abundant and span various domains. In finance, calculating the initial investment needed to reach a certain savings goal with a specific interest rate often involves such equations. For example, if you want to have $24 after a year with an annual interest rate of 20%, the initial investment n would satisfy the equation n + 0.20n = 24. In retail, determining the original price of an item after a discount can also be modeled similarly. If an item is sold for $24 after a 20% discount, the original price n would satisfy n - 0.20n = 24. In everyday life, scenarios involving sharing or distributing quantities can lead to these types of equations. For instance, if a group of friends decides to add one-fifth of their individual contributions to a common fund to reach a total of $24, the individual contribution n would satisfy n + (1/5)n = 24. These examples illustrate the versatility of equations like n + (1/5)n = 24 in representing and solving practical problems.

Importance of Algebraic Problem-Solving Skills

The importance of algebraic problem-solving skills cannot be overstated, as they are fundamental to success in various academic disciplines and professional fields. Algebra provides a framework for representing and solving problems involving unknown quantities, which is a common scenario in many real-world situations. The ability to translate a problem statement into a mathematical equation, manipulate the equation to isolate the unknown variable, and interpret the solution is a crucial skill. In STEM fields (Science, Technology, Engineering, and Mathematics), algebraic skills are essential for modeling physical phenomena, designing systems, and analyzing data. In business and finance, algebra is used for budgeting, financial planning, and investment analysis. Even in everyday life, algebraic thinking is valuable for making informed decisions, such as managing personal finances, calculating proportions, and understanding discounts and markups. Mastering algebraic problem-solving not only enhances one's mathematical abilities but also cultivates critical thinking, logical reasoning, and analytical skills that are transferable to various aspects of life. The equation n + (1/5)n = 24 is a simple yet illustrative example of how algebraic skills can be applied to solve practical problems.

Tips for Approaching Similar Problems

When approaching similar problems that involve solving for an unknown variable, several key strategies can enhance your problem-solving efficiency and accuracy. First and foremost, carefully read and understand the problem statement. Identify the unknown quantity and assign it a variable, such as n. Translate the verbal information into a mathematical equation, paying close attention to keywords and phrases that indicate mathematical operations (e.g., "added to," "of," "is"). Once the equation is set up, simplify it by combining like terms and performing necessary arithmetic operations. Isolate the variable by applying inverse operations to both sides of the equation, maintaining balance throughout the process. After finding a solution, verify it by substituting it back into the original equation to ensure it satisfies the given conditions. If the solution does not check out, revisit your steps to identify any errors. Practice is crucial for developing proficiency in algebraic problem-solving. Work through a variety of problems, gradually increasing in complexity, to build your skills and confidence. Don't hesitate to seek help from teachers, tutors, or online resources if you encounter difficulties. By following these tips and consistently practicing, you can improve your ability to solve similar algebraic problems effectively.