Ratio Of Salaries How To Calculate A And B's Income

Introduction: Unveiling the Salary Puzzle

In the realm of mathematical problem-solving, questions involving ratios and proportions often present a captivating challenge. This article delves into a specific scenario concerning the salaries of two individuals, A and B, aiming to decipher the ratio of their incomes. The problem presents us with two key pieces of information: a combined sum involving fractions of their salaries and a direct comparison highlighting the difference in their earnings. By carefully dissecting these clues and employing algebraic techniques, we can successfully navigate this puzzle and arrive at the solution. The core challenge lies in translating the given word problem into a mathematical framework, setting up equations that accurately represent the relationships described. Once this is achieved, the path to solving for the unknown ratio becomes clearer, involving algebraic manipulation and simplification. This exercise not only tests our mathematical prowess but also hones our analytical thinking and problem-solving skills. Ultimately, the ability to break down complex scenarios into manageable equations is a valuable asset in various fields, making this type of problem-solving a crucial skill to cultivate.

Setting Up the Equations: Translating Words into Math

To effectively tackle this problem, we must first convert the given information into mathematical equations. Let's denote the salary of A as 'a' and the salary of B as 'b'. The problem states that "the sum of two-fifths of the salary of A and five-eighths of the salary of B is Rs. 23000." This translates directly into the equation (2/5)a + (5/8)b = 23000. This equation forms the foundation of our solution, representing the combined salary information. The second crucial piece of information is that "the salary of A is Rs. 4000 less than the salary of B." This can be expressed as a = b - 4000. This equation provides a direct relationship between the salaries of A and B, allowing us to substitute one variable in terms of the other. These two equations now form a system of linear equations, a common mathematical structure that can be solved using various techniques. The beauty of this approach lies in its systematic nature; by carefully translating the word problem into equations, we create a clear roadmap for finding the solution. Understanding how to convert verbal descriptions into mathematical expressions is a fundamental skill in problem-solving, applicable across various disciplines. The accuracy of these equations is paramount, as any errors at this stage will propagate through the solution process. Therefore, a thorough understanding of the problem statement and careful translation are essential for success.

Solving for B's Salary: A Step-by-Step Approach

Now that we have our system of equations, the next step is to solve for the unknown variables. A common technique for solving systems of equations is substitution. We can substitute the second equation (a = b - 4000) into the first equation ((2/5)a + (5/8)b = 23000) to eliminate the variable 'a'. This substitution yields (2/5)(b - 4000) + (5/8)b = 23000. The resulting equation now contains only one variable, 'b', making it solvable. To solve for 'b', we first need to simplify the equation by distributing the (2/5) and then combining like terms. This involves multiplying (2/5) by both 'b' and -4000, resulting in (2/5)b - 1600 + (5/8)b = 23000. Next, we combine the terms with 'b', which requires finding a common denominator for the fractions (2/5) and (5/8). The common denominator is 40, so we rewrite the equation as (16/40)b + (25/40)b - 1600 = 23000. Combining these terms gives us (41/40)b - 1600 = 23000. To isolate the term with 'b', we add 1600 to both sides of the equation, resulting in (41/40)b = 24600. Finally, we multiply both sides by (40/41) to solve for 'b', which gives us b = 24000. Therefore, the salary of B is Rs. 24000. This step-by-step approach highlights the importance of methodical problem-solving, breaking down a complex equation into smaller, manageable steps. Each step builds upon the previous one, leading us closer to the solution. This process not only provides the answer but also demonstrates the power of algebraic manipulation in solving real-world problems.

Determining A's Salary: Utilizing the Relationship

Having successfully calculated the salary of B (b = 24000), we can now easily determine the salary of A using the equation a = b - 4000. This equation directly relates A's salary to B's, making the calculation straightforward. Substituting the value of b into the equation, we get a = 24000 - 4000, which simplifies to a = 20000. Therefore, the salary of A is Rs. 20000. This step demonstrates the power of having a relationship between variables. Once we solve for one variable, the relationship allows us to quickly determine the value of the other. This approach is common in many mathematical problems and highlights the importance of identifying and utilizing relationships between variables to simplify the solution process.

Calculating the Ratio: Expressing the Relationship

With the salaries of both A (Rs. 20000) and B (Rs. 24000) now known, we can calculate the ratio of their salaries. The ratio of A's salary to B's salary is expressed as a/b, which in this case is 20000/24000. To simplify this ratio, we can divide both the numerator and denominator by their greatest common divisor, which is 4000. This simplification yields the ratio 5/6. Therefore, the ratio of the salaries of A and B is 5:6. This final step demonstrates how to express the relationship between two quantities as a ratio. Ratios provide a concise way to compare values and are widely used in various fields, from finance to statistics. Understanding how to calculate and simplify ratios is a valuable skill for interpreting and analyzing data.

Conclusion: The Significance of Ratios and Problem-Solving

In conclusion, by carefully translating the given information into mathematical equations and employing algebraic techniques, we have successfully determined the ratio of the salaries of A and B to be 5:6. This exercise underscores the importance of mathematical problem-solving skills in deciphering real-world scenarios. The ability to break down complex problems into smaller, manageable steps, set up equations, and solve for unknown variables is a valuable asset in various fields. Furthermore, this problem highlights the significance of ratios in expressing relationships between quantities. Ratios provide a clear and concise way to compare values and are essential tools for analysis and decision-making. The process of solving this problem involved several key steps: translating the word problem into equations, substituting variables, simplifying expressions, and calculating the ratio. Each step required a clear understanding of mathematical concepts and techniques. By mastering these skills, we can confidently tackle a wide range of problem-solving challenges. Ultimately, the ability to think critically, analyze information, and apply mathematical principles is crucial for success in both academic and professional pursuits.