Solving Wage Ratio And Percentage Change Problems: A Step-by-Step Guide
In the realm of mathematics, particularly when dealing with ratios and percentages, it's crucial to grasp the underlying concepts to solve problems effectively. This article delves into a problem involving wage ratios and percentage changes, providing a step-by-step solution and offering insights into the logic behind each step. Our main focus will be understanding how changes in one quantity affect related quantities when they are linked by a ratio and a fixed sum. We will explore how to calculate percentage increases, determine new values after these increases, and then use this information to deduce changes in related quantities, ensuring the overall sum adheres to a specific constraint. This problem is a great example of how mathematical principles can be applied to real-world situations, such as managing finances and understanding salary adjustments. This guide aims to break down the problem into manageable parts, making the solution clear and accessible to everyone. By understanding the techniques in this problem, you can better prepare for similar situations in academic or practical contexts. Let's dive in and explore the fascinating world of ratios and percentages!
Problem Statement: Kamau and Wafula's Wages
The problem at hand involves Kamau and Wafula, whose wages are initially in the ratio of 5:3. This means that for every 5 units of currency Kamau earns, Wafula earns 3 units. The problem introduces a change: Kamau's wage increases by 12%, resulting in a new wage of Ksh. 5600. The challenge is to determine the percentage change in Wafula's wage, given that the sum of their new wages is Ksh. 9600. This requires a careful blend of ratio understanding, percentage calculations, and algebraic manipulation to arrive at the correct solution. This scenario is not just a mathematical exercise; it mirrors real-world situations involving salary adjustments and financial planning. Therefore, a thorough understanding of the solution process will provide valuable insights into managing proportional changes and maintaining budget constraints. We will explore each step of the solution methodically, ensuring that the underlying logic is clear and the calculations are accurate. The goal is to equip you with the knowledge and skills to tackle similar problems confidently, whether in an academic setting or in practical life scenarios involving financial adjustments and proportional reasoning.
Step-by-Step Solution
1. Initial Wage Ratio: Understanding the Foundation
Kamau and Wafula's wages are in the ratio 5:3. This is our foundational piece of information. This ratio signifies that their initial wages can be represented as 5x and 3x, where 'x' is a common multiplier. This approach is crucial because it allows us to deal with the wages as algebraic expressions, making it easier to manipulate and solve for unknowns. The ratio 5:3 clearly indicates the proportion of their earnings relative to each other. For every 5 units Kamau earns, Wafula earns 3 units. The introduction of 'x' transforms these proportional units into actual monetary values, bridging the gap between the abstract ratio and concrete financial figures. This step is vital for setting up the problem in a way that facilitates the subsequent calculations. Without understanding this initial ratio and how to represent it algebraically, the rest of the solution would be significantly more challenging. Recognizing the proportional relationship and expressing it using a variable is a fundamental technique in solving ratio-based problems. This lays the groundwork for calculating their initial wages and tracking how these wages change with the given percentage increase and sum constraint. In essence, understanding the initial wage ratio is the cornerstone of solving this problem, enabling a clear and logical progression towards the final answer.
2. Kamau's Wage Increase: Calculating the New Value
Kamau's wage increases by 12%, and his new wage becomes Ksh. 5600. This is a critical piece of information that allows us to calculate Kamau's initial wage. Let's denote Kamau's initial wage as 5x. A 12% increase means his wage is now 112% of the original, which can be expressed mathematically as 1.12 * 5x. Setting this equal to Ksh. 5600, we have the equation 1. 12 * 5x = 5600. Solving for x gives us x = 5600 / (1. 12 * 5) = 1000. This calculation is essential because it allows us to determine the value of 'x', which is the common multiplier in the wage ratio. Once we know 'x', we can find both Kamau's and Wafula's initial wages. The ability to calculate percentage increases and set up algebraic equations is a fundamental skill in solving problems involving proportional changes. This step highlights the importance of translating percentage changes into mathematical expressions and using equations to solve for unknown variables. Knowing Kamau's new wage after the increase provides a fixed point from which we can work backwards to find the initial conditions and then move forward to analyze Wafula's wage change. Therefore, this step is not just about finding a single value; it's about unlocking a key piece of information that unlocks the rest of the problem.
3. Determining Initial Wages: Unveiling the Starting Point
Now that we know x = 1000, we can determine the initial wages of both Kamau and Wafula. Kamau's initial wage was 5x, which is 5 * 1000 = Ksh. 5000. Wafula's initial wage was 3x, which is 3 * 1000 = Ksh. 3000. This step is crucial because it establishes the baseline from which we will measure the changes in their wages. Understanding their initial earnings is essential for calculating the percentage change in Wafula's wage later on. This stage in the solution process demonstrates the power of algebraic representation in solving ratio problems. By finding the value of 'x', we have effectively scaled the initial ratio to reflect the actual monetary values of their wages. This not only gives us a clear picture of their financial situation at the start but also sets the stage for analyzing the impact of the wage increase and the subsequent adjustments. The calculation of initial wages is a pivotal step, transforming abstract ratios into concrete financial figures, thereby making the problem more tangible and easier to solve. It also underscores the importance of accuracy in calculations, as these initial values serve as the foundation for all subsequent computations.
4. Wafula's New Wage: Solving for the Unknown
The problem states that the sum of their new wages is Ksh. 9600. We know Kamau's new wage is Ksh. 5600, so we can calculate Wafula's new wage by subtracting Kamau's new wage from the total: 9600 - 5600 = Ksh. 4000. This step is a direct application of the information provided in the problem and a necessary step in determining the change in Wafula's wage. Finding Wafula's new wage is critical because it allows us to compare his initial wage with his new wage and thereby calculate the percentage change. This part of the solution highlights the importance of using given constraints to solve for unknowns. The sum of the new wages acts as a limiting factor, enabling us to deduce Wafula's new earning based on Kamau's increase. This calculation is straightforward yet essential, demonstrating the problem-solving strategy of utilizing known quantities to find unknown ones. Without this step, we would lack the necessary information to calculate the percentage change in Wafula's wage, making it a pivotal part of the overall solution process. The ability to extract and use such information efficiently is a valuable skill in mathematical problem-solving, particularly in scenarios involving multiple variables and constraints.
5. Percentage Change in Wafula's Wage: The Final Calculation
To find the percentage change in Wafula's wage, we compare his new wage (Ksh. 4000) to his initial wage (Ksh. 3000). The increase in wage is 4000 - 3000 = Ksh. 1000. The percentage change is calculated as (increase / initial wage) * 100, which is (1000 / 3000) * 100 = 33.33%. Therefore, Wafula's wage increased by 33.33%. This final calculation brings together all the previous steps to answer the core question of the problem. The ability to accurately calculate percentage change is a fundamental skill in mathematics and has wide-ranging applications in finance, economics, and everyday life. This step underscores the importance of understanding the formula for percentage change and applying it correctly. It also highlights the significance of using the initial value as the base for the calculation, as this ensures an accurate representation of the relative change. The result provides a clear and quantifiable measure of the impact of the wage adjustments on Wafula's earnings. This final calculation is not just about arriving at a numerical answer; it's about providing a meaningful interpretation of the changes in wages, thereby completing the solution in a way that is both mathematically sound and practically relevant.
Conclusion
In conclusion, this problem demonstrates a practical application of ratios and percentages in a real-world scenario. By systematically breaking down the problem into smaller steps, we were able to calculate the percentage change in Wafula's wage after a change in Kamau's wage and a constraint on the sum of their new wages. The key takeaways from this problem include the importance of understanding ratios, calculating percentage increases, and using algebraic equations to solve for unknowns. Each step, from establishing the initial ratio to calculating the final percentage change, built upon the previous one, showcasing the interconnectedness of mathematical concepts. This problem not only tests computational skills but also problem-solving strategies, such as using given information to deduce unknown quantities and applying constraints to narrow down solutions. The ability to solve such problems is valuable not only in academic settings but also in practical situations involving financial planning and proportional reasoning. The methodical approach used here can be applied to a wide range of similar problems, reinforcing the importance of a clear, step-by-step problem-solving process. Furthermore, this exercise highlights the relevance of mathematics in everyday life, demonstrating how mathematical principles can help us understand and manage financial changes effectively. The solution process encourages critical thinking and analytical skills, essential for both academic and real-world success.
