Solving Pema's Age Problem A Step-by-Step Guide
Age-related problems are a staple in the world of mathematics, often requiring a blend of logical reasoning and algebraic skills to solve. These problems not only test our ability to formulate equations but also challenge our understanding of relationships between different variables. In this article, we will delve into a classic age puzzle involving Pema and her brother Sonam. The problem presents an intriguing scenario where we need to determine Pema's age when she is twice as old as Sonam. By carefully dissecting the information provided and employing a systematic approach, we can unravel this mathematical mystery and arrive at the correct solution. This article will provide a step-by-step guide to solving the problem, offering valuable insights into age-related mathematical challenges. Understanding age-related problems is crucial for developing analytical and problem-solving skills, which are essential not only in mathematics but also in various aspects of life.
Understanding the Problem Statement
Before we embark on the journey of solving this intriguing age problem, it is essential to dissect the problem statement meticulously. The problem states that Pema, who is currently sixteen years old, is four times as old as her brother Sonam. This initial piece of information provides us with a crucial foundation for establishing the relationship between their ages. To fully grasp the problem, we must carefully consider the wording and the implications of each statement. The problem then poses a question: How old will Pema be when she is twice as old as her brother Sonam? This question introduces a future scenario where the age dynamic between Pema and Sonam shifts. To solve this, we need to determine the time frame when Pema's age will be exactly twice that of Sonam's. The art of understanding the problem statement lies in identifying the key information and translating it into mathematical terms. This involves recognizing the variables, the relationships between them, and the ultimate goal we are trying to achieve. By paying close attention to the details and nuances of the problem, we set ourselves on the right path toward finding the solution. This meticulous approach is not only vital in mathematics but also in real-world scenarios where clear comprehension is the first step toward effective problem-solving.
Setting up the Equations
Now that we have a firm grasp of the problem statement, the next crucial step is to translate the given information into mathematical equations. This process involves identifying the unknowns, assigning variables, and formulating equations that accurately represent the relationships described in the problem. Let's denote Pema's current age as P and Sonam's current age as S. From the problem, we know that Pema is sixteen years old, so we can write P = 16. We are also told that Pema is four times as old as her brother Sonam. This translates to the equation P = 4S. To solve this, setting up the equations is a pivotal step in solving mathematical problems, especially those involving word problems. It bridges the gap between the verbal description and the mathematical representation, allowing us to apply algebraic techniques to find the solution. A well-formulated equation captures the essence of the problem, making it easier to manipulate and solve. This skill is not only valuable in mathematics but also in various fields where complex relationships need to be analyzed and quantified. By mastering the art of equation setup, we empower ourselves to tackle a wide range of challenges with confidence and precision.
Solving for Sonam's Age
With the equations set up, our next objective is to determine Sonam's current age. We have two equations: P = 16 and P = 4S. Since we know Pema's age (P) is 16, we can substitute this value into the second equation to solve for Sonam's age (S). The equation becomes 16 = 4S. To isolate S, we need to divide both sides of the equation by 4. This gives us S = 16 / 4, which simplifies to S = 4. Therefore, Sonam is currently 4 years old. This step is crucial as it provides us with another key piece of information needed to solve the main question. Solving for Sonam's age is a critical step in our problem-solving journey. It not only gives us a concrete value for one of the variables but also strengthens our understanding of the relationship between Pema's and Sonam's ages. This process highlights the power of algebraic manipulation in extracting information from equations. By systematically applying mathematical operations, we can unravel complex problems and gain valuable insights. This skill is fundamental in mathematics and has wide-ranging applications in various fields, from science and engineering to finance and economics.
Determining the Future Scenario
Now that we know Pema is 16 and Sonam is 4, we can move on to the central question: How old will Pema be when she is twice as old as Sonam? To solve this, we need to introduce a new variable to represent the number of years that will pass until this scenario occurs. Let's call this variable 'x'. In 'x' years, Pema's age will be 16 + x, and Sonam's age will be 4 + x. The problem states that at this time, Pema's age will be twice Sonam's age. This can be expressed as the equation 16 + x = 2(4 + x). This equation captures the essence of the future age relationship between Pema and Sonam. Determining the future scenario is a critical step in solving age-related problems. It requires us to think dynamically and consider how ages change over time. By introducing a variable to represent the passage of time, we can formulate equations that describe the future relationship between the individuals involved. This process highlights the importance of algebraic modeling in capturing real-world situations and making predictions. The ability to project future scenarios is not only valuable in mathematics but also in various fields such as finance, planning, and forecasting. By mastering this skill, we can make informed decisions and prepare for future possibilities.
Solving for 'x'
We have established the equation 16 + x = 2(4 + x) to represent the future scenario where Pema's age is twice Sonam's age. Now, we need to solve for 'x', which represents the number of years that will pass until this occurs. To do this, we first expand the right side of the equation: 16 + x = 8 + 2x. Next, we want to isolate 'x' on one side of the equation. We can subtract 'x' from both sides, giving us 16 = 8 + x. Then, we subtract 8 from both sides to isolate 'x': 16 - 8 = x, which simplifies to x = 8. This means that in 8 years, Pema will be twice as old as Sonam. Solving for 'x' is a crucial step in our problem-solving process. It allows us to quantify the time frame in which the specified condition will be met. This process involves applying algebraic techniques such as expansion, simplification, and isolation of variables. By mastering these techniques, we can confidently solve equations and extract valuable information. The ability to solve for unknowns is a fundamental skill in mathematics and has wide-ranging applications in various fields, from science and engineering to economics and finance.
Calculating Pema's Age
Now that we have found the value of 'x', which is 8 years, we can calculate Pema's age when she is twice as old as Sonam. We know that Pema's current age is 16, and in 8 years, her age will be 16 + x. Substituting x = 8, we get Pema's age as 16 + 8 = 24. Therefore, Pema will be 24 years old when she is twice as old as Sonam. This is the final step in solving the problem, where we use the value of 'x' to answer the original question. Calculating Pema's age is the culmination of our problem-solving efforts. It demonstrates how we can use the value of 'x' to determine Pema's age in the specified future scenario. This process reinforces the importance of careful calculation and attention to detail. By accurately substituting values and performing arithmetic operations, we can arrive at the correct solution. This skill is essential not only in mathematics but also in various aspects of life where precise calculations are required.
Conclusion
In conclusion, we have successfully solved the age problem involving Pema and Sonam. By carefully analyzing the problem statement, setting up equations, solving for unknowns, and calculating future ages, we determined that Pema will be 24 years old when she is twice as old as Sonam. This problem exemplifies the power of mathematical reasoning and algebraic techniques in solving real-world scenarios. The process involved translating verbal information into mathematical equations, manipulating these equations to find unknown values, and interpreting the results in the context of the problem. This journey through solving Pema's age problem highlights the importance of a systematic approach to problem-solving. It demonstrates how breaking down a complex problem into smaller, manageable steps can lead to a clear and accurate solution. The skills honed in this process, such as equation setup, algebraic manipulation, and logical reasoning, are invaluable not only in mathematics but also in various aspects of life. By mastering these skills, we empower ourselves to tackle challenges with confidence and precision.
Final Answer: The final answer is
