Solving Age Problems Find The Present Age Of Q
Unraveling age-related mathematical problems often presents a fascinating challenge, blending logical deduction with algebraic techniques. This article delves into a classic age-based problem, meticulously dissecting the given information to arrive at the solution. We'll explore the steps involved in translating word problems into mathematical equations and employing these equations to determine the present age of an individual, Q. Let's embark on this journey of mathematical discovery, where we'll not only solve the problem but also gain insights into effective problem-solving strategies applicable in various contexts.
The Age Enigma
The puzzle before us states: "Four times the age of P is 9 more than 7 times the age of Q. After 9 years, the ratio of P to Q will be 3:2. Find the present age of Q." This seemingly intricate problem can be elegantly solved by carefully translating the words into mathematical expressions and then manipulating these expressions to isolate the unknown variable – the present age of Q. We are given two crucial pieces of information: a relationship between the current ages of P and Q, and a ratio of their ages after a specific time period. These two pieces of information will form the foundation of our solution, allowing us to construct a system of equations that we can then solve. The challenge lies in accurately representing the given relationships algebraically and then applying the appropriate techniques to solve for the desired unknown. As we proceed, we'll break down each step, ensuring clarity and understanding at every stage. This problem not only tests our algebraic skills but also our ability to interpret and translate real-world scenarios into mathematical models, a skill that is invaluable in various fields.
Translating Words into Equations
The cornerstone of solving word problems in mathematics lies in the ability to translate verbal statements into algebraic equations. This process involves identifying the key variables, representing them with symbols, and then expressing the relationships between them using mathematical operators. In our age puzzle, we have two primary variables: the present ages of P and Q. Let's denote the present age of P as 'p' and the present age of Q as 'q'. Now, we can dissect the first statement: "Four times the age of P is 9 more than 7 times the age of Q." This translates directly into the equation 4p = 7q + 9. This equation captures the relationship between P's age and Q's age at the present time. The phrase "four times the age of P" is represented by 4p, and "7 times the age of Q" is represented by 7q. The phrase "9 more than" indicates an addition of 9. This equation serves as our first anchor point, linking the two unknowns. Next, we consider the second piece of information: "After 9 years, the ratio of P to Q will be 3:2." This implies that in 9 years, P's age will be p + 9, and Q's age will be q + 9. The ratio of their ages will then be (p + 9) / (q + 9) = 3/2. This second equation introduces the element of time and how it affects the ages of P and Q. We now have two equations with two unknowns, a classic setup for solving a system of linear equations. This translation process is crucial because it transforms the problem from a verbal description into a precise mathematical formulation, allowing us to apply algebraic techniques to find the solution.
Constructing the System of Equations
Having successfully translated the word problem into algebraic expressions, we now have a system of two equations:
- 4p = 7q + 9
- (p + 9) / (q + 9) = 3/2
This system of equations represents the core of our problem. The first equation, 4p = 7q + 9, establishes a direct relationship between the current ages of P and Q. It tells us how P's age relates to Q's age at the present time. The second equation, (p + 9) / (q + 9) = 3/2, introduces the element of time. It describes the ratio of their ages after 9 years. The beauty of a system of equations lies in its ability to capture multiple relationships simultaneously. By solving this system, we can find the values of both 'p' and 'q', thus determining the present ages of both P and Q. The challenge now is to choose an appropriate method for solving this system. We have several options at our disposal, such as substitution, elimination, or matrix methods. Each method has its own strengths and weaknesses, and the choice of method often depends on the specific structure of the equations. In this case, we might find the substitution method particularly effective, as we can easily isolate 'p' in the first equation and substitute it into the second equation. However, regardless of the method chosen, the goal remains the same: to systematically eliminate one variable and solve for the other. This system of equations is the mathematical representation of our age puzzle, and by solving it, we unlock the solution to the problem.
Solving the System: A Step-by-Step Approach
With our system of equations in place, we can now embark on the process of solving for the unknown variables. Our system consists of:
- 4p = 7q + 9
- (p + 9) / (q + 9) = 3/2
Let's employ the substitution method for its clarity and efficiency in this scenario. First, we isolate 'p' in the first equation: p = (7q + 9) / 4. This expression now represents 'p' in terms of 'q'. Next, we substitute this expression for 'p' into the second equation: [((7q + 9) / 4) + 9] / (q + 9) = 3/2. This substitution effectively eliminates 'p' from the second equation, leaving us with an equation involving only 'q'. The equation now looks more complex, but it's a crucial step in isolating the variable we want to solve for. Now, we simplify the equation by multiplying both sides by (q + 9) and by 2 to get rid of the fractions: 2 * [((7q + 9) / 4) + 9] = 3 * (q + 9). Further simplification involves multiplying out the terms and combining like terms. This process may seem tedious, but it's essential for isolating 'q'. Expanding and simplifying the equation gives us: (7q + 9) / 2 + 18 = 3q + 27. Multiplying through by 2 to clear the remaining fraction gives: 7q + 9 + 36 = 6q + 54. Combining like terms, we get: 7q + 45 = 6q + 54. Finally, subtracting 6q from both sides and subtracting 45 from both sides, we isolate 'q': q = 9. This is our breakthrough! We have successfully solved for 'q', the present age of Q. The systematic approach of substitution, simplification, and isolation has led us to the solution. However, our journey doesn't end here. We must also verify our solution to ensure its accuracy.
Verifying the Solution
Having found a potential solution, it's crucial to verify its accuracy by plugging it back into the original equations. This step ensures that our solution satisfies all the conditions of the problem and that we haven't made any algebraic errors along the way. We found that q = 9. Let's substitute this value back into our original equations:
- 4p = 7q + 9
- (p + 9) / (q + 9) = 3/2
Substituting q = 9 into the first equation, we get: 4p = 7(9) + 9, which simplifies to 4p = 63 + 9, and further to 4p = 72. Dividing both sides by 4, we find p = 18. So, according to our solution, P's present age is 18 years. Now, let's substitute p = 18 and q = 9 into the second equation: (18 + 9) / (9 + 9) = 3/2, which simplifies to 27 / 18 = 3/2. Dividing both the numerator and denominator of the left-hand side by 9, we get 3/2 = 3/2. This confirms that our solution satisfies the second equation as well. Since our values for p and q satisfy both original equations, we can confidently conclude that our solution is correct. This verification step is a cornerstone of mathematical problem-solving. It provides a safety net against potential errors and ensures that the answer we arrive at is indeed the correct one. By verifying our solution, we not only confirm its accuracy but also deepen our understanding of the problem and the relationships between the variables involved. It's a testament to the importance of precision and attention to detail in mathematics.
The Answer Unveiled
After a meticulous journey through translation, equation construction, solving, and verification, we have arrived at the solution to our age puzzle. We successfully determined that the present age of Q, denoted by 'q', is 9 years. This answer was obtained by carefully translating the given word problem into a system of algebraic equations, applying the substitution method to solve for 'q', and then verifying our solution by plugging it back into the original equations. The process highlighted the importance of several key mathematical skills: the ability to translate verbal statements into mathematical expressions, the skill of manipulating equations to isolate variables, and the critical step of verifying solutions to ensure accuracy. This problem, while seemingly simple on the surface, provides a valuable illustration of the power of algebraic techniques in solving real-world problems. It also underscores the importance of a systematic approach to problem-solving, breaking down complex problems into smaller, manageable steps. The journey from the initial puzzle to the final answer has been a testament to the elegance and effectiveness of mathematical reasoning. By carefully applying these principles, we can confidently tackle a wide range of mathematical challenges and unlock the solutions they hold.
Concluding Thoughts on Age Problems
Age problems, like the one we've dissected here, are a classic staple in the world of mathematical puzzles. They serve as an excellent vehicle for honing critical thinking, algebraic manipulation, and problem-solving skills. These types of problems often involve translating real-world scenarios into mathematical models, a skill that is highly valuable in various disciplines. The key to success in tackling age problems lies in a systematic approach: carefully reading and understanding the problem statement, identifying the unknowns, representing them with variables, translating the given information into equations, solving the equations using appropriate techniques, and, most importantly, verifying the solution. The process is not just about finding the answer; it's about developing a logical and methodical approach to problem-solving that can be applied in a wide range of contexts. Age problems also offer a glimpse into the power of algebra as a tool for modeling and solving real-world situations. By learning to translate verbal descriptions into algebraic equations, we gain the ability to tackle problems that might otherwise seem insurmountable. Moreover, the act of verifying solutions reinforces the importance of accuracy and attention to detail in mathematics. In conclusion, age problems are more than just puzzles; they are valuable exercises that help us develop essential mathematical skills and cultivate a systematic approach to problem-solving. They challenge us to think critically, apply algebraic techniques, and verify our results, all of which are crucial skills in mathematics and beyond.
