Solving Age Problems How To Calculate Future Ages
Introduction
This article explores a mathematical problem involving the ages of Mary and her niece, Abbie. Mary is currently 54 years old, and Abbie is 18 years old. The challenge is to determine in how many years Abbie's age will be 2/5 of Mary's age. This problem requires setting up an equation that represents the relationship between their ages in the future. We will begin by defining variables, setting up the equation, and then solving for the unknown number of years. This type of problem is a classic example of age-related algebraic questions that often appear in mathematics.
1. Understanding the Problem and Initial Steps
To begin solving this age-related problem, the first crucial step involves understanding the present age relationship between Mary and Abbie. Mary, the aunt, is currently 54 years old, while Abbie, her niece, is 18 years old. The core question we aim to answer is: In how many years will Abbie's age be exactly 2/5 of Mary's age? This requires a careful consideration of how both their ages will change over time. Understanding this relationship is paramount for formulating a mathematical equation that accurately models the scenario. We need to recognize that as time passes, both Mary and Abbie will age, but at the same rate. This constant rate of aging is crucial for setting up the equation correctly. The initial step involves recognizing that we are looking for a future point in time, measured in years, at which a specific age ratio between Abbie and Mary will be met. This means we will need to introduce a variable to represent the number of years that will pass. The next phase of solving the problem involves representing the unknown number of years using a variable, which will allow us to create an algebraic expression reflecting their future ages. Identifying the core question and understanding the dynamics of how their ages change relative to each other is the foundational step in tackling this mathematical puzzle. This initial understanding paves the way for the subsequent algebraic manipulation required to arrive at the solution. Focusing on the current age disparity and projecting how it might evolve is the essence of initiating the problem-solving process.
2. Representing the Unknown: Defining the Variable
To effectively solve this problem, the next critical step is to represent the required number of years with a variable. In algebraic problem-solving, using variables to denote unknown quantities is essential, and in this case, let's define 'x' as the number of years that need to pass for Abbie's age to be 2/5 of Mary's age. This variable 'x' is the key to unlocking the solution, as it allows us to express their future ages in terms of the present ages. By introducing 'x', we can formulate an equation that captures the changing relationship between Abbie’s and Mary’s ages. This algebraic representation provides a structured way to express the future ages, and, critically, the relationship we are interested in. Specifically, in 'x' years, Abbie's age will be her current age plus 'x' years (18 + x), and similarly, Mary's age will be her current age plus 'x' years (54 + x). Defining 'x' as the number of years not only simplifies the problem but also makes it easier to manipulate the ages in an equation. This variable becomes the focal point around which we can build an algebraic expression that mirrors the problem's condition. The significance of representing the unknown with a variable lies in its ability to convert a word problem into a solvable algebraic equation. Without this variable representation, it would be exceedingly difficult to mathematically express and solve the core question. Therefore, the introduction of 'x' is not merely a symbolic step but a practical necessity for solving the problem efficiently and accurately. This representation allows us to proceed to the next step, which involves setting up the equation based on the given condition.
3. Setting Up the Equation: Translating Words into Algebra
With 'x' representing the number of years, the next pivotal step is to translate the problem's core condition into a mathematical equation. The condition states that in 'x' years, Abbie's age will be 2/5 of Mary's age. To translate this into an equation, we first express Abbie's future age as (18 + x) and Mary's future age as (54 + x), as we have previously established. The phrase "Abbie's age will be 2/5 of Mary's age" can now be directly converted into an equation: 18 + x = (2/5) * (54 + x). This equation is the cornerstone of the solution, as it encapsulates the relationship we are trying to find. It’s crucial to ensure the equation accurately represents the problem's condition; any error here will lead to an incorrect solution. The equation reflects the future scenario where Abbie's age is exactly two-fifths of Mary's age. This setup involves understanding proportional relationships and translating them into algebraic expressions. Once the equation is set up, the focus shifts to solving for 'x'. This typically involves algebraic manipulations such as distributing, combining like terms, and isolating the variable on one side of the equation. The correct setup of the equation is a critical juncture in solving this mathematical problem. It is the bridge between understanding the problem conceptually and solving it mathematically. A well-formed equation allows us to apply algebraic techniques systematically and arrive at the correct answer. The equation, 18 + x = (2/5) * (54 + x), is the algebraic representation of the problem's condition, and solving this equation will provide the value of 'x', which is the number of years we seek.
4. Solving the Equation: Algebraic Manipulation
After setting up the equation 18 + x = (2/5) * (54 + x), the next crucial phase is to solve it for 'x'. This involves a series of algebraic manipulations designed to isolate 'x' on one side of the equation. The initial step often involves eliminating the fraction to simplify the equation. This can be achieved by multiplying both sides of the equation by 5, which gives us 5 * (18 + x) = 2 * (54 + x). This step clears the fraction and makes the equation easier to handle. Next, we distribute the constants on both sides of the equation. On the left side, 5 is multiplied by both 18 and x, resulting in 90 + 5x. On the right side, 2 is multiplied by both 54 and x, yielding 108 + 2x. So, the equation now looks like this: 90 + 5x = 108 + 2x. The subsequent step involves grouping the like terms together. We subtract 2x from both sides to get the 'x' terms on one side, resulting in 90 + 3x = 108. Then, we subtract 90 from both sides to isolate the term with 'x', which gives us 3x = 108 - 90, simplifying to 3x = 18. Finally, we divide both sides by 3 to solve for 'x', which yields x = 18 / 3, and thus x = 6. This value of 'x' is the solution to the equation and represents the number of years in which Abbie's age will be 2/5 of Mary's age. The process of solving the equation requires a firm understanding of algebraic principles and careful attention to detail to avoid errors in manipulation. Each step builds upon the previous one, leading to the isolation of the variable and the determination of its value. The solution, x = 6, indicates that in 6 years, Abbie's age will be 2/5 of Mary's age.
5. Verifying the Solution: Ensuring Accuracy
Once we have obtained the solution x = 6, it's imperative to verify this answer to ensure its accuracy. This verification step is a crucial part of the problem-solving process, confirming that the solution satisfies the original problem condition. To verify, we substitute x = 6 back into the expressions for Abbie's and Mary's future ages. In 6 years, Abbie's age will be 18 + 6 = 24 years old, and Mary's age will be 54 + 6 = 60 years old. The original condition stated that Abbie's age should be 2/5 of Mary's age. So, we check if 24 is indeed 2/5 of 60. To do this, we calculate (2/5) * 60, which equals (2 * 60) / 5 = 120 / 5 = 24. Since 24 is equal to 2/5 of 60, the condition is satisfied. This confirms that our solution, x = 6, is correct. The verification process involves revisiting the problem statement and substituting the calculated value to ensure it logically fits the scenario. It is a safeguard against algebraic errors or misinterpretations of the problem. By verifying the solution, we gain confidence that the answer is not only mathematically correct but also makes sense in the context of the original problem. This step underscores the importance of precision in problem-solving and ensures that the solution is both accurate and meaningful. In this case, the verification process provides a clear affirmation that in 6 years, Abbie's age will indeed be 2/5 of Mary's age, completing the problem-solving cycle.
6. Final Answer and Conclusion
After solving the equation and verifying the solution, we can confidently state the final answer. The value of 'x', which represents the number of years required for Abbie's age to be 2/5 of Mary's age, is 6. Therefore, in 6 years, Abbie's age will be 2/5 of Mary's age. This conclusion is the culmination of the entire problem-solving process, from understanding the initial conditions to algebraic manipulation and verification. The answer provides a specific time frame in which the given age relationship will be met. The process we followed is a standard approach to solving algebraic age problems. It involves defining variables, setting up equations, solving them using algebraic techniques, and verifying the solution for accuracy. This methodology is applicable to a wide range of mathematical problems, making it a valuable skill in both academic and practical contexts. In conclusion, by carefully translating the word problem into an algebraic equation and systematically solving it, we have successfully determined that in 6 years, Abbie's age will be 2/5 of Mary's age. This final answer not only solves the immediate problem but also demonstrates the power of algebraic methods in addressing quantitative questions.
FAQ
Q1: How are you going to begin solving the problem?
To begin solving the problem, you should first identify the knowns, which are Mary's and Abbie's current ages (54 and 18, respectively). The key is to understand the question: "In how many years will Abbie's age be 2/5 of Mary's age?" This requires setting up an equation that represents the future relationship between their ages. Start by defining a variable for the unknown number of years and then express their future ages in terms of this variable.
Q2: How will you represent the required number of years?
The required number of years can be represented by a variable. Let's use 'x' to denote the number of years that need to pass for Abbie's age to be 2/5 of Mary's age. This variable will be added to both Mary's and Abbie's current ages to express their future ages.
