Solving Age Puzzle Find Present Age Of B Step By Step Guide

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Let's dive into a fascinating age-related problem that requires careful analysis and logical deduction. This type of question often appears in mathematical puzzles and competitive exams, testing our ability to translate word problems into algebraic equations. Our goal is to determine the present age of B, given the information about the sum of their ages and a specific relationship between their past ages. This exploration will not only provide the solution but also enhance our problem-solving skills in mathematics.

Decoding the Age Conundrum

At the heart of this problem lies the challenge of deciphering the relationship between A and B's ages at different points in time. The statement "I am twice as old as you were when I was as old as you are" is the key to unlocking the solution. Let's break down the information piece by piece to construct a clear and solvable equation. In this comprehensive explanation, we will meticulously dissect the intricacies of the age problem. By employing a step-by-step approach, we aim to not only arrive at the correct solution but also to furnish a thorough understanding of the underlying concepts. Our focus extends beyond merely obtaining an answer; we aspire to equip you with the skills necessary to tackle similar challenges in the realm of age-related mathematical problems. To truly grasp the problem, we must first represent the unknown ages with algebraic variables. Let A's current age be denoted as 'a' and B's current age as 'b'. We are given that the sum of their ages is 49 years, which translates to our first equation:

a + b = 49

Now, let's delve into the more intricate part of the problem: "I am twice as old as you were when I was as old as you are." This statement involves comparing A's current age to B's age at a time in the past when A was the same age as B is now. To unravel this, let's consider the time difference between then and now. Let 'x' be the number of years ago when A was as old as B is now. This means that:

a - x = b

This equation tells us that 'x' years ago, A's age was equal to B's current age. Now, let's consider B's age at that time. B's age 'x' years ago would have been:

b - x

The problem states that A is currently twice as old as B was at that time. This gives us our second crucial equation:

a = 2(b - x)

Now we have a system of three equations with three unknowns (a, b, and x):

  1. a + b = 49
  2. a - x = b
  3. a = 2(b - x)

Our next step is to solve this system of equations to find the value of 'b', which represents B's current age. We can use substitution or elimination methods to achieve this. Let's use substitution. From equation (2), we can express x in terms of a and b:

x = a - b

Now, substitute this value of x into equation (3):

a = 2(b - (a - b))

Simplify the equation:

a = 2(b - a + b) a = 2(2b - a) a = 4b - 2a

Now, bring the 'a' terms to one side:

3a = 4b

This gives us a relationship between a and b. Now we can use equation (1) (a + b = 49) along with this new equation to solve for a and b. From equation (1), we can express a as:

a = 49 - b

Substitute this into the equation 3a = 4b:

3(49 - b) = 4b

Expand and simplify:

147 - 3b = 4b 147 = 7b

Now, divide by 7 to solve for b:

b = 147 / 7 b = 21

So, B's current age is 21 years. Now we can find A's current age by substituting b = 21 into equation (1):

a + 21 = 49 a = 49 - 21 a = 28

Thus, A's current age is 28 years. We can also find the value of x:

x = a - b x = 28 - 21 x = 7

This means that 7 years ago, A was 21 years old (the same age as B is now), and B was 14 years old. A's current age (28) is indeed twice B's age 7 years ago (14).

Verifying the Solution

To ensure our solution is correct, we can substitute the values of a, b, and x back into the original equations:

  1. a + b = 49 28 + 21 = 49 (Correct)
  2. a - x = b 28 - 7 = 21 (Correct)
  3. a = 2(b - x) 28 = 2(21 - 7) 28 = 2(14) 28 = 28 (Correct)

Since all equations hold true, our solution is verified.

The Answer

The present age of B is 21 years. Therefore, the correct answer is (b) 21.

This detailed solution demonstrates the importance of breaking down complex word problems into smaller, manageable parts. By carefully defining variables and constructing equations based on the given information, we can systematically solve for the unknowns. This approach is applicable to a wide range of mathematical problems, making it a valuable skill for problem-solving in general. In this section, we will further delve into the nuances of solving age-related mathematical problems. Age problems, while seemingly straightforward, often require a methodical approach to unravel the intricacies of the relationships between the ages of individuals at different points in time. The key lies in the meticulous translation of the word problem into algebraic equations, allowing us to systematically solve for the unknowns.

Key Strategies for Tackling Age Problems

  1. Variable Assignment: The initial step in solving any age problem is to assign variables to the unknown ages. Let the present age of individual A be denoted as 'a' and the present age of individual B be denoted as 'b'. If the problem involves ages at different times, introduce additional variables as necessary to represent these past or future ages.

  2. Equation Formulation: The next crucial step is to carefully analyze the given information and translate it into algebraic equations. This often involves understanding relationships such as the sum or difference of ages, or statements comparing ages at different times. For instance, if the problem states that "the sum of their ages is 50 years," we can directly translate this into the equation a + b = 50. Similarly, if the problem states that "A is twice as old as B was 10 years ago," we can represent this as a = 2(b - 10).

  3. System of Equations: Age problems often lead to a system of equations, where we have multiple equations involving the same variables. The number of equations should ideally match the number of unknown variables to ensure a solvable system. Common methods for solving systems of equations include substitution, elimination, and matrix methods. The choice of method depends on the structure of the equations and the ease of manipulation.

  4. Time Differences: Many age problems involve comparing ages at different points in time. It is crucial to understand how time affects the ages of individuals. If 'x' years have passed, then the age of each individual increases by 'x' years. If we are considering ages 'x' years ago, then the age of each individual was 'x' years less than their current age.

  5. Careful Reading and Interpretation: The most important aspect of solving age problems is to carefully read and interpret the given information. Pay close attention to the wording of the problem, as subtle differences in phrasing can lead to different equations. It is often helpful to break down complex statements into smaller parts and represent each part algebraically.

Illustrative Examples and Problem-Solving Techniques

To further illustrate these strategies, let's consider a few examples:

Example 1: A is 10 years older than B. In 5 years, A's age will be twice B's age. Find their present ages.

  • Solution: Let A's present age be 'a' and B's present age be 'b'. From the first statement, we have: a = b + 10 In 5 years, A's age will be a + 5, and B's age will be b + 5. From the second statement, we have: a + 5 = 2(b + 5) Now we have a system of two equations:

    1. a = b + 10
    2. a + 5 = 2(b + 5) Substituting equation (1) into equation (2): b + 10 + 5 = 2(b + 5) b + 15 = 2b + 10 b = 5 Now, substituting b = 5 into equation (1): a = 5 + 10 a = 15 Thus, A's present age is 15 years, and B's present age is 5 years.

Example 2: The sum of the ages of a father and his son is 60 years. 10 years ago, the father's age was three times the age of the son. Find their present ages.

  • Solution: Let the father's present age be 'f' and the son's present age be 's'. From the first statement, we have: f + s = 60 10 years ago, the father's age was f - 10, and the son's age was s - 10. From the second statement, we have: f - 10 = 3(s - 10) Now we have a system of two equations:

    1. f + s = 60
    2. f - 10 = 3(s - 10) From equation (1), we can express f as: f = 60 - s Substituting this into equation (2): 60 - s - 10 = 3(s - 10) 50 - s = 3s - 30 4s = 80 s = 20 Now, substituting s = 20 into equation (1): f + 20 = 60 f = 40 Thus, the father's present age is 40 years, and the son's present age is 20 years.

Common Pitfalls and How to Avoid Them

While solving age problems, it is easy to fall into common pitfalls that can lead to incorrect solutions. Being aware of these pitfalls can help you avoid them:

  1. Misinterpreting Time References: One common mistake is misinterpreting time references such as "years ago" or "years hence." Always clearly define the time frame you are working with and adjust the ages accordingly. For instance, if the problem refers to ages 5 years ago, remember to subtract 5 from the current ages.

  2. Incorrectly Translating Statements: Translating word statements into algebraic equations requires precision. Pay close attention to the wording and ensure that your equations accurately represent the given relationships. Double-check your equations to make sure they align with the problem statement.

  3. Not Solving for the Correct Variable: Sometimes, you may solve the system of equations correctly but fail to identify the variable that the problem is asking for. Always reread the question to ensure that you are providing the correct answer. For example, if the problem asks for B's age, make sure you solve for 'b' and not 'a'.

  4. Ignoring Units: Ensure that all ages are expressed in the same units (usually years). If the problem involves ages in different units, convert them to a common unit before setting up the equations.

  5. Not Verifying the Solution: After obtaining a solution, it is always a good practice to verify it by substituting the values back into the original equations. This helps ensure that your solution is correct and that you have not made any algebraic errors.

Advanced Techniques and Problem Variations

Beyond the basic age problems, there are more complex variations that require advanced techniques. These may involve multiple individuals, non-linear relationships, or additional constraints. To tackle these problems effectively, consider the following techniques:

  1. Introducing Additional Variables: For problems with multiple individuals or time frames, introduce additional variables to represent the unknown ages. This can help break down the problem into smaller, more manageable parts.

  2. Using Ratios and Proportions: Some problems involve ratios or proportions of ages. In such cases, set up equations using these ratios to relate the ages. For instance, if the ratio of A's age to B's age is 2:3, you can write a/b = 2/3.

  3. Considering Time Intervals: When dealing with problems involving multiple time intervals, carefully consider how ages change over these intervals. Use time diagrams or timelines to visualize the changes and set up equations accordingly.

  4. Working Backwards: In some cases, it may be easier to work backwards from a future or past scenario to the present. Start by defining the ages at the specified time and then work your way back to the present using the given information.

  5. Trial and Error: If you are struggling to set up equations, you can sometimes use trial and error to get a sense of the solution. Start with educated guesses and refine your guesses based on the given information. This can help you identify patterns and relationships that you may have missed.

By understanding these strategies, techniques, and common pitfalls, you can confidently tackle a wide range of age-related mathematical problems. The key is to practice consistently and develop a methodical approach to problem-solving.

Conclusion

In this exploration, we have successfully solved a complex age problem by translating the word problem into a system of algebraic equations. We have also discussed various strategies, techniques, and common pitfalls to avoid while solving age problems. By mastering these skills, you can approach similar mathematical challenges with confidence and precision. The art of problem-solving in mathematics lies not just in finding the answer, but in understanding the process and developing a logical approach that can be applied to various scenarios.