Dani Told Beto, "I Am Currently Twice The Age You Were When I Was Your Age, And When You Are My Age, The Sum Of Our Ages Will Be 108 Years." How Old Are Dani And Beto?
Age-related word problems have always fascinated mathematical minds, presenting a unique blend of logical reasoning and algebraic manipulation. These puzzles often involve intricate relationships between people's ages at different points in time, requiring a careful analysis of the given information to arrive at the solution. In this article, we delve into a classic age problem involving Dani and Beto, two individuals whose ages are intertwined in a captivating way. We'll break down the problem step by step, employing algebraic techniques to unravel the mystery of their ages. Let's embark on this mathematical journey and discover the ages of Dani and Beto.
The age puzzle presented involves a conversation between Dani and Beto, where they discuss their ages in the past, present, and future. The challenge lies in deciphering the complex relationships between their ages at different times and translating them into mathematical equations. The problem states: "Dani said to Beto: 'Currently, I am twice as old as you were when I was your age, and when you are my age, between the two of us, we will sum 108 years.' How old is Dani and how old is Beto?" This seemingly simple statement holds a wealth of information that needs to be carefully extracted and organized to solve the problem. The key is to identify the different time points mentioned in the conversation – the present, a time in the past when Dani was Beto's age, and a time in the future when Beto will be Dani's age – and establish the relationships between their ages at these points. By representing their ages with variables and formulating equations based on the given information, we can systematically solve for the unknowns and determine their current ages. This problem exemplifies the power of algebra in modeling real-world scenarios and provides a stimulating exercise in logical thinking and problem-solving.
To effectively tackle this age-related puzzle, we need to meticulously dissect the given information and identify the key relationships between Dani and Beto's ages. The problem presents two crucial statements: 1) Dani's current age is twice Beto's age when Dani was Beto's age; and 2) When Beto is Dani's age, the sum of their ages will be 108 years. These statements provide the foundation for building our algebraic equations. The first statement involves a comparison of ages at two different points in time – the present and a past time when Dani was the same age as Beto is now. This implies that there is a time gap between these two instances, and we need to account for this time gap when establishing the relationship between their ages. The second statement looks into the future, projecting a scenario where Beto's age matches Dani's current age. This introduces another time gap, but this time, it's a forward projection. We need to consider how both Dani and Beto's ages will change over this time period. To make sense of these relationships, let's introduce variables to represent their current ages. Let Dani's current age be 'D' and Beto's current age be 'B'. Now, we can start translating the verbal statements into algebraic expressions. The challenge lies in accurately representing the past and future ages in terms of these variables. By carefully analyzing the wording of the problem and paying attention to the time gaps, we can successfully construct the equations that will lead us to the solution.
With the problem dissected and the key relationships identified, the next step is to translate the verbal statements into algebraic equations. This is where the power of algebra comes into play, allowing us to represent the unknown ages and their relationships in a concise and manageable form. Recall that we've defined Dani's current age as 'D' and Beto's current age as 'B'. Now, let's revisit the first statement: "Dani said to Beto: 'Currently, I am twice as old as you were when I was your age.'" This statement involves a comparison of ages at two different times: the present and a past time when Dani was Beto's age. Let's determine how long ago Dani was Beto's age. The time gap is the difference between their current ages, which is D - B. So, when Dani was Beto's age, Beto's age was B - (D - B) = 2B - D. Now we can translate the first statement into an equation: D = 2(2B - D). This equation represents the relationship between Dani's current age and Beto's age when Dani was Beto's age. Next, let's consider the second statement: "and when you have my age, between the two we will add 108 years". The time it will take for Beto to be Dani's current age is D - B. In that much time, Dani's age will be D + (D - B) = 2D - B and Beto's age will be D. Thus, the sum of the ages will be (2D - B) + D = 3D - B, which the problem states is 108. This can be expressed in the equation 3D - B = 108. We now have a system of two equations with two unknowns:
- D = 2(2B - D)
- 3D - B = 108
These equations provide a mathematical representation of the age puzzle. The next step is to solve this system of equations to determine the values of D and B, which represent Dani and Beto's current ages, respectively. By employing algebraic techniques such as substitution or elimination, we can systematically solve for the unknowns and unravel the mystery of their ages.
With the equations set up, the stage is set to solve for the unknowns and reveal the ages of Dani and Beto. We have the following system of equations:
- D = 2(2B - D)
- 3D - B = 108
Let's start by simplifying the first equation:
D = 4B - 2D
Add 2D to both sides:
3D = 4B
Now, we can express D in terms of B:
D = (4/3)B
This expression provides a direct relationship between Dani's age and Beto's age. Now, we can substitute this expression for D into the second equation:
3((4/3)B) - B = 108
Simplify:
4B - B = 108
Combine like terms:
3B = 108
Divide both sides by 3:
B = 36
So, Beto's current age is 36 years old. Now that we know Beto's age, we can substitute it back into the expression for D:
D = (4/3)(36)
Simplify:
D = 48
Therefore, Dani's current age is 48 years old. We have successfully solved the system of equations and determined the ages of Dani and Beto. By carefully translating the word problem into algebraic equations and employing systematic solving techniques, we have unraveled the mystery of their ages. The solution reveals that Dani is currently 48 years old, while Beto is 36 years old. This concludes the age puzzle solution.
With the solution in hand, it's crucial to verify its accuracy and ensure that it aligns with the original problem statement. This step is essential to confirm that we haven't made any errors in our calculations or interpretations. We found that Dani is 48 years old and Beto is 36 years old. Let's revisit the original statements and see if these ages satisfy the given conditions. The first statement was: "Dani said to Beto: 'Currently, I am twice as old as you were when I was your age.'" When Dani was Beto's age (36), the time elapsed was 48 - 36 = 12 years. So, 12 years ago, Beto was 36 - 12 = 24 years old. Is Dani's current age (48) twice Beto's age at that time (24)? Yes, 48 = 2 * 24. So, the first condition is satisfied. The second statement was: "and when you have my age, between the two we will add 108 years". When Beto is Dani's age (48), the time elapsed will be 48 - 36 = 12 years. In 12 years, Dani will be 48 + 12 = 60 years old, and Beto will be 36 + 12 = 48 years old. The sum of their ages will be 60 + 48 = 108 years. So, the second condition is also satisfied. Since both conditions are met, we can confidently conclude that our solution is accurate. This verification process not only confirms the correctness of the answer but also reinforces our understanding of the problem and the relationships between the variables. The meticulous approach of solving and verifying is a hallmark of mathematical rigor and ensures the reliability of the solution.
In this article, we embarked on a mathematical journey to solve a classic age problem involving Dani and Beto. By carefully dissecting the problem, translating the verbal statements into algebraic equations, and employing systematic solving techniques, we successfully unraveled the mystery of their ages. The problem presented a unique challenge, requiring us to consider ages at different points in time and establish the relationships between them. We learned the importance of defining variables, formulating equations, and verifying the solution to ensure accuracy. The solution revealed that Dani is currently 48 years old, while Beto is 36 years old. The verification process confirmed that these ages satisfy the conditions outlined in the problem statement. This exercise demonstrates the power of mathematical reasoning in solving real-world problems. Age problems, like many other mathematical puzzles, sharpen our logical thinking, problem-solving skills, and analytical abilities. They encourage us to break down complex scenarios into smaller, manageable parts and to identify patterns and relationships. The ability to translate real-world situations into mathematical models is a valuable skill that extends beyond the realm of mathematics and applies to various fields, including science, engineering, and finance. By engaging in such problems, we not only enhance our mathematical proficiency but also develop critical thinking skills that are essential for success in various aspects of life. The age puzzle of Dani and Beto serves as a testament to the elegance and applicability of mathematics in unraveling the complexities of the world around us.