What is the relationship between Regina and Sebastián’s ages?

Understanding the relationship between Regina and Sebastián’s ages involves using algebraic equations to solve for unknown variables based on given conditions. Let’s break it down step-by-step.

Setting Up the Problem

First, let’s define the variables. Suppose Regina’s current age is $R$ and Sebastián’s current age is $S$. The relationship between their ages will depend on the conditions provided in the problem. For example, let’s consider the following scenario:

  1. Regina is 4 years older than Sebastián.
  2. In 5 years, the sum of their ages will be 40.

Step-by-Step Solution

  1. Translate the Conditions into Equations
    Based on the conditions given, we can write two equations:

    1. Regina is 4 years older than Sebastián:

    $R = S + 4$

    1. In 5 years, the sum of their ages will be 40:

    $(R + 5) + (S + 5) = 40$

  1. Simplify the Second Equation
    Simplify the second equation to make it easier to solve:

    $R + 5 + S + 5 = 40$

    $R + S + 10 = 40$

    $R + S = 30$

  1. Substitute the First Equation into the Second
    Now, substitute the expression for $R$ from the first equation into the second equation:

    $(S + 4) + S = 30$

    $2S + 4 = 30$

  1. Solve for $S$
    Isolate $S$ by subtracting 4 from both sides of the equation:

    $2S = 26$

    $S = 13$

  1. Find $R$
    Now that we know $S$, we can find $R$ using the first equation:

    $R = S + 4$

    $R = 13 + 4$

    $R = 17$

Conclusion

Based on the given conditions, Regina is 17 years old, and Sebastián is 13 years old. The relationship between their ages can be summarized as follows:

  • Regina is 4 years older than Sebastián.
  • In 5 years, the sum of their ages will be 40.

Additional Examples

To further understand how to solve age-related problems, let’s consider a few more examples:

Example 1: Different Conditions

Suppose the conditions are:

  1. Regina is twice as old as Sebastián.
  2. Five years ago, the sum of their ages was 30.

Let’s define the variables again:

$R = 2S$

Five years ago, their ages were $R – 5$ and $S – 5$, respectively. The sum of their ages five years ago was:

$(R – 5) + (S – 5) = 30$

Simplify and solve:

$R + S – 10 = 30$

$R + S = 40$

Substitute $R = 2S$ into the equation:

$2S + S = 40$

$3S = 40$

$S = frac{40}{3}$

$S text{ (Sebastián’s age) } text{ is approximately } 13.33$

$R = 2S$

$R = 2 times 13.33$

$R text{ (Regina’s age) } text{ is approximately } 26.67$

In this scenario, Regina is approximately 26.67 years old, and Sebastián is approximately 13.33 years old.

Example 2: Future Age Conditions

Suppose the conditions are:

  1. Regina is 3 years older than Sebastián.
  2. In 10 years, Regina will be twice as old as Sebastián.

Let’s define the variables again:

$R = S + 3$

In 10 years, their ages will be $R + 10$ and $S + 10$, respectively. The condition states that Regina will be twice as old as Sebastián in 10 years:

$R + 10 = 2(S + 10)$

Simplify and solve:

$R + 10 = 2S + 20$

$R = 2S + 10$

Substitute $R = S + 3$ into the equation:

$S + 3 = 2S + 10$

$3 – 10 = 2S – S$

$-7 = S$

In this scenario, Sebastián’s age is -7, which is not possible. Therefore, we need to re-evaluate the conditions or check for any errors in the given problem.

Conclusion

Solving age-related problems requires careful translation of the given conditions into algebraic equations. By defining variables, setting up equations, and solving step-by-step, we can determine the relationship between the ages of Regina and Sebastián. Practice with different scenarios will help you become more proficient in solving these types of problems.

Citations

  1. 1. Khan Academy – Solving Age Problems
  2. 2. Purplemath – Age Word Problems
  3. 3. Math is Fun – Age Problems

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