Tu Hermano Es 5 Años Mayor Que Tú Y La Suma De Sus Edades Es 23 Años. ¿Cuántos Años Tienen Cada Uno?

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Figuring out age-related problems is a common mathematical task that requires careful thinking and setting up equations correctly. In this article, we will go through a typical problem step by step. This problem involves two people: you and your brother. The key information we have is that your brother is 5 years older than you, and the sum of your ages is 23 years. Our aim is to find out how old each of you is. Let’s break this down systematically.

Setting Up the Equations

In age-related problems, the first step often involves turning the word problem into mathematical equations. To do this, we need to assign variables to the unknown quantities. Let’s use 'x' to represent your current age. Since your brother is 5 years older than you, his age can be represented as 'x + 5'.

Now, we know that the sum of your ages is 23 years. This can be written as an equation:

x + (x + 5) = 23

This equation captures the essence of the problem. The left side, x + (x + 5), represents the sum of your age and your brother's age, and the right side, 23, represents the total sum of their ages. The goal now is to solve this equation for 'x', which will give us your age. Once we know your age, we can easily find your brother’s age by adding 5 to your age. This method of translating word problems into algebraic equations is fundamental in solving many mathematical problems, not just those involving ages.

Solving the Equation

Solving the equation involves several steps, each designed to simplify the equation until we isolate the variable 'x'. Here’s how we can do it:

  1. Combine Like Terms: In our equation, x + (x + 5) = 23, we have two 'x' terms. Combining these gives us 2x + 5 = 23. This step simplifies the equation and makes it easier to work with.
  2. Isolate the Variable Term: We want to get the term with 'x' by itself on one side of the equation. To do this, we subtract 5 from both sides of the equation: 2x + 5 - 5 = 23 - 5. This simplifies to 2x = 18. Subtracting the same number from both sides maintains the equality and moves us closer to solving for 'x'.
  3. Solve for x: Now, we have 2x = 18. To find 'x', we divide both sides of the equation by 2: 2x / 2 = 18 / 2. This gives us x = 9. Thus, your age is 9 years old.

Each of these steps is crucial in the process of solving algebraic equations. By combining like terms, isolating the variable term, and finally solving for the variable, we methodically work towards the solution. Understanding these steps is essential for tackling not just age-related problems, but a wide range of mathematical challenges.

Finding Your Brother's Age

Once we have determined your age, the next step is to find your brother's age. We know from the problem statement that your brother is 5 years older than you. We've already calculated your age to be 9 years old. To find your brother's age, we simply add 5 to your age:

Brother's age = Your age + 5

Substituting your age (9 years) into the equation:

Brother's age = 9 + 5

This gives us:

Brother's age = 14

So, your brother is 14 years old. This step is straightforward but crucial, as it completes the solution to the problem. We have now found both your age (9 years) and your brother's age (14 years). This illustrates how, once we solve for one variable in an equation, we can use that information to find other related unknowns.

Checking the Solution

After finding the ages, it is always a good practice to check the solution to ensure it satisfies the original problem statement. This step helps to verify the accuracy of our calculations and reasoning. We have found that you are 9 years old and your brother is 14 years old. The problem stated two conditions:

  1. Your brother is 5 years older than you.
  2. The sum of your ages is 23 years.

Let’s check the first condition: Is your brother (14 years old) 5 years older than you (9 years old)?

14 - 9 = 5

Yes, this condition is satisfied.

Now, let’s check the second condition: Is the sum of your ages 23 years?

9 + 14 = 23

Yes, this condition is also satisfied.

Since both conditions are met, our solution is correct. This checking process not only confirms the correctness of the answer but also reinforces the understanding of the problem and the steps taken to solve it. It’s a vital part of problem-solving in mathematics and other fields.

Conclusion

In conclusion, we have successfully solved the age problem by systematically breaking it down into manageable steps. First, we translated the word problem into a mathematical equation, representing your age as 'x' and your brother's age as 'x + 5'. We then formed the equation x + (x + 5) = 23, which represents the sum of your ages. We solved this equation to find your age, which turned out to be 9 years old.

Next, we used this information to find your brother's age by adding 5 to your age, resulting in 14 years. Finally, we checked our solution against the original problem statement to ensure accuracy, confirming that your brother is indeed 5 years older than you and that the sum of your ages is 23 years. This step-by-step approach highlights the importance of careful reading, accurate equation setup, methodical problem-solving, and thorough verification in mathematics. These skills are not only valuable in solving age-related problems but are also crucial for tackling a wide range of mathematical and real-world challenges.

This process underscores the importance of algebra in solving everyday problems and the logical steps required to arrive at the correct answer. By practicing these types of problems, individuals can improve their problem-solving skills and mathematical confidence.