Age Problems, Solving Equations, And Algebraic Expressions

Let's delve into the age-related questions concerning Kojo. This section will address how to calculate someone's age in the past and project their age into the future, a fundamental concept in algebra and everyday calculations.

When dealing with age problems, the core idea is to represent the current age with a variable and then use mathematical operations to find the age in the past or future. In this case, Kojo's current age is represented by the variable n. To determine his age 5 years ago, we need to subtract 5 from his current age. This is because time moves backward, so we are essentially rewinding the clock. The expression representing Kojo's age 5 years ago is therefore n - 5. This simple subtraction allows us to find his age at any point in the past, assuming we know his current age. For instance, if Kojo is currently 20 years old, then 5 years ago, he was 20 - 5 = 15 years old. This principle applies regardless of the current age; we always subtract the number of years from the present age to find the past age.

Conversely, to determine Kojo's age 10 years from now, we need to add 10 to his current age. This is because time is moving forward, so we are essentially fast-forwarding the clock. The expression representing Kojo's age 10 years from now is n + 10. This addition allows us to project his age into the future, again based on his current age. If Kojo is currently 20 years old, then 10 years from now, he will be 20 + 10 = 30 years old. This principle of addition holds true for any future time projection, making it a versatile tool for age-related calculations. Understanding these basic mathematical operations of subtraction and addition is crucial for solving a wide range of age-related problems and building a solid foundation in algebraic thinking. By mastering these concepts, you can confidently tackle more complex age problems and apply these principles to various real-life situations where age calculations are necessary.

Solving for t: A Linear Equation

Now, let's shift our focus to solving for the variable t in the equation 3t - 2(t + 12) = 11. This involves applying the principles of algebra to isolate t and find its value. This is a classic example of solving a linear equation, a fundamental skill in algebra.

The first step in solving this equation is to simplify it by expanding the terms within the parentheses. We have the term -2(t + 12), which needs to be distributed. This means we multiply -2 by both t and 12. Multiplying -2 by t gives us -2t, and multiplying -2 by 12 gives us -24. So, the equation now becomes 3t - 2t - 24 = 11. By distributing the -2, we have eliminated the parentheses and made the equation easier to work with. This distributive property is a crucial tool in simplifying algebraic expressions and is used extensively in solving equations.

Next, we need to combine like terms on the left side of the equation. We have two terms with t: 3t and -2t. Combining these terms means adding their coefficients, which are 3 and -2. Adding 3 and -2 gives us 1, so the combined term is 1t, which is simply written as t. Therefore, the equation simplifies to t - 24 = 11. Combining like terms allows us to consolidate the variable terms, making the equation more manageable and bringing us closer to isolating the variable.

To isolate t, we need to get rid of the constant term -24 on the left side. We do this by adding 24 to both sides of the equation. Adding 24 to the left side cancels out the -24, leaving us with just t. Adding 24 to the right side gives us 11 + 24 = 35. So, the equation now becomes t = 35. This step is based on the principle of maintaining equality; whatever operation we perform on one side of the equation, we must perform on the other side to keep the equation balanced. Finally, we have found the value of t, which is 35. This means that when t is equal to 35, the original equation 3t - 2(t + 12) = 11 holds true. Solving linear equations is a fundamental skill in algebra, and mastering these steps allows you to tackle a wide variety of mathematical problems. By carefully simplifying the equation and isolating the variable, we can determine its value and gain a deeper understanding of the relationships between variables and constants.

Projecting Kofi's Age: A Future Outlook

Let's consider the final piece of our mathematical puzzle: Kofi's age in the next 10 years. This question, while seemingly simple, underscores the importance of understanding how time affects age and how we can represent future ages algebraically.

To determine Kofi's age in the next 10 years, we need to know his current age. However, the prompt only provides a context—a discussion category of mathematics—without explicitly stating Kofi's present age. This is a common scenario in mathematical problems, where we are given a situation and asked to make logical deductions or form a general expression rather than finding a specific numerical answer. In this case, we can represent Kofi's current age with a variable, say x. This allows us to create a general expression for his age in 10 years, regardless of his current age. Using a variable is a fundamental technique in algebra, as it allows us to represent unknown quantities and work with them mathematically.

If Kofi's current age is x, then his age in 10 years will be x + 10. This is because we are adding 10 years to his present age to project his future age. The expression x + 10 captures the essence of this age projection. For instance, if Kofi is currently 15 years old, then in 10 years, he will be 15 + 10 = 25 years old. Similarly, if Kofi is currently 30 years old, then in 10 years, he will be 30 + 10 = 40 years old. The addition of 10 years remains constant, while the current age, represented by x, can vary. This flexibility is one of the key advantages of using algebraic expressions.

By representing Kofi's future age as x + 10, we have created a mathematical model that can be applied to any current age. This demonstrates the power of algebra in generalizing and solving problems. Rather than focusing on a specific numerical answer, we have developed a formula that describes the relationship between Kofi's current age and his age in 10 years. This approach is particularly useful when dealing with problems where the specific values are not given, or when we want to understand the general pattern or relationship between variables. Understanding how to represent future ages algebraically is a valuable skill in mathematics and has applications in various fields, including finance, demographics, and computer science. It allows us to make predictions and analyze trends based on current data and mathematical models.

In summary, by working through these questions related to Kojo's and Kofi's ages, and solving for the variable t, we have explored fundamental concepts in algebra, including calculating past and future ages, solving linear equations, and using variables to represent unknown quantities. These skills are essential for building a strong foundation in mathematics and for tackling more complex problems in the future.