Equation Representing Grandmother's Age Relative To Sarah's Age

In this article, we will delve into a fascinating age-related problem involving Sarah and her grandmother. We are given that Sarah's grandmother is 25 years older than twice Sarah's age. Our goal is to determine the equation that accurately represents the grandmother's age (g) in relation to Sarah's age (s). This exercise not only tests our understanding of algebraic equations but also highlights how mathematical relationships can be expressed in real-life scenarios. The problem requires us to translate a verbal description into a mathematical equation, a fundamental skill in algebra. We will carefully dissect the given information, identify the key variables, and construct an equation that captures the essence of the relationship between Sarah's age and her grandmother's age. By the end of this discussion, you will have a clear understanding of how to approach similar problems and confidently formulate equations based on given conditions. The ability to translate real-world scenarios into mathematical expressions is a cornerstone of problem-solving in various fields, making this exercise highly valuable. Moreover, understanding such relationships can help in making predictions and estimations in various contexts, from financial planning to scientific research. This problem serves as an excellent example of how mathematics can be applied to everyday situations, making it more relatable and engaging. Let's embark on this mathematical journey and unravel the equation that connects Sarah's age to her grandmother's age.

Dissecting the Problem Statement

To effectively solve the problem, we must first meticulously dissect the given information. The core statement is: "Sarah's grandmother is 25 years older than twice Sarah's age." This statement contains several key components that need to be translated into mathematical symbols and operations. First, we identify the two primary variables: Sarah's age, which we will denote as 's', and her grandmother's age, which we will denote as 'g'. The phrase "twice Sarah's age" indicates that we need to multiply Sarah's age (s) by 2, resulting in 2s. The phrase "25 years older than" signifies that we need to add 25 to the quantity 2s. Therefore, the grandmother's age (g) is equal to 2s plus 25. This careful breakdown of the statement allows us to construct the equation step by step. It's crucial to pay attention to the order of operations and the precise meaning of each phrase. For instance, "25 years older than" implies addition, while "twice" implies multiplication. A clear understanding of these linguistic cues is essential for accurate translation into mathematical expressions. Furthermore, recognizing the relationship between the variables is key to forming the correct equation. In this case, the grandmother's age is dependent on Sarah's age, making 'g' the dependent variable and 's' the independent variable. This distinction helps in structuring the equation in the appropriate form. By thoroughly analyzing the problem statement, we lay the groundwork for a successful solution. This methodical approach not only helps in solving this particular problem but also equips us with a valuable strategy for tackling other mathematical challenges. Now that we have a clear understanding of the statement, let's move on to formulating the equation.

Formulating the Equation

Now that we have carefully dissected the problem statement, we are ready to formulate the equation that represents the relationship between Sarah's age and her grandmother's age. Recall that we identified Sarah's age as 's' and her grandmother's age as 'g'. The statement "twice Sarah's age" translates to 2s, and "25 years older than" means we add 25 to this quantity. Therefore, the grandmother's age (g) is equal to 2s plus 25. This can be written as the equation: g = 2s + 25. This equation accurately captures the given relationship, where the grandmother's age is determined by doubling Sarah's age and adding 25. It is crucial to ensure that the equation reflects the correct order of operations and the relationship between the variables. In this case, the equation clearly shows that 'g' is dependent on 's', meaning the grandmother's age is calculated based on Sarah's age. This equation is a linear equation, representing a straight-line relationship between the two variables. The coefficient 2 indicates the rate at which the grandmother's age increases with Sarah's age, and the constant 25 represents the additional years the grandmother has beyond twice Sarah's age. The equation g = 2s + 25 is a concise and accurate representation of the given information, allowing us to easily calculate the grandmother's age for any given value of Sarah's age. This process of translating a verbal statement into a mathematical equation is a fundamental skill in algebra and is widely applicable in various problem-solving scenarios. With the equation formulated, we can now compare it to the given options and identify the correct answer. This step will solidify our understanding and demonstrate the effectiveness of our approach.

Analyzing the Options

With the equation g = 2s + 25 successfully formulated, the next step is to analyze the provided options and identify the one that matches our equation. The given options are:

a. g = 2s + 25 b. s = 2g + 25 c. g = 25 - 2s d. s = 25 - 2g

By comparing each option to our derived equation, g = 2s + 25, it becomes clear that option a, g = 2s + 25, is the correct answer. This option perfectly matches the equation we formulated, where the grandmother's age (g) is equal to twice Sarah's age (s) plus 25. The other options can be easily ruled out. Option b, s = 2g + 25, expresses Sarah's age in terms of the grandmother's age, but it does not accurately represent the given relationship. It incorrectly states that Sarah's age is twice the grandmother's age plus 25. Option c, g = 25 - 2s, suggests that the grandmother's age is 25 minus twice Sarah's age, which contradicts the problem statement. This option implies that the grandmother's age decreases as Sarah's age increases, which is not the case. Option d, s = 25 - 2g, similarly misrepresents the relationship, indicating that Sarah's age is 25 minus twice the grandmother's age. This option also implies an inverse relationship that is not supported by the problem statement. The process of elimination further reinforces that option a is the only correct choice. By carefully comparing our formulated equation with each option, we can confidently select the answer that accurately represents the given relationship. This exercise highlights the importance of not only formulating the equation correctly but also verifying it against the provided choices. Now that we have identified the correct equation, let's delve deeper into its implications and applications.

Implications and Applications of the Equation

The equation g = 2s + 25, which accurately represents the relationship between Sarah's age and her grandmother's age, has several implications and practical applications. This equation allows us to easily determine the grandmother's age for any given value of Sarah's age. For example, if Sarah is 10 years old, we can substitute s = 10 into the equation: g = 2(10) + 25 = 20 + 25 = 45. This tells us that the grandmother is 45 years old when Sarah is 10. Similarly, if Sarah is 20 years old, the grandmother's age would be g = 2(20) + 25 = 40 + 25 = 65 years old. The equation also reveals the linear nature of the relationship. For every year Sarah ages, her grandmother ages by two years, plus the constant difference of 25 years. This constant difference is significant as it represents the age gap between the grandmother and twice Sarah's age. The equation can also be rearranged to express Sarah's age in terms of her grandmother's age. By subtracting 25 from both sides, we get g - 25 = 2s. Dividing both sides by 2, we obtain s = (g - 25) / 2. This form of the equation allows us to calculate Sarah's age if we know her grandmother's age. For instance, if the grandmother is 75 years old, Sarah's age would be s = (75 - 25) / 2 = 50 / 2 = 25 years old. This versatility of the equation makes it a valuable tool for understanding and predicting the ages of Sarah and her grandmother at different points in their lives. Furthermore, this type of age-related problem is a common application of linear equations in algebra. Understanding how to formulate and manipulate such equations is crucial for problem-solving in various mathematical and real-world contexts. The ability to translate verbal descriptions into mathematical expressions is a fundamental skill that extends beyond age-related problems and is applicable in areas such as finance, physics, and engineering.

Conclusion

In conclusion, we have successfully identified the equation that represents the relationship between Sarah's age and her grandmother's age. By carefully dissecting the problem statement, we translated the verbal description into a mathematical equation, g = 2s + 25. This equation accurately captures the given relationship, where the grandmother's age (g) is 25 years older than twice Sarah's age (s). We analyzed the provided options and confirmed that option a, g = 2s + 25, is the correct answer. This exercise highlights the importance of translating real-world scenarios into mathematical expressions, a fundamental skill in algebra and problem-solving. The equation we derived not only provides a concise representation of the age relationship but also allows us to calculate the grandmother's age for any given value of Sarah's age, and vice versa. We explored the implications and applications of the equation, demonstrating its practical utility in predicting ages at different points in time. The linear nature of the equation reveals a consistent relationship between the ages, with the grandmother aging two years for every year Sarah ages, plus the constant age difference of 25 years. This type of age-related problem is a classic example of how linear equations can be applied in everyday situations. The process of formulating and manipulating equations is a crucial skill that extends beyond mathematics and is applicable in various fields, including finance, science, and engineering. By mastering these skills, we enhance our ability to analyze and solve problems effectively. This exercise serves as a valuable lesson in translating verbal statements into mathematical expressions, a skill that will undoubtedly prove useful in future endeavors. The ability to connect mathematical concepts to real-world scenarios is a key aspect of mathematical literacy and is essential for success in various academic and professional pursuits.