Solving For Tina's Age An Equation-Based Approach
In the realm of mathematics, word problems often present real-life scenarios that require us to translate verbal information into mathematical equations. These equations then serve as powerful tools for solving for unknown quantities. In this article, we'll delve into a specific age-related problem, dissecting the given information, constructing an appropriate equation, and ultimately determining the age of a particular individual. This exercise will not only enhance our problem-solving skills but also provide valuable insights into the practical applications of algebraic concepts.
Decoding the Problem: Cole's Age and Tina's Age
Our problem revolves around the ages of two individuals, Cole and Tina. We are presented with the crucial piece of information that Cole's age is 3 years less than his sister Tina's age, which we denote by the variable t. Furthermore, we are given the concrete fact that Cole is 18 years old. The challenge lies in representing this situation using an equation and subsequently solving for Tina's age.
To effectively tackle this problem, we must first carefully analyze the relationship between Cole's age and Tina's age. The statement "Cole's age is 3 years less than Tina's age" implies that we can obtain Cole's age by subtracting 3 from Tina's age. Mathematically, this relationship can be expressed as:
Cole's age = Tina's age - 3
Now, we introduce the variable t to represent Tina's age. Substituting t into the equation, we get:
Cole's age = t - 3
Since we know that Cole is 18 years old, we can replace "Cole's age" with 18 in the equation:
18 = t - 3
This equation, 18 = t - 3, is the algebraic representation of the given situation. It encapsulates the relationship between Cole's age and Tina's age, allowing us to solve for the unknown quantity, t, which represents Tina's age.
Constructing the Equation: Translating Words into Symbols
The process of translating a word problem into a mathematical equation is a fundamental skill in algebra. It requires us to carefully dissect the given information, identify the key relationships between the quantities involved, and express these relationships using mathematical symbols.
In our problem, the statement "Cole's age is 3 years less than his sister Tina's age" is the cornerstone for constructing the equation. To translate this statement, we first represent Tina's age using the variable t. Then, we recognize that "3 years less than Tina's age" implies subtraction. Therefore, we subtract 3 from t to represent Cole's age.
This leads us to the expression t - 3, which represents Cole's age in terms of Tina's age. Since we are given that Cole is 18 years old, we can equate the expression t - 3 to 18, resulting in the equation:
t - 3 = 18
This equation is the algebraic embodiment of the given information. It succinctly captures the relationship between Cole's age and Tina's age, providing us with a powerful tool for solving for the unknown quantity, t.
The ability to construct equations from word problems is a crucial skill in mathematics. It allows us to transform real-world scenarios into mathematical models, which can then be analyzed and solved using algebraic techniques. This process involves identifying the key variables, recognizing the relationships between these variables, and expressing these relationships using mathematical symbols and operations.
Solving the Equation: Unveiling Tina's Age
Now that we have successfully constructed the equation t - 3 = 18, our next step is to solve for t, which represents Tina's age. To solve for t, we need to isolate it on one side of the equation. This can be achieved by performing the same operation on both sides of the equation, ensuring that the equality remains balanced.
In our equation, t is being subtracted by 3. To isolate t, we need to undo this subtraction. The inverse operation of subtraction is addition. Therefore, we will add 3 to both sides of the equation:
t - 3 + 3 = 18 + 3
Simplifying both sides of the equation, we get:
t = 21
This result tells us that t, which represents Tina's age, is equal to 21. Therefore, Tina is 21 years old.
The process of solving an equation involves applying algebraic principles to isolate the unknown variable. This often involves performing inverse operations on both sides of the equation to maintain balance and gradually reveal the value of the variable. In our case, we used the addition property of equality to isolate t and determine Tina's age.
Verifying the Solution: Ensuring Accuracy
After solving an equation, it's always a good practice to verify the solution to ensure its accuracy. This involves substituting the obtained value back into the original equation and checking if it satisfies the equation.
In our case, we found that t = 21. To verify this solution, we substitute 21 for t in the original equation, t - 3 = 18:
21 - 3 = 18
Simplifying the left side of the equation, we get:
18 = 18
This confirms that our solution, t = 21, is correct because it satisfies the original equation. When we substitute 21 for t, the equation holds true, indicating that Tina's age is indeed 21 years old.
Verifying the solution is a crucial step in the problem-solving process. It helps us catch any errors we might have made during the solution process and ensures that our answer is accurate and reliable. This step provides us with confidence in our solution and validates our understanding of the problem.
The Answer: Tina's Age Revealed
Through our step-by-step analysis and solution, we have successfully determined Tina's age. We began by carefully dissecting the word problem, identifying the relationship between Cole's age and Tina's age. We then translated this relationship into an algebraic equation, t - 3 = 18, where t represents Tina's age.
Solving this equation, we found that t = 21. This means that Tina is 21 years old. We further verified our solution by substituting 21 for t in the original equation, confirming its accuracy.
Therefore, the answer to our problem is that Tina is 21 years old.
This problem-solving exercise has demonstrated the power of algebraic equations in representing real-world scenarios and solving for unknown quantities. By carefully translating verbal information into mathematical symbols and applying algebraic techniques, we were able to unravel the age relationship between Cole and Tina and determine Tina's age.
Key Takeaways: Mastering Age-Related Problems
This problem serves as a valuable illustration of how algebraic concepts can be applied to solve practical problems. Here are some key takeaways from this exercise:
- Translate verbal information into mathematical equations: The ability to translate word problems into algebraic equations is crucial for solving them. This involves identifying the key variables, recognizing the relationships between these variables, and expressing these relationships using mathematical symbols and operations.
- Construct equations based on given relationships: In age-related problems, carefully analyze the relationships between the ages of different individuals. Express these relationships using algebraic expressions and equations.
- Solve equations using algebraic techniques: Once you have constructed an equation, use algebraic techniques to isolate the unknown variable and solve for its value. This often involves performing inverse operations on both sides of the equation to maintain balance.
- Verify your solution: Always verify your solution by substituting the obtained value back into the original equation. This ensures that your solution is accurate and reliable.
By mastering these key takeaways, you will be well-equipped to tackle a wide range of age-related problems and other mathematical challenges. Remember, practice is key to developing your problem-solving skills. The more you practice, the more confident and proficient you will become in applying mathematical concepts to real-world scenarios.
In conclusion, this exercise has not only helped us determine Tina's age but also provided valuable insights into the problem-solving process. By carefully analyzing the given information, constructing an appropriate equation, and applying algebraic techniques, we have successfully unraveled the age relationship between Cole and Tina. This underscores the power of mathematics in making sense of the world around us and solving practical problems.
