Solving Natalie's Age A Tricky Math Problem
Introduction
Age-related word problems often appear in mathematics, requiring us to translate real-world scenarios into algebraic equations. These problems not only test our understanding of algebraic principles but also our ability to apply logical reasoning. In this article, we will tackle a classic age problem involving Natalie and her son, Zac. The problem states that Natalie is currently five times as old as Zac. In eight years, she will be three times as old as her son. Our mission is to determine Natalie's current age by methodically setting up and solving algebraic equations. This exercise provides a practical application of algebra and highlights the importance of clear, logical thinking in problem-solving. By breaking down the problem into manageable steps and using algebraic techniques, we can successfully find the solution and understand the underlying mathematical relationships.
Setting Up the Equations
To begin solving this age problem, we must first translate the given information into algebraic equations. This involves identifying the unknowns and representing them with variables. In this scenario, the unknowns are Natalie's current age and Zac's current age. Let's denote Natalie's current age as N and Zac's current age as Z. The first piece of information we have is that Natalie is five times as old as Zac. This can be written as the equation:
N = 5Z
This equation establishes a direct relationship between Natalie's age and Zac's age. It tells us that whatever Zac's age is, Natalie's age is five times that amount. The second piece of information involves their ages in eight years. In eight years, Natalie's age will be N + 8, and Zac's age will be Z + 8. The problem states that in eight years, Natalie will be three times as old as Zac. This can be written as the equation:
N + 8 = 3(Z + 8)
This equation describes the relationship between their ages eight years into the future. It tells us that Natalie's age plus eight years will be three times Zac's age plus eight years. We now have two equations with two variables:
- N = 5Z
- N + 8 = 3(Z + 8)
These equations form a system of linear equations that we can solve to find the values of N and Z. Solving this system will give us Natalie's and Zac's current ages. The next step involves using these equations to find the values of our unknowns.
Solving the Equations
Now that we have our system of equations, the next step is to solve for the variables N and Z. We have two equations:
- N = 5Z
- N + 8 = 3(Z + 8)
One common method for solving a system of equations is substitution. Since the first equation already expresses N in terms of Z, we can substitute 5Z for N in the second equation. This will give us an equation with only one variable, which we can then solve. Substituting N with 5Z in the second equation, we get:
5Z + 8 = 3(Z + 8)
Now, we need to simplify and solve for Z. First, distribute the 3 on the right side of the equation:
5Z + 8 = 3Z + 24
Next, we want to isolate the variable Z. Subtract 3Z from both sides of the equation:
5Z - 3Z + 8 = 3Z - 3Z + 24
This simplifies to:
2Z + 8 = 24
Now, subtract 8 from both sides:
2Z + 8 - 8 = 24 - 8
This simplifies to:
2Z = 16
Finally, divide both sides by 2 to solve for Z:
2Z / 2 = 16 / 2
Z = 8
So, Zac's current age is 8 years old. Now that we have the value of Z, we can substitute it back into the first equation to find N, Natalie's age:
N = 5Z
N = 5(8)
N = 40
Therefore, Natalie's current age is 40 years old. We have successfully solved the system of equations using substitution, finding that Zac is 8 years old and Natalie is 40 years old.
Verifying the Solution
After finding a solution to a mathematical problem, it's essential to verify that the solution is correct. This involves plugging the values we found back into the original equations to ensure they hold true. In our case, we found that Natalie's current age (N) is 40 years, and Zac's current age (Z) is 8 years. Let's revisit our original equations:
- N = 5Z
- N + 8 = 3(Z + 8)
First, let's plug the values into the first equation:
40 = 5(8)
40 = 40
The first equation holds true. Now, let's plug the values into the second equation:
40 + 8 = 3(8 + 8)
48 = 3(16)
48 = 48
The second equation also holds true. Since both equations are satisfied by our values for N and Z, we can confidently say that our solution is correct. Natalie is currently 40 years old, and Zac is currently 8 years old. This verification step is crucial because it confirms that our algebraic manipulations and calculations were accurate and that our solution aligns with the problem's conditions. It reinforces the importance of double-checking our work to ensure the validity of our results.
Conclusion
In this article, we successfully solved an age-related problem using algebraic techniques. We were given that Natalie is currently five times as old as her son Zac, and in eight years, she will be three times as old as her son. To solve this, we first defined variables to represent their current ages (N for Natalie and Z for Zac) and then translated the given information into two algebraic equations:
- N = 5Z
- N + 8 = 3(Z + 8)
We then used the method of substitution to solve this system of equations. By substituting 5Z for N in the second equation, we were able to solve for Z, finding that Zac's current age is 8 years old. We then substituted this value back into the first equation to find Natalie's current age, which is 40 years old. To ensure the accuracy of our solution, we verified our results by plugging the values back into the original equations. Both equations held true, confirming that our solution is correct. This problem demonstrates the power of algebra in solving real-world scenarios. By translating word problems into algebraic equations, we can systematically find solutions. The process of setting up the equations, solving them, and verifying the solution is a fundamental skill in mathematics and is applicable in various fields. This exercise not only enhances our algebraic skills but also strengthens our logical reasoning and problem-solving abilities. Age problems like this serve as excellent examples of how mathematical concepts can be used to understand and solve everyday situations.