Solving the Family Age Puzzle

Imagine a scenario where a father's age is intricately connected to his son's age through the lens of simple yet sophisticated mathematical reasoning. This family age puzzle can be solved using algebraic techniques, making it a delightful challenge for students and enthusiasts alike. In this article, we will walk through the solution, explaining the steps and providing a clear understanding of how to approach similar problems.

Understanding the Puzzle

Let's introduce the problem: A father is three times as old as his son. In 14 years, the father will be twice as old as his son. How old is the father now?

Setting Up the Equations

The first step is to define the variables:

Let x be the son's current age. The father's current age is 3x.

Now, let's translate the second condition into an equation. In 14 years, the father's age will be 3x 14, and the son's age will be x 14. The problem states that the father will be twice as old as his son in 14 years. Therefore, we can write:

3x 14 2(x 14)

Solving the Equation

Let's simplify and solve the equation step by step:

Step 1: Expand the right side of the equation.

3x 14 2x 28

Step 2: Isolate the variable x by subtracting 2x from both sides.

3x - 2x 14 28

Step 3: Simplify.

x 14 28

Step 4: Subtract 14 from both sides to solve for x.

x 14

Now that we have determined the son's age, let's find the father's age.

The father's current age is 3x 3(14) 42 years old.

Let's verify the solution by checking both conditions:

In the present, the son is 14 years old, and the father is 42 years old (3 times 14). In 14 years, the son will be 28 years old, and the father will be 56 years old (42 14 56, which is twice 28).

Additional Verification with Simultaneous Equations

To further validate our solution, let's use a system of simultaneous equations.

Setting Up the Simultaneous Equations

Let F be the father's age and S be the son's age.

F 3S F 14 2(S 14)

Substitute F 3S into the second equation:

3S 14 2(S 14)

Simplify and solve for S:

3S 14 2S 28

3S - 2S 28 - 14

S 14

Substitute S 14 into F 3S:

F 3(14) 42

Conclusion and Proof

The solution is consistent:

The father is 42 years old, and the son is 14 years old now. In 14 years, the father will be 56 years old, and the son will be 28 years old, which satisfies the given conditions.

Final Answer

The father is currently 42 years old. This solution is robust and can be used to explore similar age puzzles in mathematics.