Ages of Alex and Jake: A Mathematical Enigma

Mathematical puzzles involving ages can often challenge our understanding of basic algebraic equations and how to manipulate them to find solutions. In this article, we will solve a given problem by breaking it down into manageable steps and applying algebraic principles. The problem is as follows:

Given Problem

Four years ago, Alex was twice as old as Jake. Four years from now, Jake will be (frac{3}{4}) of Alex's age. How old will Alex be after three years?

Step-by-Step Solution

Step 1: Define the Variables

Let J be Jake's current age and A be Alex's current age.

Step 2: Set up the Equations

From the first statement, we know that four years ago, Alex was twice as old as Jake. So, four years ago, Alex's age was A - 4 and Jake's age was J - 4. Given the condition, we can write the equation:

[ A - 4 2(J - 4) ]

Simplify the equation:

[ A - 4 2J - 8 ] [ A 2J - 4 ]

This is our first equation (1).

Step 3: Use the Second Statement to Formulate Another Equation

From the second statement, we know that four years from now, Jake will be (frac{3}{4}) of Alex's age. In four years, Jake's age will be (J 4) and Alex's age will be (A 4). We can write the equation:

[ J 4 frac{3}{4}(A 4) ]

Substitute the first equation (A 2J - 4) into this equation:

[ J 4 frac{3}{4}(2J - 4 4) ]

Simplify the equation:

[ J 4 frac{3}{4}(2J) ] [ J 4 frac{6J}{4} ] [ J 4 frac{3J}{2} ]

Now, solve for (J):

[ frac{3J}{2} - J 4 ] [ frac{3J - 2J}{2} 4 ] [ frac{J}{2} 4 ] [ J 8 ]

So, Jake's current age is 8 years.

Step 4: Find Alex's Current Age

Using the first equation (A 2J - 4), substitute (J 8):

[ A 2(8) - 4 ] [ A 16 - 4 ] [ A 12 ]

Therefore, Alex's current age is 12 years.

Step 5: Determine Alex's Age After Three Years

The question asks for Alex's age after three years. So, we add 3 years to Alex's current age:

[ A 3 12 3 15 ]

Therefore, Alex will be 15 years old after three years.

Conclusion

Through algebraic manipulation and logical reasoning, we have determined that Alex will be 15 years old after three years. This problem highlights the importance of setting up and solving equations based on given conditions to find the desired solution.

Further Reading

For more age-related problems and exercises, you might want to explore: Solving Age Problems with Algebra Algebraic Equations for Beginners Math Puzzles with Solutions