Solving Age Relationships in Algebraic Equations

Age relationships can be a fascinating area to explore, especially when expressed through algebraic equations. One such intriguing puzzle involves a mother and her son, whose ages are linked mathematically. Let's delve into solving this puzzle step-by-step.

The Problem

A mother is three times as old as her son at the moment, and in 11 years, she will be twice as old as her son. What are their present ages?

Step-by-Step Solution

To solve this puzzle, we can start by defining the current ages of the mother and the son with variables. Let:

Y 3X

In this equation, Y represents the mother's current age, and X represents the son's current age. According to the puzzle, the mother is three times as old as her son.

Y 11 2 (X 11)

This equation represents the situation in 11 years, where the mother's age will be twice that of her son. We need to use the first equation to express Y in terms of X and then substitute it into this equation.

Solving the Equations

Substitute Y with 3X in the second equation:

3X 11 2(X 11)

Expand and simplify:

3X 11 2X 22

Subtract 2X from both sides:

X 11 22

Subtract 11 from both sides:

X 11

So, the son's current age is 11 years. Now, use the first equation to find the mother's age:

Y 3X 3 × 11 33

Therefore, the mother's current age is 33 years.

Verification

To verify the solution, let's check the conditions given in the problem:

Currently, the son is 11 years old, and the mother is 33 years old. This satisfies the condition that the mother is three times as old as her son.

In 11 years, the son will be 11 11 22 years old, and the mother will be 33 11 44 years old. This satisfies the condition that the mother will be twice as old as the son.

Conclusion

The solution to the problem is that the son is currently 11 years old, and the mother is 33 years old. This puzzle not only demonstrates the power of algebraic equations but also illustrates how such problems can be solved through systematic and logical steps.