Solving the Age Puzzle: A Mathematical Deconstruction

Age puzzles

Age puzzles can be intriguing and educational, challenging our understanding of algebra and logical reasoning. Here, we explore a classic age puzzle using both narrative and algebraic methods. Follow along as we solve the puzzle step-by-step, detailing each equation and its solution.

Problem Statement

The puzzle involves a mother and daughter. The problem is:

The present age of a woman is 30 years older than her daughter. 15 years ago, she was twice as old as her daughter. How old was the daughter at that time? How old will the woman be 12 years from now?

Narrative Solution

Let's denote the present age of the daughter as x years. Consequently, the present age of the woman would be x 30 years. According to the information provided:

15 years ago, the woman's age was x 30 - 15 x 15 years. 15 years ago, the daughter's age was x - 15 years.

We know that 15 years ago, the woman was twice as old as her daughter:

x 15 2(x - 15)

Now, let's solve this equation:

x   15  2x - 30

Subtract x from both sides:

15  x - 30

Add 30 to both sides:

x   45  0

This simplifies to:

x 45

So, the present age of the daughter is 45 years, and the present age of the woman is:

x 30 45 30 75 years old.

Now, to find out how old the woman will be 12 years from now:

75 12 87 years old.

Summary:

The daughter is currently 45 years old. The woman is currently 75 years old. The woman will be 87 years old in 12 years.

Algebraic Solution

Show that:

If M D 30 and M - 15 2(D - 15), solve for D.

Start with the equations:

M D 30 M - 15 2(D - 15)

Substitute the first equation into the second:

D 30 - 15 2(D - 15) or

D 15 2D - 30, which simplifies to:

D 45

Substituting back to M:

M 45 30 75

In 12 years, the woman will be:

75 12 87

Conclusion

Through both narrative and algebraic methods, we have solved the age puzzle. The daughter is currently 45 years old, and the woman is 75 years old. In 12 years, the woman will be 87 years old. This puzzle not only tests our algebraic skills but also our ability to interpret and apply real-world scenarios to mathematical equations. Understanding such puzzles can enhance critical thinking and analytical skills in various fields.