Solving the Age Puzzle: When Will Diana Be Half as Old as Her Mother?

Mrs. Guzman is currently 32 years old, and her daughter Diana is just 8 years old. A common puzzle in mathematics and algebra is to determine how old Diana will be when she is half as old as her mother. In this article, we will explore the solution to this puzzle and walk through the steps to find the answer.

Setting Up the Problem

To solve this problem, we can set up a simple algebraic equation. Let's denote the number of years from now when Diana will be half as old as her mother as ( x ).

Current Ages

Mrs. Guzman's current age: 32 years old Diana's current age: 8 years old

Ages in ( x ) years

Mrs. Guzman's age in ( x ) years: ( 32 x ) Diana's age in ( x ) years: ( 8 x )

Setting Up the Equation

We want to find ( x ) when Diana's age is half of Mrs. Guzman's age. We can write the equation as follows:

( 8 x frac{1}{2}(32 x) )

Solving the Equation

Let's solve this equation step by step:

Multiply both sides by 2 to eliminate the fraction:

2(8 x) 32 x

Expand and simplify:

16 2x 32 x

Rearrange the equation to isolate ( x ):

2x - x 32 - 16

Simplify to find ( x ):

x 16

Calculating Diana's Age

After 16 years, Diana's age will be:

8 16 24

At that time, Mrs. Guzman's age will be:

32 16 48

Diana will be 24, which is indeed half of 48.

Conclusion

We have determined that Diana will be half as old as her mother in 16 years. This results in Diana being 24 years old, while Mrs. Guzman will be 48 years old. This problem is a classic example of using algebra to solve real-world age problems.

Further Reading

For those interested in exploring more age-related puzzles and algebraic solutions, consider delving into:

The Mathematical Puzzle: When Will Sarah Be Half as Old as Her Mother? Algebraic Solutions for Real-World Age Problems A Collection of Age Puzzles and Solutions