An Age Puzzle: A Man and His Son

Today, we will explore a classic age-related puzzle, involving a man and his son, and solve it using simple algebraic equations. This type of problem enhances our analytical skills and provides a fun challenge. The story goes:

A man is 3 times older than his son. In ten years, the sum of their ages will be 76. What are their current ages?

Defining Variables and Setting Up Equations

To solve this problem, we will define the variables for the current ages of the man and his son. Let's denote the son's current age as ( S ) and the man's current age as ( M ).

Since the man is 3 times older than his son, we can express the man's age as:

(M 3S)

In ten years, the son's age will be ( S 10 ) and the man's age will be ( M 10 ). According to the problem, the sum of their ages in ten years will be 76:

(M 10 S 10 76)

Substituting ( M 3S ) into the equation:

(3S 10 S 10 76)

Solving the Equation

Now, let's simplify and solve for ( S ):

Combine like terms: ( 3S S 10 10 76 ) ( 4S 20 76 ) Subtract 20 from both sides: ( 4S 56 ) Divide both sides by 4: ( S 14 )

So, the son's current age is 14 years. Now, we can find the man's age:

(M 3S 3 times 14 42)

Therefore, the man's current age is 42 years.

Verification

In ten years, the son's age will be ( 14 10 24 ) and the man's age will be ( 42 10 52 ). The sum of their ages in ten years is:

(24 52 76)

This confirms that our solution is correct. The man is currently 42 years old and the son is 14 years old.

Additional Insights

Let's explore another way to solve the problem:

If the son is ( x ) years old, then the man is ( 3x ) years old. In ten years, the son will be ( x 10 ) and the man will be ( 3x 10 ). The sum of their ages in ten years is:

(x 10 3x 10 76)

Simplifying the equation:

(4x 20 76)

Subtract 20 from both sides: ( 4x 56 ) Divide both sides by 4: ( x 14 )

The son is 14 years old, and the man is ( 3 times 14 42 ) years old.

When the son was born, 14 years ago, the man was ( 42 - 14 28 ) years old. This means the man was 28 years old in the year the son was born, while the son was less than one year old.