Solving the Age Puzzle: The Sum of Ages of Three Children Born at Intervals of 3 Years

Introduction: This article explores an interesting age puzzle where the sum of the ages of three children is 33 years. The children were born at intervals of 3 years each. We will use algebra and interval arithmetic to solve the problem step by step, ensuring clarity and coherence in the solution process.

Problem Statement

We need to determine the age of the eldest child given that the sum of their ages is 33 years and they were born at intervals of 3 years each.

Algebraic Approach to Solving the Problem

Let's denote the age of the youngest child as x years. Then:

Youngest child: x years Middle child: x 3 years Eldest child: x 6 years

The sum of their ages is given as:

x (x 3) (x 6) 33

Let's simplify and solve this equation step by step:

Simplify the equation: x x 3 x 6 33 3x 9 33 Subtract 9 from both sides: 3x 24 Divide both sides by 3: x 8

Now, we can find the ages of the children:

Youngest child: 8 years Middle child: 8 3 11 years Eldest child: 8 6 14 years

Therefore, the age of the eldest child is 14 years.

Alternative Method: Using Arithmetic Progression

Another approach to solving the problem involves recognizing that the ages of the three children form an arithmetic progression. In an arithmetic progression, the sum of the terms is three times the middle term. Given the sum of the ages is 33, we can find the middle term using the following steps:

The sum of the ages (33) divided by 3 gives the middle term: 33 / 3 11 The middle term is 11, so the ages of the children are 8, 11, and 14.

Therefore, the age of the eldest child is also 14 years.

Algebraic Manipulation: Another Approach

Another valid method involves expressing the ages in terms of the eldest child's age. Let n be the age of the eldest child. Then the ages of the children are:

Eldest child: n years Middle child: n - 3 years Youngest child: n - 6 years

The sum of their ages is:

n (n - 3) (n - 6) 33

Let's simplify and solve this equation:

Simplify the equation: n n - 3 n - 6 33 3n - 9 33 Add 9 to both sides: 3n 42 Divide both sides by 3: n 14

Thus, the age of the eldest child is 14 years.

Conclusion

We have explored multiple methods to solve the age puzzle and consistently arrived at the same solution: the age of the eldest child is 14 years. This puzzle not only tests basic algebraic skills but also highlights the importance of recognizing patterns and utilizing interval arithmetic effectively.