The Age Riddle: Decoding the Relationship Between Parent and Child

Mathematics, often perceived as a realm of abstract concepts, can also entertain and challenge us with interesting puzzles. One such intriguing puzzle involves determining the age of a child and their parent based on a unique relationship. Here, we explore a mathematical puzzle and solve it step-by-step to highlight the beauty of logical reasoning and algebraic manipulation.

Introducing the Puzzle

Let us consider a situation where a man was 27 years old when his child was born, and now the parent is three times the child's age. How old is the child?

Setting Up the Equations

Let's denote the present age of the son by x years. Consequently, the present age of the man (the father) is 3x years. From the problem, we have two pieces of information:

Twelve years ago, the father's age was 3x - 12, and the son's age was x - 12. The father's age 12 years ago was 27 times the son's age at that time.

Formally, we can express this relationship as:

3x - 12 27(x - 12)

Solving the Equations

Now, we solve for x step-by-step:

Expand both sides of the equation: 3x - 12 27x - 324 Move all terms involving x to one side, and constants to the other: 3x - 27x -324 12 Combine like terms: -24x -312 Divide both sides by -24 to find x: x -312 / -24 13

Thus, the son is 13 years old, and the father, being three times that age, is 39 years old.

Verification

To verify our solution, we check if the conditions of the problem are met:

Twelve years ago, the son's age was 13 - 12 1 year, and the father's age was 39 - 12 27 years. Indeed, 27 is 27 times the son's age of 1 year. This confirms the solution is correct.

Discussion on Interpretation

It's important to note that the puzzle could have slight variations. For instance, if the phrase "three times" were meant to be "one-third," the solution would differ. Additionally, interpreting "three times more" as a ratio or an increase similarly leads to different solutions:

If "three times" in the context means one-third, the son's age would be 12 years, and the father's 21 years, which doesn't fit the given condition. Interpreting "three times more" as a ratio of 3:1 or 4:1 provides different ages for the son and the father, but neither fits the exact condition of the problem.

Closing Remarks

The solution to the riddle, where the father is 39 and the son is 13, stands out as the correct answer when the problem conditions are strictly followed. This riddle illustrates the importance of precise interpretation in mathematics and the joy of solving problems that bridge the gap between real-life scenarios and abstract concepts.