The Math Behind Family Ages: Solving Age Riddles

Introduction

Mathematics can be a fascinating tool for solving everyday puzzles, especially when it comes to family age problems. These types of problems challenge our understanding of relationships between numbers and help us think logically. In this article, we will dive into a series of family age problems and solve them step-by-step to better understand the mathematical concepts involved.

Problem 1: The 42-Year-Old Man and His 12-Year-Old Son

A man is 42 years old, and his son is 12 years old. We want to find out how many years ago the man's age was six times that of his son.

Step-by-Step Solution:

Define variables: Let x be the number of years ago when the man's age was six times that of his son. Write the equations:

The man's age x years ago: 42 - x

The son's age x years ago: 12 - x

Set up the equation:

42 - x 6(12 - x)

Expand and simplify:

42 - x 72 - 6x

42 - 72 -6x x

-30 -5x

x 6

Verification:

Man's age 6 years ago: 42 - 6 36

Son's age 6 years ago: 12 - 6 6

36 6 x 6, hence the solution is correct.

Additional Problems

Problem 2: Father and Son Riddle

The original answer provided here is incorrect. Let's solve it correctly.

Solution:

Let the current age of the son be x years.

Then the father's current age would be 2x years.

12 years ago:

Son's age: x - 12

Father's age: 2x - 12

According to the problem, 12 years ago, the father's age was six times the son's age:

2x - 12 6(x - 12)

Expand and simplify:

2x - 12 6x - 72

2x - 6x -72 12

-4x -60

x 15

Therefore, the son's current age is 15 years.

And the father's current age is 30 years.

Problem 3: Sum of Ages 5 Years Ago

Let the current age of the son be x years.

Then the father's current age would be x 35 (since the total age 5 years ago was 35).

5 years ago:

Son's age: x - 5

Father's age: x 35 - 5 x 30

We know that 5 years ago, the father's age was six times the son's age:

x 30 6(x - 5)

Expand and simplify:

x 30 6x - 30

x - 6x -30 - 30

-5x -60

x 12

Therefore, the son's current age is 12 years, and the father's current age is 47 years.

Conclusion

Solving age problems is a great way to practice mathematical reasoning and algebraic manipulations. Whether it's determining the age of a father and son, or the sum of their ages at a previous point, understanding these types of problems helps build strong foundational skills in mathematics. By applying these techniques, you can solve complex age riddles with confidence.