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Mathematical Puzzle: Determining Ages

Introduction

Dr. Johnson encountered a family in the park, consisting of a grandfather, a father, and a son. Curious about their ages, he posed a question to them.

The Family’s Age Statements

The grandfather responded, “All together we are 100 years old.” The father added, “My son and I together are 45 years old, and my son is 25 years younger than I am.”

Solving the Age Problem

To find their individual ages, we can define the following variables:
– X1 = age of the grandfather
– X2 = age of the father
– X3 = age of the son

Based on their statements, we can create the following equations:
1. X1 + X2 + X3 = 100 (Total age of the family)
2. X2 + X3 = 45 (Combined age of the father and son)
3. X2 – X3 = 25 (Age difference between the father and son)

Calculating the Ages

By solving this system of equations, we find:
– X1 = 55 (Grandfather’s age)
– X2 = 35 (Father’s age)
– X3 = 10 (Son’s age)

Conclusion

Thus, the ages of the family members are 55, 35, and 10 years old, respectively.