Age Problem: Dr. Johnson and His Wife
Combined Age
Dr. Johnson and his wife have a total combined age of 91 years.
Age Relationship
Currently, Dr. Johnson is twice as old as his wife was when he was the same age as she is now. The objective is to determine their respective ages.
Defining Variables
Let Dr. Johnson’s age be represented as D, and his wife’s age as W. The relationships can be summarized with the following equations:
1. D + W = 91 (Equation 1)
Age Difference
The age difference between Dr. Johnson and his wife is expressed as D – W. The key relationship states:
Dr. Johnson’s current age (D) equals twice his wife’s age when he was her current age. This can be formulated as:
2 × (W – (D – W)) = D
Solving the Equation
Expanding this equation:
2 × (W – D + W) = D
2 × (2W – D) = D
4W – 2D = D
Rearranging gives:
4W = 3D
W = (3D) ÷ 4
Substituting into Equation 1
Substituting W in Equation 1 yields:
D + (3D ÷ 4) = 91
4D + 3D = 364
7D = 364
Calculating Ages
Solving for D results in:
D = 52
Using this value to find W:
W = 39
Conclusion
Dr. Johnson is 52 years old, while his wife is 39 years old.