Let the present age of A be A years and the present age of B be B years.
According to the problem:
- Five years ago, A’s age was A – 5 and B’s age was B – 5.
- At that time, the relationship was: A – 5 = 4(B – 5)
Expanding the equation:
- A – 5 = 4B – 20
- This simplifies to: A = 4B – 15 (Equation 1)
Now, let’s consider the second condition:
- Five years from now, A’s age will be A + 5 and B’s age will be B + 5.
- The relationship at that time will be: A + 5 = 2(B + 5)
Expanding this equation gives us:
- A + 5 = 2B + 10
- Simplifying this results in: A = 2B + 5 (Equation 2)
Now we have a system of two equations:
- Equation 1: A = 4B – 15
- Equation 2: A = 2B + 5
Since both equations equal A, we can set them equal to each other:
- 4B – 15 = 2B + 5
Simplifying this equation:
- 4B – 2B = 5 + 15
- This leads to: 2B = 20
- Thus, B = 10 years
Now, we can substitute the value of B back into one of our equations to find A’s age:
- Using Equation 2: A = 2(10) + 5 = 20 + 5
- A = 25 years
Therefore, the present ages are:
- A is 25 years old.
- B is 10 years old.