The sum of the ages of A and B (in years) is 22. The product of their ages two years back was 77. Which one of the following is the value of the difference of their current ages?

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  1. 2
  2. 3
  3. 4
  4. 5

Answer (Detailed Solution Below)

Option 3 : 4
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The Correct answer is Option 3. 

Key PointsLet's denote the current ages of A and B as x and y respectively. According to the problem:
⇒ x+y=22
⇒ The product of their ages two years ago was 77, so 
⇒ (x−2)(y−2)=77.
Expanding the second equation:
⇒ (x−2)(y−2)=xy−2x−2y+4=77

Substitute 
⇒ x+y=22 into the equation:

⇒ xy = 117 

So, we have:
⇒ x+y=22
⇒ xy=117
⇒ The quadratic equation whose roots are 
x and y is:
⇒ 𝑡−(x+y)t+xy=0
⇒ 𝑡−22t+117=0

⇒ Solve this quadratic equation using the quadratic formula t

⇒ So, the ages of A and B are 13 years and 9 years.

⇒ The difference between their current ages is: 13−9=4

Thus, the correct answer is: 3) 4

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