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 ?
[amp_mcq option1=”2″ option2=”3″ option3=”4″ option4=”5″ correct=”option3″]
This question was previously asked in
UPSC CAPF – 2024
Given: a + b = 22.
Two years back, their ages were (a-2) and (b-2).
Given: (a-2)(b-2) = 77.
Expanding the second equation: ab – 2a – 2b + 4 = 77
ab – 2(a + b) + 4 = 77.
Substitute the first equation (a + b = 22) into this:
ab – 2(22) + 4 = 77
ab – 44 + 4 = 77
ab – 40 = 77
ab = 117.
We need to find the difference of their current ages, which is |a – b|.
We use the identity: (a – b)² = (a + b)² – 4ab.
Substituting the values we have:
(a – b)² = (22)² – 4(117)
(a – b)² = 484 – 468
(a – b)² = 16.
Therefore, |a – b| = √16 = 4.
The difference in their current ages is 4 years.
Factoring the quadratic: (x – 9)(x – 13) = 0.
The roots are x = 9 and x = 13.
So, the ages are 9 and 13 years.
Let’s check the conditions:
Sum of ages = 9 + 13 = 22 (Correct).
Ages two years back were 9-2 = 7 and 13-2 = 11.
Product of ages two years back = 7 * 11 = 77 (Correct).
The difference in their current ages is |13 – 9| = 4.