Correct Answer:
B. 8 years
The correct answer is 8 years. We can find Ali's present age by setting up an equation based on the given condition about the reciprocals of his past and future ages.
Step-by-Step Solution
Let Ali's present age be x years.
- His age 2 years ago was: x − 2
- His age 4 years from now will be: x + 4
The sum of the reciprocals of these two ages is given as 1/4:
1/(x − 2) + 1/(x + 4) = 1/4
To solve, we find a common denominator:
- (x + 4) + (x − 2) over (x − 2)(x + 4) = 1/4
- (2x + 2) / ((x − 2)(x + 4)) = 1/4
Cross‑multiply:
- 4(2x + 2) = (x − 2)(x + 4)
- 8x + 8 = x² + 2x − 8
- Bring all terms to one side: 0 = x² − 6x − 16
Factor the quadratic:
- x² − 6x − 16 = (x − 8)(x + 2) = 0
- So x = 8 or x = −2
Since age cannot be negative, Ali's present age is 8 years.
Verification
Check the condition with x = 8:
- Age 2 years ago = 6; reciprocal = 1/6
- Age 4 years from now = 12; reciprocal = 1/12
- Sum = 1/6 + 1/12 = (2 + 1)/12 = 3/12 = 1/4 ✔
Why Other Options Are Incorrect
- 6 years: Past age = 4, future = 10; sum of reciprocals = 1/4 + 1/10 = 0.35, not 0.25.
- 10 years: Past = 8, future = 14; sum = 1/8 + 1/14 ≈ 0.196, not 0.25.
- 12 years: Past = 10, future = 16; sum = 1/10 + 1/16 = 0.1625, not 0.25.
Therefore, the only age that satisfies the equation is 8 years, making it the correct answer.