Question: The sum of the reciprocals of Rehman’s ages (in years) 3 years ago and 5 years from now is \\frac{1}{3}). Find his present age.
Video Explanation
Explanatory Answer
Let the present age of Rehman be x.
3 years ago his age would have been (x - 3).
5 years from now his age would be (x + 5).
Given: Sum of reciprocals of age 3 years ago and 5 years from now is \\frac{1}{3})
Age 3 years ago was (x - 3). So, reciprocal of age 3 years ago = \\frac{1}{x - 3}) ---------------- (1)
Age 5 years from now is (x + 5). So, reciprocal of age 5 years from now = \\frac{1}{x + 5}) ----------------(2)
Sum of (1) and (2) = \\frac{1}{x - 3}) + \\frac{1}{x + 5}) = \\frac{1}{3})
Because x cannot be 3 or -5 multiply the equation by (x - 3) (x + 5)
(x + 5) + (x – 3) = \\frac{1}{3})(x - 3) (x + 5)
2x + 2 = \\frac{1}{3})(x – 3) (x + 5)
Multiply the equation by 3
3(2x + 2) = (x – 3) (x + 5)
6x + 6 = x2 + 5x - 3x - 15
x2 - 4x - 21 = 0
Find the values of x by factorizing the quadratic equation
x2 - 7x + 3x - 21 = 0
x(x – 7) + 3(x – 7) = 0
(x – 7) (x + 3) = 0
x = 7 and x = -3
x is the present age of Rehman. Present age cannot be negative.
Hence, Rehman’s present age is 7 years.