The correct answer is option (A):
14 years
Let's break down this age problem step by step. We'll use variables to represent the unknowns and set up equations based on the given information.
First, let's denote Rajesh's current age as R and Shiv's current age as S.
From the problem, "Eight years ago, Rajesh was half as old as Shiv." This translates to the equation:
R - 8 = (1/2)(S - 8)
Simplifying this equation, we get:
2R - 16 = S - 8
2R - S = 8 (Equation 1)
Next, we are told, "the ratio of their ages after 4 years becomes 3:4." This means:
(R + 4) / (S + 4) = 3/4
Cross-multiplying this gives us:
4(R + 4) = 3(S + 4)
4R + 16 = 3S + 12
4R - 3S = -4 (Equation 2)
Now, we have two equations with two variables. We can solve this system of equations. Let's multiply Equation 1 by -3:
-6R + 3S = -24 (Equation 3)
Now add Equation 2 and Equation 3:
4R - 3S = -4
-6R + 3S = -24
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-2R = -28
Dividing both sides by -2:
R = 14
Therefore, Rajesh's current age is 14 years old. We didn't need to solve for Shiv's age to answer the question, but we could substitute R = 14 back into either equation to find S = 20.