Answer
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Hint: We need to have a clear understanding of the given question to proceed further. According to the question statement, try to form an equation and proceed further.
Let the present age of Ramesh be $x$ and the present age of Rahim be $y$.
It is given that the ratio of ages of Ramesh and Rahim =5:7
$ \Rightarrow \frac{x}{y} = \frac{5}{7}$ .... (1)
It is given that the age of Ramesh would be twice the age of Rahim when Ramesh is 9 years older than his present age and Rahim is 9 years younger than his present age.
$ \Rightarrow \left( {x + 9} \right) = 2\left( {y - 9} \right)$
Simplifying the above equation we get
$ \Rightarrow 2y - x = 27$
Solving for $y$ (or we can also solve for $x$)
From equation (1) we can substitute $x = \left( {\frac{5}{7}} \right)y$, then
$ \Rightarrow 2y - \left( {\frac{5}{7}} \right)y = 27$
$ \Rightarrow y\left( {\frac{{14 - 5}}{7}} \right) = 27$
$$ \Rightarrow y = \left( {\frac{{27 \times 7}}{9}} \right)$$
$ \Rightarrow y = 21$
Substituting $y$ value in equation (1) we get
$ \Rightarrow x = \frac{5}{7} \times 21 = 15$
$\therefore $The age of Ramesh is 15 years and the age of Rahim is 21 years.
Note: While solving the problems based on ages we have to read and understand the problem carefully. We have to identify the logical relations given to form equations in terms of variables. Then solve the equations to get the values for variables which will give us their ages.
Let the present age of Ramesh be $x$ and the present age of Rahim be $y$.
It is given that the ratio of ages of Ramesh and Rahim =5:7
$ \Rightarrow \frac{x}{y} = \frac{5}{7}$ .... (1)
It is given that the age of Ramesh would be twice the age of Rahim when Ramesh is 9 years older than his present age and Rahim is 9 years younger than his present age.
$ \Rightarrow \left( {x + 9} \right) = 2\left( {y - 9} \right)$
Simplifying the above equation we get
$ \Rightarrow 2y - x = 27$
Solving for $y$ (or we can also solve for $x$)
From equation (1) we can substitute $x = \left( {\frac{5}{7}} \right)y$, then
$ \Rightarrow 2y - \left( {\frac{5}{7}} \right)y = 27$
$ \Rightarrow y\left( {\frac{{14 - 5}}{7}} \right) = 27$
$$ \Rightarrow y = \left( {\frac{{27 \times 7}}{9}} \right)$$
$ \Rightarrow y = 21$
Substituting $y$ value in equation (1) we get
$ \Rightarrow x = \frac{5}{7} \times 21 = 15$
$\therefore $The age of Ramesh is 15 years and the age of Rahim is 21 years.
Note: While solving the problems based on ages we have to read and understand the problem carefully. We have to identify the logical relations given to form equations in terms of variables. Then solve the equations to get the values for variables which will give us their ages.