Problem on ages 2

let $R$ be the age of Ram and $G$ be the age of his Grandpa in year $2002$

$R=\frac{1}{2}G$ $\color{#D61F06}(1)$

The sum of their ages at the end of year $2002$ is $2(2002)-3854=150$ . So

$R+G=150$ $\color{#D61F06}(2)$

Substituting $\color{#D61F06}(1)$ in $\color{#D61F06}(2)$ , we get

$G=100$

Now, solving for $R$ , we have

$R=50$

Finally, $1$ year after $2002$ ,

$R+1=$ $\boxed{\color{#20A900}51}$

Consider the age of grandpa as x. So, age of Ram is 0.5x. Then the sum of the years is 2002 - x + 2002 - 0.5x = 3854 .... x = 100. How the age of Ram in 2003 is 0.5x + 1 = 0.5(100) +1 = 51.