Solution:
Let present ages of Simmi and Niti be a and b years respectively.
Ten years hence, the ratio between Simmi's age and Niti's age = 7 : 9
According to question
$$\frac{{a + 10}}{{b + 10}} = \frac{7}{9}$$
By cross multiplying we get
$$\eqalign{
& \Rightarrow 9a + 90 = 7b + 70 \cr
& \Rightarrow 7b - 9a = 20......(i) \cr
& {\text{Also}},\frac{{a - 2}}{{b - 2}} = \frac{1}{3} \cr} $$
By cross multiplying we get
$$\eqalign{
& \Rightarrow 3a - 6 = b - 2 \cr
& \Rightarrow 3a - b = 4.....(ii) \cr} $$
Multiplying equation (ii) by 3
$$9a - 3b = 12.....(iii)$$
Adding equation (ii) and (iii) we get
$$\eqalign{
& - 9a + 7b = 20 \cr
& 4b = 32 \cr
& b = 8{\text{ year}} \cr} $$
From equation (ii) we get
$$\eqalign{
& a = \frac{{4 + b}}{3} \cr
& \,\,\,\,\, = \frac{{4 + 8}}{3} \cr
& \,\,\,\,\,\, = \frac{{12}}{3} \cr
& \,\,\,\,\,\, = 4\,{\text{year}} \cr} $$
Since, Abhay is 4 years older to Niti
So, Abhay's present age = 8 + 4 = 12 years
