Seven years ago, the ages (in years) of A and B were in the ratio 4 : 5 and 7 years hence, their ages will be in the ratio 5 : 6. What will be the ratio of their ages 5 years from now?

Let Age of A = a, And ages B = b 

then according to question $$\dfrac {a-7}{b-7} = \dfrac {4}{5}$$

$$\Rightarrow  5a - 35 = 4b - 28 $$

$$\Rightarrow 5a -4b = 7 $$   ........ equestion (A)

again second case $$\dfrac{a + 7 }{b+7} = \dfrac {5} {6} $$

$$\Rightarrow 6a + 42 = 5b + 35 $$

$$\Rightarrow 6a - 5b = 7  $$    ......... equestion (B)

then Substract B from A 

so $$ 5a - 4b + 6a - 5b  = 0 $$

$$\Rightarrow  11 a = 9b $$

$$\Rightarrow a = \dfrac {9}{11}b $$  ...... equestion (C)

put the value in the equestion B

so 55 b - 54b = 77 

b = 77  so  we  put the value of b = 77 in equestion (C)

$$a = \dfrac {9}{11}\times 77 $$

a = 63 

then again  $$\dfrac {a +5}{b+ 5}$$

$$\Rightarrow \dfrac {63+5}{77 +5}$$

$$\Rightarrow \dfrac {68}{82}$$

$$\Rightarrow \dfrac {34}{41}$$ Ans