Let the present ages of A and B be x years and y years respectively.

Then, A’s age 5 years ago = (x – 5) years
and B’s age 5 years ago = (y – 5) years

According to the first condition, 

five years ago, A was thrice as old as B.

$\therefore$(x – 5) = 3(y – 5)

⇒ x – 5 = 3y – 15

⇒ x – 3y = –10  ...(i)

According to the second condition, 

ten years later, A shall be twice as old as B.

$\therefore$(x + 10) = 2(y + 10)

⇒ x + 10 = 2y + 20

⇒ x – 2y = 10  ...(ii)

Substracting equations (i) from (ii), we get:

y = 20 years

Now substituting the value of y in equation (ii), we get:

⇒ x – 2(20) = 10

⇒ x – 40 = 10

⇒ x = 50 years
Hence, A’s present age is 50 years.