In the figure there are 3 semicircles touching each other internally and one circle touching 2 of them externally and third one internally. Radius of complete circle is

Hey, thank you for asking the question
without the diagram the explanation is difficult but i will try to explain as best as possible

let center of bigger semicircle(radius=3) is R1 ,the next bigger one (radius=2) is R2 and the smaller one(radius=1) is R3 and complete circle one is R (assume its radius is r )

Now consider the triangle R2 R3 R and draw a line from R to R1 it acts as a cevian on this trisngle lengths of triangle

R2 R3 = 3

R2 R =2+R

R3 R=1+R

R1 R=3-R  [bigger circle is normal and complete circle are same so radius of both coincide]

R2 R1=1

R3 R1=2

now apply cosine rule at point R1

(2+r)²-1²-(3-r)²/2*1*(3-r)   = -[(1+r)²-2²-(3-r)²/2*2*(3-r)]

solving this we get   r=  12/14=6/7

p+q=13