If p:q is the sub duplicate of p-x:q-x then x is

Solve by equation which is

P/Q = P-X/Q-X

PQ-PX = PQ-QX

PX-QX = 0

X = 0

If p:q is the sub duplicate ratio of p- x: q-x  then p-x:q-x is the duplicate ratio of p:q , ie

p²:q² = p-x:q-x

p²q - xp² = pq² -xq²

p²q - pq² = xp² - xq²

pq(p-q) = x(p²-q²)

pq(p-q) = x(p+q)(p-q)

x = pq/(p+q)

sad that an article could not solve this sum.