Question

If 2s:3s is the duplicate ratio of 2s-p:3t-p then

Answer 1

Hello,
you have that:
2s / 3t = ( 2s -p)² / (3t -p)²;
solving is obtained:
2s( 3t -p )² = 3t( 2s - p )²;
2s( 9t² - 6tp + p² ) = 3t(4s² -4sp+ p²);
18st²-12spt+2p²s= 12s²t-12spt+3p²t;
18st²-12s²t= 3p²t - 2p²s;
6st(3t -2s) = p²(3t - 2s);
6st = p²
then:
p² = 6st

Bye :-)

Answer 2

It seems there is a typing mistake in the given ratio. It could be 2 s : 3 t.  Duplicate ratio  of x : y = x² : y².

Given  2 s : 3 t  = (2 s - p)² : (3 t - p)²
Or,       2s / (3t) =  (2s-p)² / (3 t - p)²

We cross-multiply the terms from both sides. 

         2s (3t-p)² = 3t (2s-p)²
=>     2s(9t² - 6 tp + p²) = 3t (4s² - 4sp + p²)
=>     18 st² - 12 stp + 2 sp² = 12 ts² - 12 tsp + 3tp²
=>     18 st² - 12 ts² + 2 sp² - 3 tp² = 0
=>     6 t s (3 t - 2 s) + p² (2 s - 3 t) = 0
 =>    (3 t - 2 s) (6 t s - p²) = 0

=>   So   3t = 2s    or    p² = 6 t s
      This is the answer.