2. If a + b = x + y  then ab ≠ xy. If ab  ≠ xy which is smaller ?  what will be its difference ?

 If the two numbers in the pair are widely separated, its product will be smaller.  e.g., in 7+11 = 5 +13 , 5 and 13 are widely separated than 7 and 11,  hence  the product 5 x 13 = 65 is smaller than the product 7 x 11= 77. 

Let a + b = x+y = 2k

a = (k-n) and b = (k+n) , ab = k² – n²

x = (k+m) and y = (k-m), xy = k² -  m² 

which shows that n= {(b-a)/2] and m = [(x-y)/2]

(a+b)²  =  (x+y)²

 

a² + b² + 2ab =  x²  + y²  + 2xy

(k-n)²  + (k+n)²  + 2ab = (k +m)²  + (k – m)²  + 2xy

 

2(k² + n² ) + 2ab =  2(k² + m² ) + 2xy

Or ab – xy = m²  -  n²  = [(x-y)/2]² – [(b-a)/2]²