If x is subtracted from each of the numbers 20, 37, 54 and 105, then the numbers so obtained in this order are in proportion. What is the mean proportional between (7x - 5) and (x + 1)?

8

The proportion : 

(20 - x) : (37 - x) :: (54 - x) : (105 - x)

Product of middle terms = Product of extreme terms 

(37 - x)(54 - x) = (20 - x)(105 - x)

1998 - 91x + x² = 2100 - 125x + x²

34x = 102

   x = 3

7x - 5 = 16, x + 1 = 4

Mean proportion to a, b = $\sqrt{a\; \times\; b}$

Mean proportion of 19, 4 = $\sqrt{16\; \times\; 4}$

                                    = 8