When x is added to each of 22, 26, 19 and 21, then the numbers so obtained, in this order, are in proportion. Then, if 2x : y :: y : (4x - 8), and y > 0, what is the value of y?

Given:

Numbers: 22, 26, 19, 21

When x is added, they are in proportion: (22 + x) : (26 + x) :: (19 + x) : (21 + x)

Second proportion: 2x : y :: y : (4x - 8), with y > 0

Formula used:

If a : b :: c : d, then a/b = c/d or ad = bc

If a : b :: b : c, then b² = ac (for mean proportion)

Calculations:

First, find x using the first proportion:

(22 + x)/(26 + x) = (19 + x)/(21 + x)

⇒ (22 + x)(21 + x) = (26 + x)(19 + x)

⇒ 22 × 21 + 22x + 21x + x² = 26 × 19 + 26x + 19x + x²

⇒ 462 + 43x + x² = 494 + 45x + x²

⇒ 462 + 43x = 494 + 45x

⇒ 43x - 45x = 494 - 462

⇒ -2x = 32

⇒ x = -16

Now, use the second proportion with the value of x:

2x : y :: y : (4x - 8)

Since y is the mean proportional, y² = (2x)(4x - 8)

Substitute x = -16:

⇒ y² = (2 × -16) × (4 × -16 - 8)

⇒ y² = (-32) × (-64 - 8)

⇒ y² = (-32) × (-72)

⇒ y² = 2304

⇒ y = \(\sqrt{2304}\)

⇒ y = 48 (Since y > 0)

∴ The value of y is 48.