When x is added to each of 8, 14, 20 and 30 the numbers obtained are in proportion. What is the mean proportional between the numbers (4x - 4) and (8x + 1)?

Question: When x is added to each of 8, 14, 20 and 30 the numbers obtained are in proportion. What is the mean proportional between the numbers (4x - 4) and (8x + 1)?

Solution:
If x is added to each of 8, 14, 20 and 30 then, the number so obtained in this form a proportion
According to the question
(8 + x) : (14 + x) : : (20 + x) : (30 + x)
⇒ (8 + x)/(14 + x) = (20 + x)/(30 + x)
Using Componendo and Dividendo Rule
⇒ (8 + x + 14 + x)/(8 + x - 14 - x) = (20 + x + 30 + x)/(20 + x - 30 - x)
⇒ (22 + 2x)/(- 6) = (50 + 2x)/(- 10)
⇒ (11 + x)/3 = (25 + x)/5
⇒ 5(11 + x) = 3(25 + x)
⇒ 55 + 5x = 75 + 3x
⇒ 5x - 3x = 75 - 55
⇒ 2x = 20
∴ x = 20/2 = 10

As we know,
Mean proportion of two numbers a and b = √(a × b).

Mean proportion:
√{(40 - 4).(80 + 1)}
= √(36 × 81)
= 6 × 9
= 54