When x is added to each of 9, 15, 21 and 31, the numbers so obtained are in proportion. What is the mean proportional between the numbers (3x–2) and (5x+4) ?

35

ATQ ,

⇒ \(\frac{9 + x}{15 + x}\) = \(\frac{21 + x}{31 + x}\)

Using componendo and dividendo 

⇒ \(\frac{(15 + x)\;+\;(9 + x)}{(15 + x)\;-\;(9 + x)}\) = \(\frac{(31 + x)\;+\;(21 + x)}{(31 + x)\;-\;(21 + x)}\)

⇒ \(\frac{24 + 2x}{6}\) = \(\frac{52 + 2x}{10}\)

⇒ 60 + 5x = 78 + 3x

⇒ x = 9

⇒ Mean proportional of a and b = \(\sqrt {(a)(b)}\)

⇒ Mean proportional = \(\sqrt {(3x - 2)(5x + 4) }\) = \(\sqrt {25 \times 49}\) = 35