Question : When $x$ is subtracted from each of 19, 28, 55 and 91, the numbers so obtained in this order, are in proportion. What is the mean proportional between ($x$ + 9) and $x$²?

Option 1: 27

Option 2: 32

Option 3: 28

Option 4: 24

Correct Answer: 28

Solution : When $x$ is subtracted from each of 19, 28, 55, and 91, the obtained numbers are in proportion.
Using the concept of Componendo and Dividendo Rule
If $\frac ab=\frac cd$, then $\frac{a+b}{a-b}=\frac{c+d}{c-d}$
Given numbers are 19, 28, 55, 91
According to the question,
$(19 - x), (28 - x), (55 - x)\text{ and }(91 - x)$ are proportional
⇒ $\frac{19−x}{28−x}=\frac{55−x}{91−x}$
Appling Componendo-Dividendo method, we get
⇒ $\frac{(19−x)+(28−x)}{(19−x)−(28−x)}=\frac{(55−x)+(91−x)}{(55−x)−(91−x)}$
⇒ $\frac{47−2x}{−9}=\frac{146−2x}{−36}$
⇒ $188 - 8x = 146 - 2x$
⇒ $6x = 42$
⇒ $x = 7$
⇒ mean proportional = $\sqrt{7 + 9) × 7^2}$
= 28
Hence, the correct answer is 28.