Question : 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 $(7 x-5)$ and $(x+1) $?
Option 1: 8
Option 2: 9
Option 3: 6
Option 4: 12
Correct Answer: 8
Solution :
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.
So, $\frac{(20-x)}{(37-x)} = \frac{(54-x)}{(105-x)}$
$⇒[(20-x)(105-x)]=[(54-x)(37-x)]$
$⇒34x=102$
$⇒x = 3$
The mean proportional between $(7x-5)$ and $(x+1)$ is the square root of their product. So, the mean proportional is $\sqrt{(7x-5)(x+1)}$.
Substituting $x = 3$ into the above expression, we get $\sqrt{(7\times3-5)(3+1)} = \sqrt{(16)(4)} = 8$
Hence, the correct answer is 8.
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