If the A. M. of two numbers a and \(b\) is equal to \(\sqrt{(10) \text { times }}\) their G. M. then \([(a-b) /(a+b)]=\) (A) \([\sqrt{(10) / 3]}\) (B) \(3 \sqrt{(10)}\) (C) \([9 /(10)]\) (D) \([3 / \sqrt{(10)}]\)

To find the A.M. of two numbers (a and b), we use the following formula: \[A.M. = \frac{a+b}{2}\]

To find the G.M. of two numbers (a and b), we use the following formula: \[G.M. = \sqrt{ab}\]

We are given that the A.M. of two numbers a and b is equal to the square root of 10 times their G.M. So, we can write \[\frac{a+b}{2} = \sqrt{10} \times \sqrt{ab}\] Now, we need to simplify this equation to get the relationship between 'a' and 'b'.

To simplify the equation, we can square both sides of the equation: \[\left(\frac{a+b}{2}\right)^2 = 10ab\] Expanding and simplifying this equation further: \[(a+b)^2 = 40ab\] \[a^2 + 2ab + b^2 = 40ab\] \[a^2 - 38ab + b^2 = 0\]

To get the value of (a-b) / (a+b), we will now square the equation: \[(a-b)^2 = a^2 - 2ab + b^2\] Since we found in Step 4 that \(a^2 - 38ab + b^2 = 0\), we can now replace \(a^2 - 2ab + b^2\) in the above equation: \[(a-b)^2 = 38ab\] Divide both sides by 38: \[\frac{(a-b)^2}{38} = ab\] Now, we want to find the value of (a-b) / (a+b). Let's divide both sides of the equation by ab: \[\frac{(a-b)^2}{38ab} = 1\] Notice that (a-b)^2 / (a+b)^2 can be written as ((a-b)/(a+b))^2. Therefore, we can write the equation as: \[\frac{(a-b)^2}{38(ab)} = \frac{(a-b)^2}{(a+b)^2}\] Now, we have: \[\frac{(a-b)^2}{(a+b)^2} = \frac{1}{38}\] Taking the square root of both sides: \[\frac{(a-b)}{(a+b)} = \sqrt{\frac{1}{38}}\] So, the value of (a-b) / (a+b) is: \[\frac{(a-b)}{(a+b)} = \sqrt{\frac{1}{38}}\] Comparing this result with the given options, we find that none of the options matches our solution.