Question

If the ratio of A.M. between two positive real numbers a and b to their H.M. is \[m:n\], then \[a:b\] is
A. \[\frac{{\sqrt {m - n} + \sqrt n }}{{\sqrt {m - n} - \sqrt n }}\]
B. \[\frac{{\sqrt n + \sqrt {m - n} }}{{\sqrt n - \sqrt {m - n} }}\]
C. \[\frac{{\sqrt m + \sqrt {m - n} }}{{\sqrt m - \sqrt {m - n} }}\]
D. None of these

Answer 1

Hint
A positive real number is one that is bigger than zero, as well as a rational number or integer. Any real number other than zero or a negative one is not positive. Consequently, 0 is not actually positive. In reality, real numbers can be almost any number you can imagine. This can contain fractions, rational and irrational numbers, whole numbers or integers, and entire numbers. Real numbers include the number zero and can be either positive or negative.
A negative zero exists; it just so happens to be equal to the standard zero. The numbers that appear on the number line are real numbers. Natural numbers, integers, rational, and irrational numbers are all included in this. All numbers, whether they are positive or negative, big or small, whole or decimal, are numbers.
Formula used:
A.M is equal to \[\frac{{a + b}}{2}\]
And, H.M is equal to \[\frac{{2ab}}{{a + b}}\]

Complete step-by-step solution
Let us assume that
A.M is equal to \[\frac{{a + b}}{2}\]
And, H.M is equal to \[\frac{{2ab}}{{a + b}}\]
The given statement is
\[\frac{{AM}}{{HM}} = \frac{{{m}}}{n}\] which is then equal to
\[ = > \frac{{\frac{{a + b}}{2}}}{{\frac{{2ab}}{{a + b}}}} = \frac{m}{n}\]
This equation can also be written as
\[\frac{{{{(a + b)}^2}}}{{4ab}} = \frac{m}{n}\] ---(1)
Subtract both sides with one,
\[\frac{{{{(a + b)}^2}}}{{4ab}} - 1 = \frac{m}{n} - 1\]
By solving it becomes,
\[\frac{{{{(a + b)}^2} - 4ab}}{{4ab}} = \frac{{m - n}}{n}\]
\[\frac{{{{(a -b)}^2}}}{{4ab}} = \frac{{m - n}}{n}\] ---(2)
Divide the equations (1) and (2)
\[\frac{{{{(a + b)}^2}}}{{{{(a - b)}^2}}} = \frac{m}{{m - n}}\]
\[\frac{{(a + b)}}{{(a - b)}} = \frac{{\sqrt m }}{{\sqrt {m - n} }}\]
By the components of dividend, the equation becomes
\[\frac{{a + b + (a - b)}}{{(a + b) - (a - b)}} = \frac{{\sqrt m + \sqrt {m - n} }}{{\sqrt m - \sqrt {m - n} }}\]
\[ = > \frac{{2a}}{{2b}} = \frac{{\sqrt m + \sqrt {m - n} }}{{\sqrt m - \sqrt {m - n} }}\]
Which is finally calculated as
\[ = > \frac{{a}}{{b}} = \frac{{\sqrt m + \sqrt {m - n} }}{{\sqrt m - \sqrt {m - n} }}\]
Therefore, the correct option is C.

Note
The equation \[ G.M = > {(\sqrt {(A.M)(H.M)} )^2}\]
 can be used to show the relationship between AM, GM, and HM. The arithmetic mean (AM) and the harmonic mean (HM) are multiplied to create the geometric mean (GM). The value of AM is greater than that of GM and HM, as shown by a number of calculations and demonstrated by specialists who utilize AM, GM, and HM in statistics. In comparison to HM and AM, GM is worth more than the latter. Compared to AM and GM, HM has a lower value.