Question

If the ratio of two numbers be \[9:1\] , then the ratio of geometric and harmonic means between them will be
A. \[1:9\]
B. \[5:3\]
C. \[3:5\]
D. \[2:5\]

Answer 1

Hint
Before solving the problem, we have to know about ratio and harmonic means.
A particular kind of numerical average is the harmonic mean. It is determined by multiplying the total number of observations by each number in the series' reciprocal. All rates must be positive because the harmonic mean does not accept rates that have a negative or zero value.
The harmonic mean of the model's precision and recall is known as the F-score, which is a method of combining the model's precision and recall. The harmonic mean of recall and precision is the F1 score.
Formula used:
H.M between two number a and b is
\[H = \frac{{2ab}}{{a + b}}\]
G.M between two number a and b is
\[G = \sqrt {ab} \]

Complete step-by-step solution
The given equation is,
\[\frac{a}{b} = \frac{9}{1}\]
The equation can also be written as,
\[a = 9b\]
The formula used to solve the problem is,
\[H = \frac{{2ab}}{{a + b}}\]
The right side of the ratio is,
\[G = \sqrt {ab} \]
Then, the ratio becomes
\[H:G = \frac{{2ab}}{{a + b}}:\sqrt {ab} \]
This then is calculated as,
\[ = > \frac{{2.9{b^2}}}{{10b}}:3b\]
\[ = > \frac{3}{5}\]
So, the ratio of geometric and harmonic means between them is calculated as \[G:H = 5:3\]
Therefore, the correct option is B.

Note:
The Geometric Mean (GM) in mathematics is the average value or mean that, by calculating the product of the values of the set of numbers, denotes the central tendency of the numbers. In essence, we multiply the numbers together and calculate their nth root, where n is the total number of data values. The characteristics of the geometric mean are as follows: For a particular data collection, the geometric mean is always smaller than the arithmetic mean. The ratio of the geometric means of two series is equal to the ratio of the related geometric mean of those two series.