Question

If the ratio of H.M. and G.M between two numbers a and b is \[4:5\], then the ratio of the two numbers will be
A. \[1:2\]
B. \[2:1\]
C. \[4:1\]
D. \[1:4\]

Answer 1

Hint
The equation \[AM \times HM{\rm{ }} = {\rm{ }}G{M^2}\] can be used to show the relationship between AM, GM, and HM. Here, the square of the geometric mean is equal to the product of the arithmetic mean (AM) and harmonic mean (HM) (GM). In light of all the alternatives, it follows that the two numbers in the provided series must always be equal to one another. Take the nth root of the multiplied numbers after multiplying the numbers collectively, where n is the total number of data values.
A mathematical progression is said to be geometric if the ratio between any two consecutive terms is always the same. Simply put, it indicates that the next number in the series is determined by multiplying a fixed number by the number before it.
Formula use:
To find the ratio of two numbers,
H.M of a and b \[ = \frac{{2ab}}{{(a + b)}}\]
G.M of a and b \[ = \sqrt {(ab)} \]
 \[AM \times HM{\rm{ }} = {\rm{ }}G{M^2}\]
Complete step-by-step solution
To find the ratio of two numbers,
H.M of a and b \[ = \frac{{2ab}}{{(a + b)}}\]
G.M of a and b \[ = \sqrt {(ab)} \]
Ratio of HM and GM \[ = 4:5\]
By dividing both the equations, it becomes
\[ = > \frac{{(a + b)}}{{\sqrt {ab} }} = \frac{5}{2}\]
\[ = > \frac{a}{{\sqrt {(ab)} }} + \frac{b}{{\sqrt {(ab)} }} = \frac{5}{2}\]
\[ = > \sqrt {\frac{a}{b}} + \sqrt {\frac{b}{a}} = \frac{5}{2}\]
Substitute \[\sqrt {\frac{a}{b}} = x\]
\[ = > x + \frac{1}{x} = \frac{5}{2}\]
\[ = > 2{x^2} - 5x + 2 = 0\]
By solving, the equation becomes
\[x = \frac{1}{2}\]or \[2\]
As \[a > b\], \[\sqrt {(\frac{a}{b})} = 2\]
\[ = > \frac{a}{b} = 4\]
So, the ratio of two numbers is calculated as \[4:1\]
Therefore, the correct option is C.

Note
By typically dividing two figures, ratios contrast them.. . You are multiplying information A by information B, as evidenced by this. When two numbers are compared quantitatively, the ratio tells us how many times one value can fit inside another.
The number multiplied (or divided) at each point in a geometric series is referred to as the "common ratio" because you will always receive this value if you divide (that is, if you determine the ratio of)