If the ratio of `H.M.` and `G.M.` between two numbers `a` and `b` is `4:5`, then find the ratio of the two number ?

Options

A. `4:1`
B. `3:2`
C. `3:4`
D. `2:3`

Explanation

To solve the problem, we need to find the ratio of two numbers \( a \) and \( b \) given that the ratio of their Harmonic Mean (H.M.) to their Geometric Mean (G.M.) is \( 4:5 \).

### Step-by-Step Solution:

1. **Understanding the Means**:
The Harmonic Mean (H.M.) of two numbers \( a \) and \( b \) is given by:
\[ H.M. = \frac{2ab}{a + b} \]
The Geometric Mean (G.M.) of two numbers \( a \) and \( b \) is given by:
\[ G.M. = \sqrt{ab} \]

2. **Setting Up the Ratio**:
According to the problem, we have:
\[ \frac{H.M.}{G.M.} = \frac{4}{5} \]
Substituting the formulas for H.M. and G.M. into the equation, we get:
\[ \frac{\frac{2ab}{a + b}}{\sqrt{ab}} = \frac{4}{5} \]

3. **Simplifying the Equation**:
This can be simplified to:
\[ \frac{2ab}{(a + b)\sqrt{ab}} = \frac{4}{5} \]
Cross-multiplying gives:
\[ 5 \cdot 2ab = 4(a + b)\sqrt{ab} \]
Simplifying further:
\[ 10ab = 4(a + b)\sqrt{ab} \]

4. **Dividing by \( \sqrt{ab} \)**:
Dividing both sides by \( \sqrt{ab} \) (assuming \( ab \neq 0 \)):
\[ 10\sqrt{ab} = 4\left(\frac{a + b}{\sqrt{ab}}\right) \]
Let \( x = \frac{a}{b} \), then \( \sqrt{ab} = \sqrt{b^2x} = b\sqrt{x} \) and \( a + b = b(x + 1) \):
\[ 10b\sqrt{x} = 4\left(\frac{b(x + 1)}{b\sqrt{x}}\right) \]
This simplifies to:
\[ 10\sqrt{x} = 4\left(\frac{x + 1}{\sqrt{x}}\right) \]

5. **Cross-Multiplying Again**:
Cross-multiplying gives:
\[ 10\sqrt{x} \cdot \sqrt{x} = 4(x + 1) \]
Which simplifies to:
\[ 10x = 4x + 4 \]

6. **Solving for \( x \)**:
Rearranging gives:
\[ 10x - 4x = 4 \implies 6x = 4 \implies x = \frac{4}{6} = \frac{2}{3} \]

7. **Finding the Ratio**:
Since \( x = \frac{a}{b} \), we have:
\[ \frac{a}{b} = \frac{2}{3} \]
Therefore, the ratio \( a:b \) is:
\[ a:b = 2:3 \]

### Final Result:
The ratio of the two numbers \( a \) and \( b \) is \( 2:3 \).