Let G be the geometric mean of two positive numbers a and b, and M be the arithmetic mean of `1/a and 1/b`if `(1/M):G ` is `4:5`, then `a:b` can be

Options

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

Answer

To solve the problem, we will follow the steps outlined in the video transcript, breaking them down into a clear, step-by-step solution.

### Step-by-Step Solution:

1. **Define the Means**:
- Let \( G \) be the geometric mean of two positive numbers \( a \) and \( b \). Thus,
\[ G = \sqrt{ab} \]
- Let \( M \) be the arithmetic mean of \( \frac{1}{a} \) and \( \frac{1}{b} \). Therefore,
\[ M = \frac{\frac{1}{a} + \frac{1}{b}}{2} = \frac{b + a}{2ab} \]

2. **Set Up the Ratio**:
- We are given that \( \frac{1}{M} : G = 4 : 5 \). This can be expressed as:
\[ \frac{1}{M} = \frac{4}{5} G \]

3. **Substitute the Values**:
- Substitute \( M \) and \( G \) into the ratio:
\[ \frac{2ab}{a + b} = \frac{4}{5} \sqrt{ab} \]

4. **Cross Multiply**:
- Cross multiplying gives:
\[ 5 \cdot 2ab = 4 \cdot (a + b) \sqrt{ab} \]
- Simplifying this results in:
\[ 10ab = 4(a + b) \sqrt{ab} \]

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

6. **Multiply Through by \( \sqrt{x} \)**:
- Multiply through by \( \sqrt{x} \):
\[ 10x = 4(x + 1) \]
- Expanding gives:
\[ 10x = 4x + 4 \]
- Rearranging gives:
\[ 6x = 4 \quad \Rightarrow \quad x = \frac{2}{3} \]

7. **Finding the Ratio \( a:b \)**:
- Since \( x = \frac{a}{b} \), we have:
\[ \frac{a}{b} = \frac{2}{3} \]
- Therefore, the ratio \( a:b \) can also be expressed as:
\[ a:b = 2:3 \]

### Final Answer:
The ratio \( a:b \) can be \( 2:3 \).