Let G be the geometric mean of two positive numbers a and b, and M be the arithmetic mean of $$\frac{1}{a}$$ and $$\frac{1}{b}$$. If $$\frac{1}{M}$$ : G is 4 : 5, then a : b can be:

We are given two positive numbers $$a$$ and $$b$$. The geometric mean $$G$$ of $$a$$ and $$b$$ is defined as $$G = \sqrt{ab}$$. The arithmetic mean $$M$$ of $$\frac{1}{a}$$ and $$\frac{1}{b}$$ is given by $$M = \frac{\frac{1}{a} + \frac{1}{b}}{2}$$. Simplifying $$M$$, we get:

$$M = \frac{\frac{1}{a} + \frac{1}{b}}{2} = \frac{\frac{b + a}{ab}}{2} = \frac{a + b}{2ab}.$$

The problem states that $$\frac{1}{M} : G = 4 : 5$$, which means the ratio of $$\frac{1}{M}$$ to $$G$$ is 4 to 5. This can be written as:

$$\frac{\frac{1}{M}}{G} = \frac{4}{5} \quad \Rightarrow \quad \frac{1}{M \cdot G} = \frac{4}{5} \quad \Rightarrow \quad M \cdot G = \frac{5}{4}.$$

Substituting the expressions for $$M$$ and $$G$$, we have:

$$\left( \frac{a + b}{2ab} \right) \cdot \sqrt{ab} = \frac{5}{4}.$$

Simplify the left side:

$$\frac{a + b}{2ab} \cdot \sqrt{ab} = \frac{a + b}{2 \sqrt{ab} \cdot \sqrt{ab}} \cdot \sqrt{ab} = \frac{a + b}{2 \sqrt{ab}}.$$

Note that $$\sqrt{ab} = \sqrt{a} \sqrt{b}$$, so:

$$\frac{a + b}{2 \sqrt{ab}} = \frac{a + b}{2 \sqrt{a} \sqrt{b}}.$$

Set $$k = \sqrt{\frac{a}{b}}$$. This implies $$k^2 = \frac{a}{b}$$, so $$a = k^2 b$$. Substitute this into the equation:

$$\frac{k^2 b + b}{2 \sqrt{(k^2 b) \cdot b}} = \frac{b(k^2 + 1)}{2 \sqrt{k^2 b^2}} = \frac{b(k^2 + 1)}{2 \cdot k b} = \frac{k^2 + 1}{2k}.$$

We now have:

$$\frac{k^2 + 1}{2k} = \frac{5}{4}.$$

Solve for $$k$$:

Multiply both sides by $$2k$$:

$$k^2 + 1 = \frac{5}{4} \cdot 2k = \frac{10k}{4} = \frac{5k}{2}.$$

Multiply both sides by 2 to eliminate the denominator:

$$2(k^2 + 1) = 5k \quad \Rightarrow \quad 2k^2 + 2 = 5k \quad \Rightarrow \quad 2k^2 - 5k + 2 = 0.$$

Solve this quadratic equation using the quadratic formula:

$$k = \frac{5 \pm \sqrt{(-5)^2 - 4 \cdot 2 \cdot 2}}{2 \cdot 2} = \frac{5 \pm \sqrt{25 - 16}}{4} = \frac{5 \pm \sqrt{9}}{4} = \frac{5 \pm 3}{4}.$$

This gives two solutions:

$$k = \frac{5 + 3}{4} = \frac{8}{4} = 2 \quad \text{and} \quad k = \frac{5 - 3}{4} = \frac{2}{4} = \frac{1}{2}.$$

Recall that $$k = \sqrt{\frac{a}{b}}$$, so:

• If $$k = 2$$, then $$\sqrt{\frac{a}{b}} = 2$$, so $$\frac{a}{b} = 4$$, giving $$a : b = 4 : 1$$.
• If $$k = \frac{1}{2}$$, then $$\sqrt{\frac{a}{b}} = \frac{1}{2}$$, so $$\frac{a}{b} = \frac{1}{4}$$, giving $$a : b = 1 : 4$$.

Now, check the options:

• A. $$1 : 4$$
• B. $$1 : 2$$
• C. $$2 : 3$$
• D. $$3 : 4$$

The ratio $$a : b = 1 : 4$$ (from $$k = \frac{1}{2}$$) corresponds to option A. The ratio $$4 : 1$$ is not listed in the options.

Verify $$a : b = 1 : 4$$:

Let $$a = 1$$, $$b = 4$$. Then:

$$G = \sqrt{1 \cdot 4} = \sqrt{4} = 2,$$

$$M = \frac{\frac{1}{1} + \frac{1}{4}}{2} = \frac{1 + 0.25}{2} = \frac{1.25}{2} = 0.625,$$

$$\frac{1}{M} = \frac{1}{0.625} = 1.6,$$

Ratio $$\frac{1}{M} : G = 1.6 : 2 = 1.6 \div 2 = 0.8 = \frac{4}{5}$$, which matches $$4 : 5$$.

Other options do not satisfy:

• Option B ($$1:2$$): $$G = \sqrt{2} \approx 1.414$$, $$M = \frac{1 + 0.5}{2} = 0.75$$, $$\frac{1}{M} \approx 1.333$$, ratio $$\approx 1.333 : 1.414 \approx 0.943 \neq 0.8$$.
• Option C ($$2:3$$): $$G = \sqrt{6} \approx 2.45$$, $$M = \frac{0.5 + \frac{1}{3}}{2} = \frac{5/6}{2} \approx 0.4167$$, $$\frac{1}{M} \approx 2.4$$, ratio $$\approx 2.4 : 2.45 \approx 0.98 \neq 0.8$$.
• Option D ($$3:4$$): $$G = \sqrt{12} \approx 3.464$$, $$M = \frac{\frac{1}{3} + \frac{1}{4}}{2} = \frac{7/12}{2} \approx 0.2917$$, $$\frac{1}{M} \approx 3.4286$$, ratio $$\approx 3.4286 : 3.464 \approx 0.99 \neq 0.8$$.

Thus, $$a : b = 1 : 4$$ is the only ratio from the options that satisfies the condition.

Hence, the correct answer is Option A.