Let the harmonic mean and geometric mean of two positive numbers be in the ratio 4:5. Then the two numbers are in ratio
(a) 4:1
(b) 1:4
(c) 5:4
(d) none of these

Answer

To solve the problem where the harmonic mean (H) and geometric mean (G) of two positive numbers are in the ratio 4:5, we can follow these steps:

### Step 1: Define the Means
Let the two positive numbers be \( a \) and \( b \).

The harmonic mean \( H \) is given by:
\[ H = \frac{2ab}{a + b} \]

The geometric mean \( G \) is given by:
\[ G = \sqrt{ab} \]

### Step 2: Set Up the Ratio
We are given that:
\[ \frac{H}{G} = \frac{4}{5} \]

Substituting the expressions for \( H \) and \( G \):
\[ \frac{\frac{2ab}{a + b}}{\sqrt{ab}} = \frac{4}{5} \]

### Step 3: Simplify the Equation
This simplifies to:
\[ \frac{2ab}{(a + b)\sqrt{ab}} = \frac{4}{5} \]

Cross-multiplying gives:
\[ 10ab = 4(a + b)\sqrt{ab} \]

### Step 4: Rearranging the Equation
Rearranging the equation, we have:
\[ 10ab = 4a\sqrt{ab} + 4b\sqrt{ab} \]
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 \( \sqrt{ab} = t \):
\[ 10t = 4\left(\frac{a}{t} + \frac{b}{t}\right) \]
This simplifies to:
\[ 10t = 4\left(\frac{a + b}{t}\right) \]
Multiplying both sides by \( t \):
\[ 10t^2 = 4(a + b) \]

### Step 5: Expressing in Terms of Ratios
We can rewrite the equation as:
\[ \frac{10}{4} = \frac{a + b}{t^2} \]
This leads to:
\[ \frac{5}{2} = \frac{a + b}{ab} \]
Rearranging gives:
\[ 5ab = 2(a + b) \]

### Step 6: Letting \( a \) and \( b \) be in a Ratio
Let \( a = kx \) and \( b = ky \) for some \( k \). Then:
\[ 5(kx)(ky) = 2(kx + ky) \]
This simplifies to:
\[ 5k^2xy = 2k(x + y) \]
Dividing both sides by \( k \) (assuming \( k \neq 0 \)):
\[ 5kxy = 2(x + y) \]

### Step 7: Solving for the Ratio
Now we can express the ratio \( \frac{a}{b} = \frac{x}{y} \). From the equation:
\[ 5xy = 2x + 2y \]
Rearranging gives:
\[ 5xy - 2x - 2y = 0 \]
Factoring this equation leads to:
\[ (5y - 2)(x - 2) = 0 \]
This gives us the solutions for \( x \) and \( y \).

### Step 8: Finding the Ratios
From the factorization, we can find the ratios of \( a \) and \( b \):
- If \( 5y - 2 = 0 \) then \( y = \frac{2}{5} \)
- If \( x - 2 = 0 \) then \( x = 2 \)

Thus, the ratio \( \frac{a}{b} = \frac{x}{y} = \frac{2}{\frac{2}{5}} = 5:2 \).

### Conclusion
The two numbers are in the ratio \( 5:4 \).