Let x be the HM and y be the GM of two positive numbers m and n. If `5x = 4y`, then which one of the following is correct?

Options

A. 5m = 4n
B. 2m = n
C. 4m = 5n
D. m = 4n

Answer

To solve the problem, we need to find the relationship between the two positive numbers \( m \) and \( n \) given that the harmonic mean (HM) \( x \) and geometric mean (GM) \( y \) satisfy the equation \( 5x = 4y \).

### Step-by-Step Solution:

1. **Define the Harmonic Mean (HM) and Geometric Mean (GM)**:
- The harmonic mean \( x \) of two numbers \( m \) and \( n \) is given by:
\[ x = \frac{2mn}{m+n} \]
- The geometric mean \( y \) of two numbers \( m \) and \( n \) is given by:
\[ y = \sqrt{mn} \]

2. **Substitute HM and GM into the given equation**:
- We are given that \( 5x = 4y \). Substituting the expressions for \( x \) and \( y \):
\[ 5 \left( \frac{2mn}{m+n} \right) = 4 \sqrt{mn} \]

3. **Simplify the equation**:
- Multiply both sides by \( (m+n) \) to eliminate the fraction:
\[ 10mn = 4\sqrt{mn}(m+n) \]

4. **Rearranging the equation**:
- Divide both sides by \( \sqrt{mn} \) (assuming \( m, n > 0 \)):
\[ 10\sqrt{mn} = 4(m+n) \]

5. **Divide both sides by 2**:
- Simplifying gives:
\[ 5\sqrt{mn} = 2(m+n) \]

6. **Square both sides**:
- To eliminate the square root, square both sides:
\[ 25mn = 4(m+n)^2 \]

7. **Expand the right-hand side**:
- Expanding \( (m+n)^2 \):
\[ 25mn = 4(m^2 + 2mn + n^2) \]

8. **Rearranging the equation**:
- Rearranging gives:
\[ 25mn = 4m^2 + 8mn + 4n^2 \]
- This can be rearranged to:
\[ 4m^2 - 17mn + 4n^2 = 0 \]

9. **Using the quadratic formula**:
- This is a quadratic equation in terms of \( m \):
\[ m = \frac{-(-17n) \pm \sqrt{(-17n)^2 - 4 \cdot 4 \cdot 4n^2}}{2 \cdot 4} \]
- Simplifying gives:
\[ m = \frac{17n \pm \sqrt{289n^2 - 64n^2}}{8} \]
- Further simplifying:
\[ m = \frac{17n \pm \sqrt{225n^2}}{8} \]
- Thus:
\[ m = \frac{17n \pm 15n}{8} \]

10. **Finding the values of \( m \)**:
- This gives two possible values:
- \( m = \frac{32n}{8} = 4n \)
- \( m = \frac{2n}{8} = \frac{n}{4} \)

### Conclusion:
The relationship between \( m \) and \( n \) can be expressed as:
- \( m = 4n \) or \( n = \frac{m}{4} \)

Given the options, the correct answer is:
- \( m = 4n \)