If the AM and GM of two numbers are 5 and 4 respectively, then what is the HM of those numbers?

Options

A. `(5)/(4)`
B. `(16)/(5)`
C. `(9)/(2)`
D. 9

Explanation

To find the harmonic mean (HM) of two numbers given their arithmetic mean (AM) and geometric mean (GM), we can use the following relationships:

1. The formula for the arithmetic mean (AM) of two numbers \( a \) and \( b \) is:
\[ AM = \frac{a + b}{2} \]
Given that \( AM = 5 \), we can write:
\[ \frac{a + b}{2} = 5 \]
Multiplying both sides by 2 gives:
\[ a + b = 10 \quad \text{(Equation 1)} \]

2. The formula for the geometric mean (GM) of two numbers \( a \) and \( b \) is:
\[ GM = \sqrt{a \cdot b} \]
Given that \( GM = 4 \), we can write:
\[ \sqrt{a \cdot b} = 4 \]
Squaring both sides gives:
\[ a \cdot b = 16 \quad \text{(Equation 2)} \]

3. Now, we can use the values from Equation 1 and Equation 2 to find the harmonic mean (HM). The formula for the harmonic mean of two numbers \( a \) and \( b \) is:
\[ HM = \frac{2ab}{a + b} \]

4. Substituting the values from Equation 1 and Equation 2 into the HM formula:
\[ HM = \frac{2 \cdot 16}{10} \]
Simplifying this gives:
\[ HM = \frac{32}{10} = 3.2 \]

Therefore, the harmonic mean of the two numbers is \( 3.2 \).