To find the Harmonic Mean (H.M.) of two items when the Arithmetic Mean (A.M.) and Geometric Mean (G.M.) are given, we can use the following relationships:
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The formula for A.M. of two numbers \( a \) and \( b \) is: \[ A.M. = \frac{a + b}{2} \]
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The formula for G.M. of two numbers \( a \) and \( b \) is: \[ G.M. = \sqrt{a \cdot b} \]
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The formula for H.M. of two numbers \( a \) and \( b \) is: \[ H.M. = \frac{2ab}{a + b} \]
From the problem, we know:
- \( A.M. = 5 \)
- \( G.M. = 4 \)
Using the A.M. formula: \[ \frac{a + b}{2} = 5 \implies a + b = 10 \quad \text{(1)} \]
Using the G.M. formula: \[ \sqrt{a \cdot b} = 4 \implies a \cdot b = 16 \quad \text{(2)} \]
Now we have two equations:
- \( a + b = 10 \) (Equation 1)
- \( a \cdot b = 16 \) (Equation 2)
We can use these two equations to find \( H.M. \).
Substituting the values into the H.M. formula:
Using Equation (1) and (2): \[ H.M. = \frac{2ab}{a + b} = \frac{2 \cdot 16}{10} = \frac{32}{10} = 3.2 \]
Therefore, the Harmonic Mean (H.M.) of the two items is: \[ \boxed{3.2} \]
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