If AM and HM of two positive numbers are 9 and 4 respectively, then their GM is?
Understand the Problem
The question asks for the geometric mean (GM) of two positive numbers given their arithmetic mean (AM) is 9 and harmonic mean (HM) is 4. To solve, we first need to determine the relationship between AM, HM, and GM and then calculate the GM using the known values.
Answer
$6$
Answer for screen readers
The geometric mean (GM) is $6$.
Steps to Solve
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Understand the relationships between AM, HM, and GM The three means—Arithmetic Mean (AM), Geometric Mean (GM), and Harmonic Mean (HM)—have the following relationships: $$ AM \geq GM \geq HM $$
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Set up equations for the means Let the two positive numbers be ( a ) and ( b ).
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The arithmetic mean (AM) is given by: $$ AM = \frac{a + b}{2} $$ Thus, from the problem, we have: $$ \frac{a + b}{2} = 9 $$
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The harmonic mean (HM) is given by: $$ HM = \frac{2ab}{a + b} $$ From the problem, we have: $$ \frac{2ab}{a + b} = 4 $$
- Substituting AM into the HM equation From the first equation, we can express ( a + b ): $$ a + b = 18 $$
Substituting this value into the HM equation: $$ \frac{2ab}{18} = 4 $$
- Solve for ( ab ) Multiply both sides by 18: $$ 2ab = 72 $$
Thus, $$ ab = 36 $$
- Use the values of ( a + b ) and ( ab ) to find GM The GM is calculated using: $$ GM = \sqrt{ab} $$ Substituting the value we found: $$ GM = \sqrt{36} = 6 $$
The geometric mean (GM) is $6$.
More Information
The geometric mean is often used in scenarios where you want to average ratios or percentages, making it a useful measure in finance and growth rates.
Tips
- Confusing the formulas for AM, HM, and GM. It's important to remember each formula's definition.
- Not properly substituting values when forming equations, leading to incorrect calculations.
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