Question

If G.M.is 18 and A.M. is 27 then find the value of H.M.
A. \[\dfrac{1}{18}\]
B. \[\dfrac{1}{12}\]
C. \[12\]
D. \[9\sqrt 6 \]

Answer 1

Hint: In this question, we need to find the value of H.M. which is the harmonic mean. For this, we have to use the relation between geometric mean (G.M.), arithmetic mean (A.M.), and harmonic mean (H.M.) which is mentioned below. After that, we will simplify the equation so that we will get the desired result.

Formula used: The following relation will be used to solve this question.
Here, we can say that the square of the geometric mean (G.M.) is simply the product of the arithmetic mean (AM) and harmonic mean (HM).
\[{\left( {{\text{G}}{\text{.M}}{\text{.}}} \right)^2} = \left( {{\text{A}}{\text{.M}}{\text{.}}} \right) \times \left( {{\text{H}}{\text{.M}}{\text{.}}} \right)\]

Complete step-by-step solution:
We know that \[A.M. = 27\]and \[{\text{G}}{\text{.M}}{\text{.}} = 18\]
But we know that \[{\left( {{\text{G}}{\text{.M}}{\text{.}}} \right)^2} = \left( {{\text{A}}{\text{.M}}{\text{.}}} \right) \times \left( {{\text{H}}{\text{.M}}{\text{.}}} \right)\]
By using the above relation, we get
\[{\left( {{\text{G}}{\text{.M}}{\text{.}}} \right)^2} = \left( {{\text{A}}{\text{.M}}{\text{.}}} \right) \times \left( {{\text{H}}{\text{.M}}{\text{.}}} \right)\]
But we know that \[A.M. = 27\]and \[{\text{G}}{\text{.M}}{\text{.}} = 18\]
Thus, we get
\[{\left( {{\text{18}}} \right)^2} = \left( {{\text{27}}} \right) \times \left( {{\text{H}}{\text{.M}}{\text{.}}} \right)\]
By simplifying, we get
\[324 = \left( {{\text{27}}} \right) \times \left( {{\text{H}}{\text{.M}}{\text{.}}} \right)\]
Now, we will find the value of H.M. from the above equation.
\[\dfrac{{324}}{{27}} = \left( {{\text{H}}{\text{.M}}{\text{.}}} \right)\]
Thus, by simplifying we get
\[\left( {{\text{H}}{\text{.M}}{\text{.}}} \right) = 12\]
Hence, the value of H.M. is 12 G.M.is 18 and A.M. is 27.

Therefore, the correct option is (C).

Additional information: The most widely used measures of central tendency are AM (Arithmetic Mean), GM (Geometric Mean), as well as HM (Harmonic Mean). The relationship between AM, GM, and HM can be derived using basic progression or mathematical sequence expertise. Her, we can say that a Mathematical Sequence is an array or group of items that follow a predefined pattern.

Note: Many students generally make mistakes in finding the relation between geometric mean, arithmetic mean, and harmonic mean. This relation plays a significant role to find the value of the harmonic mean.