Question

If the A.M. and H.M. of two numbers is \[27\]and \[12\] respectively, then G.M. of the two numbers will be
A. \[9\]
B. \[18\]
C. \[24\]
D. \[36\]

Answer 1

Hint:
We'll use the AM, GM, and HM formulas for two numbers, which are \[AM = \frac{{a + b}}{2}\]and \[HM = \frac{{2ab}}{{a + b}}\]. Simply enter the A.M. and H.M. values in the L.H.S. of the equation and change a few terms.
Since, A is the Arithmetic mean of the numbers \[a,b\]
\[\therefore A = \frac{{a + b}}{2}\]
Since, H is the Harmonic mean of the numbers \[a,b\]
\[\therefore H = \frac{{2ab}}{{a + b}}\]
Formula used:
A is the Arithmetic mean of the numbers \[a,b\]
\[\therefore A = \frac{{a + b}}{2}\]
Since, H is the Harmonic mean of the numbers \[a,b\]
\[\therefore H = \frac{{2ab}}{{a + b}}\]

Complete step-by-step solution:
We have been given in the question that A.M. and H.M. of two numbers is \[27\] and \[12\] respectively.
Let \[{\rm{a}}\]and \[{\rm{b}}\]be two numbers, then
Use the formula for arithmetic mean,
\[{\rm{AM}} = \frac{{{\rm{a}} + {\rm{b}}}}{2}\]
Substitute the values for AM as given in the question:
\[ \Rightarrow 27 = \frac{{{\rm{a}} + {\rm{b}}}}{2}\]
Now, we have to move \[2\] from the denominator to the left side of the equation and multiply, we get
\[ \Rightarrow {\rm{a}} + {\rm{b}} = 54\]
Now, use the formula of Harmonic mean:
\[{\rm{HM}} = \frac{{2{\rm{ab}}}}{{{\rm{a}} + {\rm{b}}}}\]
Substitute the value of HM as given in the question:
\[ \Rightarrow 12 = \frac{{2{\rm{ab}}}}{{54}}\]
Now, we have to move\[54\]in the denominator to the left side of the equation and multiply with\[12\], and divide by \[2\], we get
 \[ \Rightarrow {\rm{ab}} = 324\]
Now, use the formula for GM and calculate the required value:
\[{\rm{GM}} = \sqrt {{\rm{ab}}} \]
Substitute the value of\[ab\]obtained in the previous step, we get
 \[ = \sqrt {324} = 18\]
Therefore, the G.M. of the two numbers will be\[{\rm{GM}} = 18\].
Hence, the option B is correct.
Note:
Student gets confused mostly, because one of the most crucial ideas is the arithmetic and harmonic means, which are applied to almost all types of issues. The term "arithmetic mean" refers to a value that is determined by dividing the total number of values in a set by the sum of its elements. The inverse of the arithmetic mean of the given data values is the harmonic mean. It is based on all of the set's values.