Question

The harmonic mean between two numbers is \[14\frac{2}{5}\] and geometric mean is \[24\]. The greater number between them is
A. 72
B. 54
C. 36
D. None of these

Answer 1

Hint:
We must use formulas for various types of means when answering questions of this nature. We are aware that the arithmetic mean (AM), geometric mean (GM), and harmonic mean are the three Pythagorean means (HM). Additionally, we are aware that the Arithmetic Mean (AM) equals \[\frac{{a + b}}{2}\], the Geometric Mean (GM) equals \[\sqrt {ab} \] and the Harmonic Mean (HM) equals \[\frac{{2ab}}{{(a + b)}}\]if a and b are two positive numbers.
Formula use:
Arithmetic mean: \[AM = \frac{{a + b}}{2}\]
Geometric mean: \[GM = \sqrt {ab} \]
Harmonic mean: \[HM = \frac{{2ab}}{{a + b}}\]
In this, case we should remember this formulas to process with the problem. We will get the required solution.
Complete step-by-step solution
We have been given that,
Harmonic mean of two numbers is \[14\frac{2}{5}\]--- (1)
Geometric mean is \[24\]--- (2)
Now, let us assume that \[{\bf{a}}\] and \[b\] be the numbers.
We have been already know that,
Geometric Mean \[ = \sqrt {{\rm{ab}}} \]
\[{\rm{GM}} = \sqrt {{\rm{ab}}} \]
On substituting the value of geometric mean as per the given question is,
\[24 = \sqrt {{\rm{ab}}} \]
Now, we have to square on both sides, we get
\[576 = {\rm{ab}}\]--- (3)
Now, solve for \[b\] from the above equation, we get
\[\frac{{576}}{{\rm{a}}} = {\rm{b}}\]
For Harmonic mean:
Harmonic mean\[ = \frac{{2ab}}{{a + b}}\]
\[ \Rightarrow {\rm{HM}} = \frac{{2{\rm{ab}}}}{{{\rm{a}} + {\rm{b}}}}\]
On substituting the value of HM as per the equation (1), we get
\[ \Rightarrow 14\frac{2}{5} = \frac{{2{\rm{ab}}}}{{{\rm{a}} + {\rm{b}}}}\]
\[14\frac{2}{5} = \frac{{14 \cdot 5 + 2}}{5} = \frac{{72}}{5}\]
\[ \Rightarrow \frac{{72}}{5} = \frac{{2{\rm{ab}}}}{{{\rm{a}} + {\rm{b}}}}\]
Now, we have to solve for \[a + b\], we get
\[ \Rightarrow {\rm{a}} + {\rm{b}} = \frac{{10{\rm{ab}}}}{{72}}\]
On substituting the value of \[{\rm{ab}}\] as per the equation (3), we get
\[ \Rightarrow {\rm{a}} + {\rm{b}} = \frac{{10 \times 576}}{{72}}\]--- (4)
From the above equation, simplify the right side of the equation:
\[\frac{{576}}{{72}} = 8\]
Multiply the numbers \[10 \times 8\]:
\[{\rm{a + b = 80}}\]
Now we have to replace the value:
\[a + \frac{{576}}{a} = 80\]
Multiply both sides by \[a\]:
\[aa + \frac{{576}}{a}a = 80a\]
Simplify the above equation, we get
\[{a^2} + 576 = 80a\]
Now we have to group the variables on one side:
\[{a^2} - 80a = 576\]
Now rewriting in quadratic form:
\[{a^2} - 80a - 576 = 0\]
Factor the above quadratic equation, we have
\[(a - 8)(a - 72) = 0\]
Now, equate both the factors to zero, we get
\[a = 8,a = 72\]
Therefore, the greater number between them is \[a = 72\].
Hence, the option A is correct.
Note:
An A.P is a set of numbers where the difference between any two consecutive terms is the same. A G.P is a set of numbers where any two consecutive terms have the same ratio. A H.P is a set of numbers that contains the reciprocals of the terms in an A.P. Student must remember these to solve these types of problems easily.