# If ratio of AM and GM of two numbers is 5/4, what are the numbers and what is their HM?

Oct 6, 2017

Numbers are $2$ and $8$. Their H.M. is $\frac{16}{5}$.

#### Explanation:

Let the two numbers be $a$ and $b$.

Their A.M. is $\frac{a + b}{2}$ and as G.M. is $\sqrt{a b}$

and as ratio between A.M. and G.M. is $\frac{5}{4}$

$\frac{\frac{a + b}{2}}{\sqrt{a b}} = \frac{5}{4}$

or $2 a + 2 b = 5 \sqrt{a b}$

or $4 {a}^{2} + 4 {b}^{2} + 8 a b = 25 a b$

i.e. $4 {a}^{2} + 4 {b}^{2} - 17 a b = 0$

or $\left(4 a - b\right) \left(a - 4 b\right) = 0$

i.e. either $\frac{a}{b} = 4$ or $\frac{b}{a} = 4$

Hence numbers are of the form $x$ and $4 x$

and their G.M. is $2 x$ and H.M.@ will be $\frac{2 \cdot x \cdot 4 x}{x + 4 x}$ or $\frac{8 {x}^{2}}{5} x$ or $\left(8 \frac{x}{5}\right)$
Further, as difference of G.M. and H.M. is $\frac{4}{5}$

$2 x - \frac{8}{5} x = \frac{4}{5}$

or $\frac{2}{5} x = \frac{4}{5}$ and $x = 2$

Hence numbers are $2$ and $8$.

Observe their A.M. is $5$, G.M. is $4$ and H.M. is $\frac{16}{5}$.

@ If $h$ is H.M. between $a$ and $b$,

$\frac{1}{a} , \frac{1}{h} , \frac{1}{b}$ are in arithmetic sequence and hence

$2 \left(\frac{1}{h}\right) = \frac{1}{a} + \frac{1}{b} = \frac{a + b}{a b}$

and $\frac{1}{h} = \frac{a + b}{2 a b}$ i.e. $h = \frac{2 a b}{a + b}$