Question

kindly solve if the harmonic mean and geometric mean of two positive numbers are in the ratio 4:5 then the two numbers are in the ratio ?​

Answer 1

Answer:

Numbers are in ratio 4:1

Step-by-step explanation:

Given :

The harmonic mean and geometric mean of two positive numbers are in the ratio = 4:5

Both are positive numbers

To find : The ratio in which the numbers are

Let the numbers be Ka and a

so, a> 0 as number is positive.

Apply relation for G.M

$g.m = {(ka \times a)}^{ \frac{1}{2} } \\ \\ = a \sqrt{ka}$

Now apply relation for H.M

$h = \frac{2(ka)a}{ka + a} \\ \\ = \frac{2(ka)a}{a(k + 1)} \\ \\ = \frac{2ka}{k + 1}$

Now using the given Ratio relation

$\frac{h.m}{g.m} = \frac{4}{5}$

Put the values

$\huge{ \frac{ \frac{2ka}{k + 1} }{ a\sqrt{k} } = \frac{4}{5} } \\ \\ \huge{ \frac{2ka}{(k + 1) \times (a \sqrt{k}) } = \frac{4}{5} }\\ \\ \huge{ \frac{2 \sqrt{k} }{k + 1} = \frac{4}{5} } \\ \\ \huge{5(2 \sqrt{k} ) = 4(k + 1)} \\ \\ \huge{ 10 \sqrt{k} = 4k + 4 }$

Now divide by 2 both sides

$5 \sqrt{k} = 2k + 2 \\ \\ 2k - 5 \sqrt{k} + 2 = 0$

°•° Solve this Quadratic equation

•°• Apply Quadratic Formula

$\sqrt{k} = \frac{5 \pm \sqrt{ {5}^{2} - {4}^{2} } }{2 \times 2} \\ \\ = \frac{5 \pm \sqrt{25 - 16} }{4} \\ \\ = \frac{5 \pm \sqrt{9} }{4} \\ \\ = \frac{5 \pm 3}{4}$

Finally, we get

$\sqrt{k} = \frac{5 \pm 3}{4}$

Two values of k are

•Taking +ve sign

$\sqrt{k} = \frac{5 + 3}{4} = \frac{8}{4} = 2 \\ \\ \sqrt{k} = 2 \\ \\ k = 4$

Next root,

•Taking -ve sign

$\sqrt{k} = \frac{5 - 3}{4} = \frac{1}{2} \\ \\ \sqrt{k} = \frac{1}{2} \\ \\ k = \frac{1}{4}$

Now A/c to question

•°•Required Ratio $= \frac{4a}{ a } = \frac{4}{1}$

•°• The numbers are in ratio 4:1

$\rule{200}{1}$

$\boxed{\underline{\bold{\mathsf{Formula \: used}}}}$

◾ Geometric mean =${(a \times b) }^{ \frac{1}{2} }$

◾Harmonic mean $= \frac{2ab}{a + b}$

◾Quadratic formula =$\frac{b \pm \sqrt{ {b}^{2} - 4ac } }{2a}$

Where a,b c are respective co-efficients