Question

find the G.M of positive numbers whose A.M and H.M are 8 and 18 respectively​

Answer 1

Answer:

\huge{\pink{\underline{\mathfrak{your\:answer}}}}

youranswer

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\bold{G.M\:=\:6}G.M=6

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step-by-step explanation:

Let the two positive numbers be 'x' and 'y'.

Now,

it is given that,

Arithmetic mean, A.M = 12

= > \frac{x + y}{2} = 12=>

2

x+y

=12

=> x + y = 12 × 2

=> x + y = 24..................(i)

Now,

given that,

Harmonic mean, H.M = 3

= > \frac{2}{ \frac{1}{x} + \frac{1}{y} } = 3=>

x

1

+

y

1

2

=3

doing cross multiplication,

we get,

= > \frac{1}{x} + \frac{1}{y} = \frac{2}{3}=>

x

1

+

y

1

=

3

2

Taking L.C.M of denominators and simplifying,

we get,

= > \frac{x + y}{xy} = \frac{2}{3}=>

xy

x+y

=

3

2

But, from eqn (i),

(x+y) = 24,

so putting the value,

we get,

= > \frac{24}{xy} = \frac{2}{3}=>

xy

24

=

3

2

On cross-multiplication,

we get,

= > xy = \frac{24 \times 3}{2}=>xy=

2

24×3

= > xy = 36=>xy=36

But,

we know that,

Geometric mean, G.M of two numbers is the square root of their product.

so,

G.M of 'x' and 'y' is \sqrt{xy}

xy

But,

xy = 36

therefore,

G.M = \sqrt{36}

36

=> G.M = 6

Hence,

Geometric mean,G.M of the numbers = 6

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