Question

If AM and GM of two numbers are 10 and 8 respectively.Find the numbers?

Answer 1

$<b>$Here a = 10 and b = 8

We know that AM of the two numbers a and b is

$\bf AM = \dfrac{a + b}{2}$

AM of the given numbers a and b is given 10

$\bf \dfrac{a + b}{2} = 10$

$\bf{a + b = 20------------(1)}$

We know that GM of the numbers a and b is

$\mathbf{GM = \sqrt{ab} }$

GM of the given numbers a and b is Given 8

$\mathbf{GM = { \sqrt{ab} } = 8 }$

On Squaring both sides we have

$\mathbf{( { \sqrt{ab} )}^{2} = {8}^{2}}$

$\mathbf{ab = 64 ---------(2)}$

So

a + b = 20--------------(1)

ab = 64 ---------------(2)

From eq.(2)

a = 64/b

Putting the value of a in eq.(1) we get

a + b = 20

$\bf \dfrac{64}{b} + b = 20$

$\bf \frac{64 + {b}^{2} }{b} = 20$

$\bf 64 + {b}^{2} = 20 b$

$\bf{b}^{2} - 20b + 64 = 0$

$\bf b(b - 16) - 4(b - 16) = 0$

$\bf(b - 4)(b - 16) = 0$

So b - 4 = 0

or b -16 = 0

b = 4

b = 16

Now putting b = 4 in

$\mathbf{a = \frac{64}{b} }$

$\mathbf{a = \frac{64}{4} }$

$\textbf {a = 16}$

Putting b = 16 in

$\mathbf {a = \dfrac{64}{b} }$

$\mathbf{a = \dfrac{64}{16}}$

$\mathbf{a = 4}$

Hence the numbers are

a = 4 , b = 16

or b = 4 , a = 16

Thanks

Have a colossal day ahead

Be Brainly

Answer 2

Answer:

$\huge{\mathcal{\underline{\green{AnSwER}}}}$

⇒AM = $\frac{a + b}{2}$

⇒ GM = √ab

Given AM = 10, GM = 8.

⇒ $\frac{a + b}{2}$ = 10

⇒ a + b = 20

⇒ a = 20–b

⇒ $\sqrt{(20 - b)b}$

⇒ 20b – b²= 64

⇒ b² – 20b + 64 = 0

⇒ b²– 16b – 4b + 64 = 0

⇒ b(b – 16) – 4(b – 16) = 0

⇒ b = 4 or b = 16

⇒ If b = 4

then a = 16

⇒ If b = 16

then a = 4.

Hence, the numbers are 4 and 16