Question

If A.M and G.M of two positive numbers a and b are 10 and 8 respectively . Find the numbers

Answer 1

Answer:

We have,

2

a+b

=10 and

ab

=8⇒a+b=20 and ab=64

Clearly,a and b are roots of the equation

x

2

−(a+b)x+ab=0

or, x

2

−20x+64=0

⇒(x−16)(x−4)=0⇒x=4,16⇒a=4,b=16 or a=16,b=4

Answer 2

$\huge\underline\bold\purple{★ QUESTION :}$

If A.M and G.M of two positive numbers a and b are 10 and 8 respectively . Find the numbers

$\huge\underline\bold\purple{★ SOLUTION :}$

We know that AM of two numbers a & b is

$AM = \frac{a + b}{2}$

↪ AM of two numbers a & b is “ 10 ”.

So,

$⟹ \frac{a + b}{2} = 10$

$⟹a + b = 10 \times 2$

$⟹a + b = 20 \: .....(1)$

Also, GM of two numbers a & b is

$GM = \sqrt{ab}$

↪ GM of two numbers is “ 8 ”.

$⟹ \sqrt{ab} = 8$

• [squaring on both sides ]

$⟹ { (\sqrt{ab} )}^{2} = {(8)}^{2}$

$⟹ab = 64$

$⟹a = \frac{64}{b} \: ......(2)$

Now, substitute the value of a in (1)

$⟹ \frac{64}{b} + b = 20$

$⟹ \frac{64 + {b}^{2} }{b} = 20$

$⟹64 + {b}^{2} = 20b$

$⟹ {b}^{2} - 20b + 64 = 0$

$⟹ {b}^{2} - 16b - 4b + 64 = 0$

$⟹b(b - 16) - 4(b - 16) = 0$

$⟹(b - 16)(b - 4) = 0$

$⟹b - 16 = 0;b - 4 = 0$

$⟹b = 16;b = 4$

Substitute the value of b in (2)

$⟹a = \frac{64}{4}$

$⟹a = 16$

And also, substitute the other value of b in (2)

$⟹a = \frac{64}{16}$

$⟹a = 4$

$\boxed{∴ a = 4\;,b = 16}$

(or)

$\boxed{∴ a = 16\;,b = 4}$

Step-by-step explanation:

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