Question

Find the two positive numbers whose arithmetic mean is 34 and the geometric mean is 16.

Answer 1

Let the two positive nos be x and y.
Their arithmetic mean is 34,
(x+y)/2=34
x+y=68
Again, their geometric mean is 16
root over (xy)=16
xy=256
y=256/x
x+(256/x)=68
x^2+256=68x
x^2-68x+256=0
x^2-(64+4)x+256=0
x^2-64x-4x+256=0
x(x-64)-4(x-64)=0
(x-64)(x-4)=0
either x=64 or x=4
either y=4 or y=64

The two positive nos are 64 and 4.

Answer 2

Step-by-step explanation:

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

We know,

♠ Arithmetic mean(A.M.) = (a + b)/2 = 34 (given)

♠ Also, Geometric mean(G.M.) = √ab = 16 (given)

We get :

(a + b) = 2*34 = 68           ...(i)

ab = 16² = 256

Now,

(a - b)² = (a + b)² - 4ab

⇒(a - b)² = (68)² - 4*256

⇒(a - b)² = 4624 - 1024

⇒(a - b)² = 3600

⇒(a - b) = 60                    ...(ii)

From eq. (ii) :

a = 60 + b                         ...(iii)

Putting this value in eq. (i) :

(a + b) = 68

⇒60 + b + b = 68

⇒60 + 2b = 68

⇒2b = 8

⟹b=4

Putting b = 4 in eq. (iii) :

a = 60 + b

⇒a = 60 + 4

⟹a=64

∴ So, the numbers are 64 and 4.