Question

If the arithmetic mean is 34 and geometric mean is 16 what is the greatest number

Answer 1

Let the numbers be x, y 
Then arithmatic mean = (x+y)/2 =34 
→x+y =68 
Also geometric mean =√(xy)=16 
oy xy=16²=256 
Hence 
x(68−x)=256 
or x²−68x+256=0 
(x−64)(x−4)=0 
Hence x=64 or x=4 
and y=4 or 64 
Larger number is 64

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.

the greatest number is 64.