Question

The arithmetic mean of 2 numbers is 34 and their geometric mean is 16. One of the numbers will be

Answer 1

Let the two numbers be x and y.
Now,
Arithmetic mean = 34
$= > \frac{x + y}{2} = 34$

$= > x + y = 34 \times 2 = 68$
=> x= 68-y

Now,
Geometric Mean =16

That means,
x/16= 16/y
=> 16*16= xy
=>xy= 256
=>(68-y)×y =256
=>68y-y2=256
=> y2 - 68y + 256 = 0
=> y2 - 64y - 4y +256=0
=> y(y-64) - 4(y-64)= 0
=> y= 64 or y=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.