Question

If 34 and 16 are the arithmetic means and geometric means of two positive numbers respectively. Find the numbers.

• A. 125 and 5
• B. 64 and 4
• C. 60 and 68
• D. 16 and 256

Answer 1

The correct option is B 64 and 4

Let the two numbers be m and n. Then,
Arithmetic Mean = 34
$\Rightarrow \frac{m+n}{2}=34$
$\Rightarrow m+n=68$
And,
Geometric Mean = 16
$\sqrt{mn}=16$
$\Rightarrow mn=256$ ............................... (i)
Therefore, $(m-n{)}^2=(m+n{)}^2-4mn$
$\Rightarrow (m-n{)}^2=(68{)}^2-4\times 256=3600$
$\Rightarrow m-n=60$ ............................... (ii)
On solving (i) and (ii), we get m = 64 and n = 4.
Hence, the required numbers are 64 and 4.