# Let the harmonic mean and geometric mean of two positive  numbers be the ratio 4 : 5. Then the        two number are in the ratio………….

Deepak Patra
askIITians Faculty 471 Points
9 years ago
Hello Student,
Let a and b be two positive two positive numbers.
Then, H. M. = 2ab/a + b. and G. M. = √ab
ATQ HM : GM = 4 : 5
∴ 2ab/(a + b) √ab = 4/5
$\Rightarrow$2√ab / a + b = 4/5 $\Rightarrow$a + b + 2√ab/a + b 2√ab = 5 + 4/ 5 – 4
$\Rightarrow$(√a + √b)2 /(√a - √b)2 = 9/1 $\Rightarrow$√a + √b/√a - √b 3, -3
$\Rightarrow$2√a/2√b = 3 + 1/3 – 1, -3 + 1/ -3 – 1
$\Rightarrow$√a/√b = 2, 1/2 $\Rightarrow$a/b = 4, 1/4 a : b = 4 : 1 or 1 : 4
ALTERNATE SOLUTION:
Left for two + ve no. ‘s a and b, a/b = m
Then G = √ab = b√m and H = 2ab/a + b = 2nb√m/b + bm
∴ H/G = 4/5 $\Rightarrow$2√m /m + 1 = 4/5 $\Rightarrow$5√m = 2m + 2
$\Rightarrow$2m - 5√m + 2 = 0 $\Rightarrow$(√m – 2) (√m – 1/2) = 0
$\Rightarrow$m = 4 or 1/4 ∴ req. ratio = 4 : 1 or 1 : 4.

Thanks
Deepak Patra
Arun Kumar IIT Delhi
askIITians Faculty 256 Points
9 years ago
Hello Student,
$\\\sqrt{ab}={2ab \over a+b}*{5 \over 4} \\=>4(a+b)=10\sqrt{ab} \\=>4(\sqrt{a \over b}+\sqrt{b \over a})=10 \\=>t+1/t=5/2$
Please solve and then square the possible value to get the answer.
Thanks & Regards
Arun Kumar
Btech, IIT Delhi