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

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

Grade:upto college level

3 Answers

Deepak Patra
askIITians Faculty 471 Points
7 years ago
Hello Student,
Please find the answer to your question
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
\Rightarrow2√ab / a + b = 4/5 \Rightarrowa + 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
\Rightarrow2√a/2√b = 3 + 1/3 – 1, -3 + 1/ -3 – 1
\Rightarrow√a/√b = 2, 1/2 \Rightarrowa/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 \Rightarrow2√m /m + 1 = 4/5 \Rightarrow5√m = 2m + 2
\Rightarrow2m - 5√m + 2 = 0 \Rightarrow(√m – 2) (√m – 1/2) = 0
\Rightarrowm = 4 or 1/4 ∴ req. ratio = 4 : 1 or 1 : 4.

Thanks
Deepak Patra
askIITians Faculty
Arun Kumar IIT Delhi
askIITians Faculty 256 Points
7 years ago
Hello Student,
Please solve and then square the possible value to get the answer.
Thanks & Regards
Arun Kumar
Btech, IIT Delhi
Askiitians Faculty
manas yendluri
24 Points
3 years ago
i dont know if this is valid but if i am wrong plz do correct me2√ab / a + b = 4/5in this step we can compare the ratios.. hence 2√ab=4 and a + b =5... solving these and substituting will give u either 4:1 or 1:4

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