Grade upto college level Algebra

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

Shane Macguire

12 Years agoGrade upto college level

3 Answers

Deepak Patra

12 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

$\Rightarrow$ 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
askIITians Faculty

Arun Kumar

12 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
Askiitians Faculty

manas yendluri

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

Shane Macguire

3 Answers

Deepak Patra

Arun Kumar

manas yendluri

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