What are the 2 numbers with GM= 4 and HM=16/5?

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justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

Let the two numbers be A and B. Now it is given that their GM is 4.

This gives us sqrt (A*B) = 4

=> AB = 4^2 = 16

We also know that their HM is 16/5, so 2*AB / (A+B) = 16/5

=> 2*AB / (A+B) = 2*16 / (A+B) = 16/5

=> (A + B) = 2*16*5/16 = 10

=> A + B = 10

=> A = 10 - B

AB = (10 - B)*B = 16

=> 10B - B^2 = 16

=> B^2 - 10B + 16 = 0

=> B^2 - 8B - 2B + 16 = 0

=> B( B - 8) - 2(B -8) = 0

=> ( B - 2)(B - 8) = 0

=> B is 2 and 8

=> A is 8 and 2

The two numbers are 8 and 2.

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neela | High School Teacher | (Level 3) Valedictorian

Posted on

Let a and b be the two numbers.

Then the geometric mean (GM)of the numbers is (ab)^(1/2) and the harmonic mean( (HM)  of the numbers is  2/(1/a+1/b) = 1ab/(a+b).

Given that GM  and HM of two numbers = 4 and 16/5 respectively.

So (ab)^(1/2) = 4, Or ab = 16...(1)  And 2ab/(a+b) = 16/5....(2).

So we substitute ab = 16 in (2) and we get: 2*16/(a+b) = 16/5. Or a+b = 32*5/16 = 10.

(a-b)^2 = (a+b)^2- 4ab = (10)^2 - 4*16 = 36

Therefore  a+b = 10.....(3) and a-b= 6...(4).

(3)+(4): 2a = 10+6 = 16, so a= 16/2 = 8.

(3)- (4): 2b = 10-6 = 4. So  b = 4/2 = 2.

Therefore a = 8 and b = 2.

Therefore the two numbers are 8 and 2.

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