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If the harmonic mean between two positive numbers is to their geometric mean as $12 : 13$, then the numbers could be in the ratio:

  1. $12: 13$ 
  2. $1/12: 1/13$
  3. $4:9$ 
  4. $2:3$
asked in Quantitative Aptitude by (4.8k points) 38 105 176 | 27 views

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We know :

a) Harmonic mean of two positive numbers a and b  =  2ab/(a + b)

b) Geometric mean of two positive numbers a and b  =  √ab

So given , H.M. / G.M.   =  12 / 13

==>   [2ab / (a+b)] / √ab     =     12 / 13

==>   √ab / (a+b)           =    6 / 13

==>   ab / (a+b)2           =    36 / 169

==>   a = 4k  and  b = 9k

Hence the ratio  a : b   =  4 : 9

Hence C) should be the correct option.

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