Let the harmonic mean and geometric mean of two positive numbers be in the ratio 4:5 Thenthe ratio numbers - Maths - Sequences and Series

Question

L e t   t h e   h a r m o n i c   m e a n   a n
d   g e o m e t r i c     m e a n   o f  
t w o   p o s i t i v e   n u m b e r s   b e  
i n   t h e   r a t i o   4 : 5 T h e n t h e  
r a t i o   n u m b e r s   a r e r   i n   t h
e   r a t i o ( 1 ) 1 : 4 ( 2 ) 1 : 2 ( 3 ) 4 : 5 ( 4 ) n o n
e   o f   t h e s e

Punit Sharma · Accepted Answer

Dear Student,

Let the two terms as a,bas harmonic mean=2aba+band geometric mean=absince the ratio of harmonic mean and geomtric mean is 4:5hence, 2aba+b:ab=4:5or, 2aba+bab=45or, 2ab(a+b)ab=45or, 2(ab×ab)(a+b)ab=45or, 2aba+b=45or, aba+b=25or, 52=a+bab=aab+bab=a×aa×b+b×ba×b=ab+baLet, ab=x, hence,52=x+1xor, 52=x2+1xor, 2x2+2=5xor, 2x2-5x+2=0or, 2x2-4x-x+2=0or, 2x(x-2)-1(x-2)=0or, (2x-1)(x-2)=0hence, x= 2 ; 12as
 x=abHence, either ab=2 ; or ab=12thus, we get,ab=4 or 14 or, a:b=4:1or, a:b=1:4

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L e t t h e h a r m o n i c m e a n a n d g e o m e t r i c m e a n o f t w o p o s i t i v e n u m b e r s b e i n t h e r a t i o 4 : 5 . T h e n t h e r a t i o n u m b e r s a r e r i n t h e r a t i o ( 1 ) 1 : 4 ( 2 ) 1 : 2 ( 3 ) 4 : 5 ( 4 ) n o n e o f t h e s e

$L e t t h e h a r m o n i c m e a n a n d g e o m e t r i c m e a n o f t w o p o s i t i v e n u m b e r s b e i n t h e r a t i o 4 : 5 . T h e n t h e r a t i o n u m b e r s a r e r i n t h e r a t i o (1) 1 : 4 (2) 1 : 2 (3) 4 : 5 (4) n o n e o f t h e s e$