The sum of two numbers is 20 and there Geometric mean is 20% lower than their arithmetic mean .Find the ratio of the numbers?kindly post solution as well

1) 4:1
2) 9:1
3) 1:1
4) 17:3

• Let the two numbers be x and y
  Their sum is 20 ---> x + y = 20 .... (1)
  AM = (x+y)/2 and GM = √xy
  Their GM is 20% less than their AM
   (x+y)/2-(x+y)/10=√xy
  ⇒ 4(x+y)/10=√xy
  ⇒(4×20)/10=√xy
  ⇒8=√xy
  ⇒xy=64
  (x-y)^2=(x+y)^2-4xy
  ⇒(x-y)^2=400-256
  ⇒(x-y)^2=144
   x – y = 12 ………… (2)
  Solving equations (1) and (2)
  We get x = 16 and y = 4
  Two numbers are 16 and 4.
  
  
  
• 12 years agoHelpfull: Yes(12) No(3)
• a+b = 20
  (a+b)/2 = 10
  ab = 8^2 = 64
  a,b = 16,4 or 4,16
  see options --> 4:1
• 12 years agoHelpfull: Yes(9) No(5)
• 1 more method.....
  Sum of both numbers is 20
  So if one is x, then the other is (20-x).......... (1)
  GM = SQRT ((20-x).x)....... (2)
  AM = 1/2 (20) = 10.......... (3)
  
  Now , GM is 20% lower than AM ... which means GM is 80% of AM
  GM = 80% of 10 = 8 ..... from (3)
  
  So, 8 = SQRT ((20-x).x)... from (2) & above
  Squaring both sides, we get quadratic equation as follows:
  x ^2 - 20x + 64 = 0
  Solving, we get values as 16 & 4
  Ratio is 4:1... so answer = 1
• 12 years agoHelpfull: Yes(5) No(0)
• ans is 4:1
  suppose the numbers are 16 and 4. their sum is 20 and arithmetic mean is 10 and GM is 8 and by solving the we get that gm is 205 less than am
• 11 years agoHelpfull: Yes(1) No(0)
• x=16,y=4,
  Ans.1) 4:1
  G.M=(16*4)^1/2=8,
  A.M=16+4/2=10,
  G.M is 20% lower than A.M
  Therefore,ratio is 16/4=4:1
• 12 years agoHelpfull: Yes(0) No(0)
• Let the ratio is k:1, one is kx other x.
  Therefore (kx+x)=20(This condition is redundant)
  ((kx+x)/2)*(4/5)= sqrt.k*x
  2k - 5*sqrt.k+2=0 This implies sqrt.k=2or1/2. That is k:1=4:1 or 1:4
• 11 years agoHelpfull: Yes(0) No(0)
• Ans is 4:1
• 11 years agoHelpfull: Yes(0) No(0)
• 4:1 is the ratio and the numbers are 16 and 4.
  These satisfies all the conditions mentioned.
• 9 years agoHelpfull: Yes(0) No(0)