Maths · Relation between A.M and G.M
If the ratio of HM and GM of two quantities is then the ratio of the number is ?
If the ratio of HM and GM of two quantities is \( 12: 13, \) then the ratio of the number is ?
- A. 1: 2
- B. 2: 3
- C. 3: 4
- D. None of these
Step-by-step solution
Let the two numbers be a and b. Then HM = 2ab/(a+b), GM = √(ab). Given HM/GM = 12/13, so 2√(ab)/(a+b) = 12/13 => (a+b)/√(ab) = 13/6. Let r = √(a/b), then r + 1/r = 13/6 => 6r² - 13r + 6 = 0 => (2r-3)(3r-2)=0 => r = 3/2 or 2/3 => a/b = r² = 9/4 or 4/9. Thus the ratio is 9:4 or 4:9, which is not among the given options.