If two positive numbers are in the ratio 3+2 2 : 3-2 2 , then the ratio between their A.M (arithmetic mean) and G.M (geometric mean) is

Step-by-step solution

Let the two positive numbers be a and b . Given the ratio a:b = (3+2 2 ) : (3-2 2 ) . Let a = k(3+2 2 ) and b = k(3-2 2 ) for some constant k > 0 . The Arithmetic Mean (A.M.) is given by A = a+b 2 = k(3+2 2 + 3-2 2 ) 2 = 6k 2 = 3k . The Geometric Mean (G.M.) is given by G = ab = k^2(3+2 2 )(3-2 2 ) . Using the identity (x+y)(x-y) = x^2 - y^2 , we have (3+2 2 )(3-2 2 ) = 3^2 - (2 2 )^2 = 9 - 8 = 1 . Thus, G = k^2 1 = k . The ratio of A.M. to G.M. is A G = 3k k = 3:1 . Answer: 3: 1