Question

If the AM and GM for two numbers are 6.50 and 6 respectively then the two numbers are

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Answer 1

Answer:

If GM is 6 then the product of the two numbers is 36.

Other products that give 36 are 4 X 9, 3 x 12 and 2 x 18.

Among these that give the A.M. of 6.5 are 4 and 9.

Answer 2

Answer:

The two numbers are 4 and 9.

Step-by-step explanation:

Let the two numbers be x and y.

Given AM of two numbers = 6.5

          GM of two numbers = 6

    We know AM of x and y is (x+y)/2

                     GM of x and y is $\sqrt{xy}$

      By the given condition

                    (x+y)/2 = 6.5 ----------(i)

                and $\sqrt{xy}$ = 6     ---------(ii)

                   consider equation (i)

                    (x+y)/2 = 6.5

                         x+y = 6.5×2

                         x+y = 13 -----------(iii)

                  consider equation (ii)  

                         $\sqrt{xy}$ = 6

                 by squaring on both sides we get

                           xy = 6²

                           xy = 36

                             x = 36/y -------------(iv)

                 now consider equation (iii)

                          x+y = 13

                substitute x = 36/y in above equation

                   (36/y)+y = 13

                  (36+y²)/y = 13

                       36+y² = 13y

                 36+y²-13y = 0

                 y²-13y+36 = 0

              y²-9y-4y+36 = 0

              y(y-9)-4(y-9) = 0

                   (y-9)(y-4) = 0

                  y = 9 or y = 4

             substitute y = 9 in equation(iii)

                               x+y = 13

                               x+9 = 13

                                   x = 13-9

                                    x = 4

              substitute y = 4 in equation(iii)

                              x+y = 13

                              x+4 = 13

                                  x = 13-4

                                  x = 9

     When x = 9 then y = 4   and

                x = 4 then y = 9

      Hence, the two numbers are 4 and 9.

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