Permutations and Combinations

Permutations and Combinations

Question

The A.M. between two positive numbers is 34 and their GM. is 16. Find the numbers.

Answer

Let a and b be two numbers
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#6 {main}</pre>                              ...(i)
       G.M. = 16 rightwards double arrow space square root of ab space equals space 16 space space space space rightwards double arrow space ab space equals space 256                                  ...(ii)
From (i), b = 68 - a
Using in (ii), we get
                     a(68 - a) = 256        rightwards double arrow            68 straight a minus straight a squared space equals space 256
rightwards double arrow   straight a squared minus 68 straight a plus 256 equals 0               rightwards double arrow    straight a squared minus 64 straight a minus 4 straight a plus 256 equals 0
rightwards double arrow   a(a - 64) - 4 (a - 64) = 0         rightwards double arrow        (a - 64) (a - 4) = 0
rightwards double arrow                                a = 4, 64
When                              a = 4,         b = 68 - 4 = 64
and                                a = 64,         b = 68 - 64 = 4  

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