Find two numbers whose arithmetic mean is 12.5 and geometric mean is 10.

(a) 20, 4
(b) 15, 10
(c) 20, 5
(d) None of these

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    Answer: (c) 20, 5 

    Solution:-
    Let the two numbers be X and Y. 
    The arithmetic mean of two numbers is 12.5 and geometric mean is 10. 
    Then, (X + Y)/2 = 12.5  &  sqrt.(X Y) = 10
     X + Y = 12.5 x 2 &  X Y = 100
    X + Y = 25  &  X Y = 100
    X = 25 – Y  …………… (1) 
    Then, (Y – 25) Y = 100 
    Y^2 – 25 Y = 100
    Y^2 – 25 Y – 100 = 0 
    Y^2 – 20 Y –  5 Y – 100 = 0 
    Y (Y – 20)  – 5 (Y –  20) = 0 
    (Y – 5) (Y – 20) = 0 
    Y – 5 = 0  & Y – 20 = 0 
    Y = 5  & Y = 20 

    Put the value of Y in equation (1), 
    Then, X = 25 – 5 = 20. 

    Then, the two numbers are 20 and 5. 

    Hence, the correct answer is option (c) 20, 5. 

    Anurag Mishra Professor Answered on 4th May 2016.
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