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Question

The difference between two numbers is 48 and the difference between their arithmetic mean and their geometric mean is 18. Then, the greater of two numbers is


A

96

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B

60

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C

54

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D

49

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Solution

The correct option is D

49


Explanation for the correct option :

Step-1 : Assumption

Let the two numbers be x and y where x>y.

Then their arithmetic mean will be A.M.=x+y2 and the geometric mean will be G.M.=xy.

Step-2 : Constructing two linear equations using two conditions given in the question.

Given that the difference of the two numbers is 48. So, we have

x-y=48 1

Also, given that the difference between their arithmetic mean and the geometric mean is 18. So, we obtain

x+y2-xy=18 2

Step-3 : Solving the two equations using the substitution method

From equation 1, we get x=y+48.

Substituting x=y+48 in 2, we get

y+48+y2-(y+48)y=18ā‡’y+24-18=(y+48)yā‡’y+6=y2+48yā‡’y+62=y2+48yā‡’y2+12y+36=y2+48yā‡’36y=36ā‡’y=1

Then,

x=y+48=1+48=49

āˆ“x=49,y=1 is the solution of 1 and 2.

So, the greater of the two numbers is 49

Hence, option (D) is the correct answer.


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