Question

The difference between two numbers is 48 and the difference between arithmetic mean and their geometric mean is 18. Then, the greater of two numbers is:
(1) 96
(2)60
(3)54
(4)49.

Answer 1

Hint: Since in a given problem the difference between two numbers and difference between arithmetic mean and their geometric mean is given, we can solve the given problem by using the formula of arithmetic mean and geometric mean.

Complete step-by-step solution:
Let the two numbers be x and y
Now it is given that the difference between the numbers is 48 therefore we can write
\[x - y = 48 - - - \left( a \right)\]
We can write \[x - y = \left( {\sqrt x - \sqrt y } \right)\left( {\sqrt x + \sqrt y } \right)\]with this the above equation becomes
\[\left( {\sqrt x - \sqrt y } \right)\left( {\sqrt x + \sqrt y } \right) = 48 - - - \left( 1 \right)\]
We know that the formula for
arithmetic mean of x and y is \[\dfrac{{x + y}}{2}\]
geometric mean of x and y is \[\sqrt {xy} \]
given that difference between arithmetic mean and their geometric mean is 18
therefore, we can write
\[\dfrac{{x + y}}{2} - \sqrt {xy} = 18\]
Take LCM and cross multiply we get
\[x + y - 2\sqrt {xy} = 36\]
LHS of the above equation can be written as
\[{\left( {\sqrt x - \sqrt y } \right)^2} = 36\]
On simplification we get
\[\sqrt x - \sqrt y = \pm 6 - - - \left( 2 \right)\]
Substitute equation 2 in 1 we get
\[\sqrt x + \sqrt y = \pm 8 - - - \left( 3 \right)\]
Now add equation 2 and 3 we get
\[\sqrt x + \sqrt y + \sqrt x - \sqrt y = 14\]
\[ \Rightarrow 2\sqrt x = 14\]
\[ \Rightarrow \sqrt x = 7\]
\[ \Rightarrow x = {7^2} = 49 - - - \left( 4 \right)\]
Now using 4 in equation (a) we get
\[y = 49 - 48 = 1\]
Therefore, \[x = 49\] and \[y = 1\] in which the greater value is 49
Therefore, the correct answer is option (4)49.


Note: In Mathematics, the geometric mean is the average value or mean which signifies the central tendency of the set of numbers by finding the product of their values.
 In other words, the geometric mean is defined as the nth root of the product of n numbers. It is noted that the geometric mean is different from the arithmetic mean. Because, in arithmetic means, we add the data values and then divide it by the total number of values. But in geometric mean, we multiply the given data values and then take the root with the radical index for the total number of data values.