Question

If the arithmetic mean of two numbers a and b, a>b>0, is five times their geometric mean, then $\dfrac{{a + b}}{{a - b}}$ is equal to:
A $\dfrac{{7\sqrt 3 }}{{12}}$
B $\dfrac{{2\sqrt 2 }}{4}$
C $\dfrac{{\sqrt 6 }}{2}$
D $\dfrac{{5\sqrt 6 }}{{12}}$

Answer 1

Hint: In this question, we have two numbers a and b such that their arithmetic mean is 5 times their geometric mean. So, we need to calculate the value of $\dfrac{{a + b}}{{a - b}}$. For that we know the formula of arithmetic mean: $\dfrac{{a + b}}{2}$ and that of geometric mean : $\sqrt {ab} $, then according to the according using the relation that arithmetic mean is 5 times geometric mean, with which we can easily find the value of $\dfrac{{a + b}}{{a - b}}$.

Complete step by step answer:
We have been provided with two numbers a and b,
So, firstly we need to find the arithmetic mean using the formula: $\dfrac{{a + b}}{2}$,
So, the arithmetic mean= $\dfrac{{a + b}}{2}$,
Now we will be finding the geometric mean using the formula: $\sqrt {ab} $,
So, the geometric mean = $\sqrt {ab} $,
Now we have been provided with a relation that arithmetic mean is 5 times the geometric mean,
So, the equation would be: $\dfrac{{a + b}}{2} = 5\sqrt {ab} $,
Solving it further: $a + b = 10\sqrt {ab} $,
Now, we will be squaring both sides: ${(a + b)^2} = 100ab$,
Now we will be finding the value of: ${(a - b)^2} = {(a + b)^2} - 4ab = 96ab$,
Now, as we need to find the value of $\dfrac{{a + b}}{{a - b}}$:$\sqrt {{{\left( {\dfrac{{a + b}}{{a - b}}} \right)}^2}} = \sqrt {\dfrac{{{{(a + b)}^2}}}{{{{(a - b)}^2}}}} $,
The value of $\dfrac{{a + b}}{{a - b}}$ would come out to be: $\sqrt {\dfrac{{100}}{{96}}} = \dfrac{{5\sqrt 6 }}{{12}}$.

Therefore, we can say that option (d) is correct.

Note:
In this question, ${(a - b)^2}$ needs to be found out by using the formula: ${(a - b)^2} = {(a + b)^2} - 4ab = 96ab$ so that we can put this value in $\sqrt {{{\left( {\dfrac{{a + b}}{{a - b}}} \right)}^2}} $ to find out $\dfrac{{a + b}}{{a - b}}$. Also, the formula of arithmetic mean and geometric mean must be used correctly. If we use other techniques, then the solution might get difficult to solve, so for such types of questions, this approach can be used.