Question

The arithmetic mean of two numbers is 6 and their geometric mean is $G$ and harmonic mean is $H$ and they satisfy the relation ${G^2} + 3H = 48$. Then which of the following is correct?
A. ${G^2} = 32$
B. $H = \dfrac{{16}}{3}$
C. $G = 16$
D. $G = 64$

Answer 1

Hint: The formula of arithmetic mean of two numbers is $\dfrac{{a + b}}{2}$, geometric mean of two numbers is $\sqrt {ab}$ and harmonic mean of two numbers is $\dfrac{{2ab}}{{a + b}}$. Apply these formulas, and then use the given conditions to find the required value.

Complete step-by-step solution:
Given that the arithmetic mean of two numbers is 6.

We have also given that the geometric mean of these two numbers is $G$ and harmonic mean is $H$ and they satisfy the relation ${G^2} + 3H = 48$.

Let us assume that the two numbers are $a$ and $b$.

We know that the arithmetic mean is the sum of collection of numbers divided by the count of numbers in the collection.

Since there are only two numbers, we have given that the formula to find the arithmetic mean of two numbers is $\dfrac{{a + b}}{2}$.

Using this formula to find the arithmetic mean, we get

$\dfrac{{a + b}}{2} = 6$

Multiplying both sides in the above equation by 2, we get

$$\Rightarrow 2\left( {\dfrac{{a + b}}{2}} \right) = 6 \times 2 \\\ \Rightarrow a + b = 12 \\\$$

We also know that the formula to find the geometric mean of two numbers is $\sqrt {ab}$.

Using this formula to find the geometric mean, we get

$G = \sqrt {ab}$

Squaring both sides in the above equation, we get

${G^2} = ab$
We know that the formula to find the harmonic mean of two numbers is $\dfrac{{2ab}}{{a + b}}$.

Thus, we have $H = \dfrac{{2ab}}{{a + b}}$.
Substituting the above value of ${G^2}$ and $H$ in the given relation ${G^2} + 3H = 48$, we get

$$\Rightarrow ab + 3H = 48 \\\ \Rightarrow ab + 3 \times \dfrac{{2ab}}{{a + b}} = 48 \\\ \Rightarrow ab + \dfrac{{ab}}{2} = 48 \\\ \Rightarrow \dfrac{{2ab + ab}}{2} = 48 \\\ \Rightarrow \dfrac{{3ab}}{2} = 48 \\\ \Rightarrow 3ab = 96 \\\ \Rightarrow ab = \dfrac{{96}}{3} \\\ \Rightarrow ab = 32 \\\$$

Substituting the above value in the equation ${G^2} = ab$, we get

${G^2} = 32$

Now, we will substitute this value of ${G^2}$ in ${G^2} + 3H = 48$.

$$\Rightarrow 32 + 3H = 48 \\\ \Rightarrow 3H = 48 - 32 \\\ \Rightarrow 3H = 16 \\\ \Rightarrow H = \dfrac{{16}}{3} \\\$$

Therefore, $H = \dfrac{{16}}{3}$.

Hence, the option B is correct.

Note: In solving these types of questions, you should be familiar with the formulas of arithmetic mean of two numbers, geometric mean of two numbers and harmonic mean of two numbers. Then use the given conditions and values given in the question, and substitute in the formulas, to find the required values. Also, we are supposed to write the values properly to avoid any miscalculation.