Question

The harmonic mean between two numbers is $14\frac{2}{5}$and the geometric mean is $24$. The greater number between them is

• A. $72$
• B. $54$
• C. $36$
• D. None of these

Answer 1

The correct option is A

$72$

The explanation for the correct answer.

Find the greater number.

Given that: Harmonic mean of two numbers is $14\frac{2}{5}$, and the Geometric mean is $24$.

Let $\mathrm{a}$ and $\mathrm{b}$ be the numbers.

Geometric Mean $=\sqrt{\mathrm{ab}}$

$$\begin{gathered} \mathrm{GM}=\sqrt{\mathrm{ab}} \\ 24=\sqrt{\mathrm{ab}} \\ 576=\mathrm{ab} \\ \frac{576}{\mathrm{a}}=\mathrm{b} \end{gathered}$$

Harmonic mean $=\frac{2\mathrm{ab}}{\mathrm{a}+\mathrm{b}}$

$$\begin{gathered} \Rightarrow \mathrm{HM}=\frac{2\mathrm{ab}}{\mathrm{a}+\mathrm{b}} \\ \Rightarrow 14\frac{2}{5}=\frac{2\mathrm{ab}}{\mathrm{a}+\mathrm{b}} \\ \Rightarrow \frac{72}{5}=\frac{2\mathrm{ab}}{\mathrm{a}+\mathrm{b}} \\ \Rightarrow \mathrm{a}+\mathrm{b}=\frac{10\mathrm{ab}}{72} \\ \Rightarrow \mathrm{a}+\mathrm{b}=\frac{10\times 576}{72}(\mathrm{GM}=576) \\ \Rightarrow \mathrm{a}+\mathrm{b}=80 \\ \Rightarrow \mathrm{a}+\frac{576}{\mathrm{a}}=80 \\ \Rightarrow {\mathrm{a}}^2+576=80\mathrm{a} \\ \Rightarrow {\mathrm{a}}^2-80\mathrm{a}=576 \\ \Rightarrow {\mathrm{a}}^2-80\mathrm{a}-576=0 \\ \Rightarrow (\mathrm{a}-8)(\mathrm{a}-72)=0 \\ \Rightarrow \mathrm{a}=8,\mathrm{a}=72 \end{gathered}$$

Greater number is $72$.

Hence option $\left(a\right)$ is the correct answer.