Explanation:
A Given,
Harmonic mean \(=14 \frac{2}{5}\), Geometric mean \(=24\)
Let, the numbers are \(a\) and \(b\).
Then, Harmonic mean \(=\frac{2 a b}{a+b}\)
So, \(14 \frac{2}{5}=\frac{2 a b}{a+b} \Rightarrow \frac{72}{5}=\frac{2 a b}{a+b}\)
Now, geometric mean, \(\mathrm{G}=\sqrt{\mathrm{ab}}\)
\(24=\sqrt{\mathrm{ab}}\)
\(\mathrm{ab}=(24)^2\)
\(\mathrm{b}=\frac{(24)^2}{\mathrm{a}}\)
On putting the value \(a b\) in equation (i) -
\(\frac{72}{5}=\frac{2 \times 24 \times 24}{a+b}\)
\(\Rightarrow \frac{3}{5}=\frac{2 \times 24}{a+b}\)
\(a+b=2 \times 8 \times 5\)
\(\Rightarrow a+b=80\)
On putting the value of \(b\) in equation (iii),
\(a+\frac{24 \times 24}{a}=80 \Rightarrow \quad a^2+576=80 a\)
\(a^2-80 a+576=0 \Rightarrow a^2-72 a-8 a+576=0\)
\(a(a-72)-8(a-72)=0\)
\((a-8)(a-72)=0\)
\(a=8,72\)
\(\text { So, greater number is } 72 .\)So, greater number is 72 .
