Question

the harmonic mean and geometric mean of two no. r in ratio 4:5.
find ratio of those two no.????????

Answer 1

i think i can work this out, but harmonic mean of two numbers is
\[\frac{1}{\frac{1}{x}+\frac{1}{y}}\] and geometric mean is \[\sqrt{xy}\] and if the ratio is 4:5 we know
\[4\sqrt{xy}=\frac{5}{\frac{1}{x}+\frac{1}{y}}\]

Answer 2

\[\frac{5}{\frac{1}{x}+\frac{1}{y}}=\frac{5xy}{x+y}\] so we get
\[4\sqrt{xy}=\frac{5xy}{x+y}\] now maybe square both sides to get
\[16xy=\frac{25xy}{(x+y)^2}\]
then '
\[\frac{16}{25}=\frac{1}{(x+y)^2}\]
\[\frac{5}{4}=x+y\]

Answer 3

probably a snappier way to do this.

Answer 4

bt the ans wer given is 1/4

Answer 5

@satellite: you forgot to square the term 5xy.

Answer 6

And also the harmonic mean is 2/(1/x+1/y)

Answer 7

So you get
\[16xy=\frac{100x^2y^2}{(x+y)^2}\]
\[16=\frac{100xy}{(x+y)^2}\]
\[4x^2+4y^2-17xy=0\]
\[4(y/x)^2-17(y/x)+4=0\]

Which gives y/x=4 or 1/4