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

1. anonymous

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 $4\sqrt{xy}=\frac{5}{\frac{1}{x}+\frac{1}{y}}$

2. anonymous

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

3. anonymous

probably a snappier way to do this.

4. anonymous

bt the ans wer given is 1/4

5. anonymous

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

6. anonymous

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

7. anonymous

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