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

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

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

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Other answers:

bt the ans wer given is 1/4
@satellite: you forgot to square the term 5xy.
And also the harmonic mean is 2/(1/x+1/y)
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

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