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There are two numbers. If the arithmetic mean between the numbers is twice the geometric mean. Find the ratio of the numbers.

(Note: If the numbers are x and y, then you must find x:y. )

#NumberTheory

Note by इश्वर बस्याल
3 years, 4 months ago

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Comments

The arithmetic mean of xx and yy will be x+y2\frac{x+y}{2}, and the geometric mean of xx and yy will be xy\sqrt{xy}. So we have:

x+y2=2xy\frac{x+y}{2}=2\sqrt{xy}

x+y=4xyx+y=4\sqrt{xy}

x+y=16xyx+y=\sqrt{16xy}

x2+2xy+y2=16xyx^2+2xy+y^2=16xy

x2+y2=14xyx^2+y^2=14xy

x2=14xyy2x^2=14xy-y^2

x2=y(14xy)x^2=y(14x-y)

x2y=14xy\frac{x^2}{y}=14x-y

xy=14xyx\frac{x}{y}=\frac{14x-y}{x}

xy=14yx\frac{x}{y}=14-\frac{y}{x}

Thus, x:yx:y is equal to 14yx14-\frac{y}{x}

Did I do that right?

David Stiff - 3 years, 4 months ago

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The ratio should not contain variables.In your answer, it has variables x,yx,y.

Hint: Solve the quadratic equation

Vilakshan Gupta - 3 years, 4 months ago

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Oh, okay. Then x:yx:y would equal 7+4313.937+4\sqrt{3}\approx13.93, right?

David Stiff - 3 years, 4 months ago

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@David Stiff Yes, but there is a one more answer to this.

Vilakshan Gupta - 3 years, 4 months ago

@David Stiff Hmm that is correct. But I have mentioned the ratio of the numbers which means all the ratios. I hope you get the jest.

इश्वर बस्याल - 3 years, 4 months ago

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@इश्वर बस्याल That was silly...

x:y=7±43x:y=7\color{#D61F06}{\pm}\color{#333333}{4\sqrt{3}}

David Stiff - 3 years, 4 months ago
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