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$$\frac{l+w}{l}=\frac{l}{w}$$

Find $$\frac{l}{w}$$, your answer should be the golden ratio that is a fraction with radicals in simplest form.

(BTW, the answer is definitely NOT $$\frac{l+w}{l}$$)

This is a fun math problem I solved yesterday night.

Apr 28, 2019
edited by CalculatorUser  Apr 28, 2019

#1
0

multiply both by lw

l2+w2=l2​^w

subtract l2

w2=w

subtract w

w=0

??

Apr 29, 2019
#2
+2415
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You incorrectly multiplied by lw,

it should be lw+w^2=l^2

You also CAN'T subtract l^2

CalculatorUser  Apr 29, 2019
#3
+23342
+2

$$\dfrac{l+w}{l}=\dfrac{l}{w}$$

I assume:

$$\text{Let }\textbf{Golden Ratio} =\varphi$$

$$\begin{array}{|rcll|} \hline \dfrac{l+w}{l} &=& \dfrac{l}{w} \\\\ \dfrac{l}{l}+ \dfrac{w}{l} &=& \dfrac{l}{w} \\\\ 1 + \dfrac{w}{l} &=& \dfrac{l}{w} \quad | \quad \dfrac{l}{w} = \varphi,\ \dfrac{w}{l} = \dfrac{1}{\varphi} \\\\ 1 + \dfrac{1}{\varphi} &=& \varphi \quad |\quad \cdot \varphi \\\\ \varphi + 1 &=& \varphi^2 \\ \varphi^2 -\varphi - 1 &=& 0 \\\\ \varphi &=& \dfrac{1\pm \sqrt{1-4\cdot (-1) }}{2} \\ \varphi &=& \dfrac{1\pm \sqrt{5}}{2} \\\\ \mathbf{\varphi} &\mathbf{=}& \mathbf{\dfrac{1+ \sqrt{5}}{2}} \\ \hline \end{array}$$

Apr 29, 2019