Excellent Ratio

Algebra Level 2

$\large \frac{a+b}{a}=\frac{a}{b}=x$

Given $a$ and $b$ are real numbers and $x$ is positive, find $x$ .

3.5

\frac{\sqrt{2}}{3}

\sqrt{\pi}

\frac{1+\sqrt{5}}{2}

1 solution

Carlos Martínez Quintero
Feb 18, 2016

Begin by simplifying the left fraction: $\frac{a+b}{a}=1+\frac{b}{a}=1+\frac{1}{x}.$

Therefore, $1+\frac{1}{x}=x.$

Multiply by x: $x+1=x^2$

Rearrange: $x^2-x-1=0$

Use the quadratic formula: $x=\frac{1+\sqrt{5}}{2}$

(note: Remember that, at the beginning, x is a positive ratio, so, you add).

The golden ratio!

Swapnil Das - 5 years, 2 months ago