Excellent Ratio

Algebra Level 2

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

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

3.5 3.5 2 3 \frac{\sqrt{2}}{3} π \sqrt{\pi} 1 + 5 2 \frac{1+\sqrt{5}}{2}

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1 solution

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

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

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

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

Use the quadratic formula: x = 1 + 5 2 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

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