Mathematico | Mathematico

This shape is called a golden rectangle:

Golden rectangle

It has the property that

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

The value of $\frac{b}{a}$ is called $φ$ and is known as the golden ratio, a famous number.

Given that, in simplest form,

φ = \frac{p + \sqrt{q}}{r}, p, q, r \in N

Find $p + q + r$ .

Hint

Split the right side into two fractions:

\begin{aligned} \frac{b}{a} & = \frac{a + b}{b} \\ \frac{b}{a} & = \frac{a}{b} + \frac{b}{b} \end{aligned}

Now notice that

\frac{a}{b} = \frac{1}{(\frac{b}{a})}

Replace $\frac{b}{a}$ with $φ$ and solve.

Hint

You should get

φ = 1 + \frac{1}{φ}

Multiply both sides by $φ$ to get rid of the fraction and then use the quadratic formula.

Solution

By definition,

\begin{aligned} \frac{b}{a} & = \frac{a + b}{b} \\ \frac{b}{a} & = \frac{a}{b} + \frac{b}{b} \\ \frac{b}{a} & = \frac{1}{\frac{b}{a}} + 1 \\ φ & = \frac{1}{φ} + 1 \\ φ^{2} & = 1 + φ \\ φ^{2} - φ - 1 & = 0 \end{aligned}

We're now ready to use the formula

\begin{aligned} φ & = \frac{1 \pm \sqrt{1 - 4 (1) (- 1)}}{2} \\ = \frac{1 \pm \sqrt{5}}{2} \end{aligned}

However, $φ > 0$ because it is the ratio of two lengths. Therefore

φ = \frac{1 + \sqrt{5}}{2}