In the figure below, the rectangle has dimensions a x b and is tiled by squares. This is the smallest possible rectangle that can be tiled by squares in this manner.
a) write the continued fraction for $a/b$
b) find the value of $a+b$
Can someone please explain how i'd break this up? For part a, I am thinking that $b=6x$
$x=4y$
So $b=24y$
Is this correct? I've never seen a question like this before.
I don't know how to solve this, can someone please help? thanks
For a rectangle with side lengths $a$ and $b$, with $a\gt b$, a square dissection of this form can be converted to a simple continued fraction for $a/b$ by counting the number of squares at each stage, in decreasing size, and using those values in the continued fraction.
(Note: a simple continued fraction is one where the numerators are all $1$.)
We have:
$\color{red}1\;$ white square
$\color{red}5\;$ light grey squares
$\color{red}1\;$ black square
$\color{red}4\;$ dark grey squares
Which leads to the following continued fraction: $$\frac{a}{b} = \color{red}1 + \cfrac{1}{\color{red}5 + \cfrac{1}{\color{red}1 + \cfrac{1}{\color{red}4}}}$$
To calculate $a$ and $b$ in this instance it's easy enough to evaluate the continued fraction:
$$\frac{a}{b} = 1 + \cfrac{1}{5 + \cfrac{1}{1 + \cfrac{1}{4}}} = 1 + \cfrac{1}{5 + \cfrac{1}{\frac{5}{4}}} = 1 + \cfrac{1}{5 + \cfrac{4}{5}} = 1 + \cfrac{1}{\frac{29}{5}} = 1 + \frac{5}{29} = \frac{34}{29}$$
So we have $a/b = 34/29$, and hence $a+b=63$.