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# Continued Fractions

Let's look at another way to use fractions in balance puzzles and systems of equations. A continued fraction is a type of iterative fraction that can be either finite or infinite. The continued fraction below is infinite because the ellipsis (...) indicates that the pattern of arithmetic goes on forever.

$\large 1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{...}}}}$

Nevertheless, this expression describes a finite numerical value that we will solve for in this quiz!

For now, just make a prediction: approximately how big do you think the value of this expression is?

Now let's look at a simpler version of the iterative fraction that we just saw.

If both $$a$$ and $$b$$ are integers and $$0 < a < b,$$ which expression is larger? $\textbf{A. }\ 1 + \frac{1}{1 + \frac{1}{1 + a}}\qquad \textbf{B. }\ 1 + \frac{1}{1 + \frac{1}{1 + b}}$

Let's take another look at this infinite fraction and determine what it equals: $\large 1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{...}}}} =\, ?$ For now, let's assume that $$\large 1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{...}}}} = x.$$

Notice that the part of the expression in red is the same as the entire expression that equals $$x:$$ $\large 1 + \frac{1}{\color{red}{1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{...}}}}} = x.$ If we know that the red part of the fraction equals $$x,$$ is it possible to simplify our fraction, using substitution to this equation: $1 + \frac{1}{x} = x?$

Now we know that $\large 1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{...}}}} = x$ and $$1 + \frac{1}{x} = x.$$

What is the closest approximation of $$x\,?$$

What is the value of this expression? $\large \frac{6}{1 + \frac{6}{1 + \frac{6}{1 + \frac{6}{...}}}}$

If $$x, y,$$ and $$z$$ are positive integers, what is the value of $$y\,?$$ $\large x + \frac{1}{y + \frac{1}{z}} = \frac{10}{7}$

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