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# Balancing Scales

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 \(x\,?\)

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