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<li>Find all positive integers $x$, $y$ and $z$ such that: $$x +\cfrac{1}{y + \cfrac{1}{z}} = N = \frac{10}{7}$$</li>
<li>Show that when $N=10/7$ is replaced by $N=8/5$ it is impossible to find positive integer values of $x$, $y$ and $z$ for which the finite continued fraction on the left hand side is equal to $N$. Find another fraction (rational number) $N$ for which the same is true.</li>
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<li>The key here is that $x$ has to be the integer part of $N$ because the 'continued fraction' part of the expression gives a value less than one.<br></br>
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As $y$ and $z$ are positive integers (whole numbers), $y + 1/z &gt; 1$ and $1/(y+1/z) &lt; 1$ so we know that this must equal $3/7$ and $x = 1$.<br></br>
<br></br>
Hence $y + 1/z = 7/3$. Again $y$ has to be the integer part of $7/3$ so $y = 2$ and $z = 3$.<br></br>
</li>
<li>As in the first part, if $N = 8/5$, then we must have $x = 1$ and $y + 1/z = 5/3$.<br></br>
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To make $y$ and $z$ positive integers we must have $1/z &lt; 1$ and $y = 1$.<br></br>
<br></br>
It then follows that $1/z = 2/3$ so it is impossible to find positive integer values for $x$, $y$ and $z$.</li>
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<h3><span style="font-weight: bold;">Why do this problem?</span></h3>
For experience of reasoning about the integer part of a number and working with fractions.<br></br>
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<h3><span style="font-weight: bold;">Possible approach</span></h3>
Challenge the students to invent their own problems of this type.<br></br>
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<h3><span style="font-weight: bold;">Key question</span></h3>
What is the integer part of $N$?<br></br>
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What is the integer part of $N$?<br></br></mdoxml>
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Not Continued Fractions
Which rational numbers cannot be written in the form x + 1/(y +
1/z) where x, y and z are integers?
Fractions, Decimals, Percentages, Ratio and Proportion
Continued fractions
Fractions, Decimals, Percentages, Ratio and Proportion
Calculating with fractions
Algebra
Inequality/inequalities