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Lewis Carroll posed the following problem:

Two travelers spent from 2 o’clock until 9 walking along a level road up a hill and home again; their pace on the level being $x$ miles per hour, uphill $y$ mph, and downhill $2y$ mph. Find the distance walked.

In Carroll’s formulation $x$ and $y$ were given integers. Making use of the additional assumption that the original problem was solvable, find the distance walked.

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  • $\begingroup$ Suppose we have $a$ miles of level road and $b$ miles uphill (or downhill). Write down an equation involving variables $x,y,a,b$ and using the fact that total time $=7$ hours. $\endgroup$ – Lozenges Mar 10 at 18:44
  • $\begingroup$ You seem to be introducing a variant of "Knot I" (first chapter) from the problem collection A Tangled Tale. $\endgroup$ – hardmath Mar 10 at 19:26
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Let $a$ be the length of the level section in miles

Let $b$ be the length of the hill section in miles

Then $d=2(a+b)$ is the total distance walked

The total time taken is 7 hours, which allows us to set up the equation ...

$$ \frac{2a}x +\frac b{y}+\frac b{2y}=7 \\ \frac 2x a+\frac 3{2y} \bigg( \frac d2-a \bigg)=7 \\ \bigg(\frac 2x- \frac 3{2y} \bigg) a+ \bigg( \frac 3{4y} \bigg ) d=7 $$

So the problem will be solvable with $d=\frac{28y}3 $provided that $$\bigg(\frac 2x- \frac 3{2y} \bigg) =0 \implies 4y=3x $$

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