# Does this question provide enough information to answer?

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.

• 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. – Lozenges Mar 10 at 18:44
• You seem to be introducing a variant of "Knot I" (first chapter) from the problem collection A Tangled Tale. – hardmath Mar 10 at 19:26

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
$$\frac{2a}x +\frac b{y}+\frac b{2y}=7 \\ \frac 2x a+\frac 3{2y} \bigg( \frac d2-a \bigg)=7 \\ \bigg(\frac 2x- \frac 3{2y} \bigg) a+ \bigg( \frac 3{4y} \bigg ) d=7$$
So the problem will be solvable with $$d=\frac{28y}3$$provided that $$\bigg(\frac 2x- \frac 3{2y} \bigg) =0 \implies 4y=3x$$