A crew can row a certain course upstream in 84 minutes;
they can row the same course down stream in 9 minutes less than
they can row it in still water: how long would they take to row
down with the stream.
The equations I came up with are:
$$\frac{d}{x – y} = 84$$
$$\frac{d}{x + y} = \frac{d}{x} – 9$$
Where $d$ is the distance covered in their course, $x$ is the rate of the people rowing the boat, and $y$ is the rate of the stream.
I tried substituting $d$ with $84(x -y)$, and that rendered the equation $3x^2 – 25xy + 28y^2 = 0$. As far as I know, this equation isn't solvable. Any attempts I make to obtain other equations or results leads to this same equation.
If there's any alternate method to solving this, I would love to know.
Thanks.
Best Answer
That equation is perfectly solvable. You're implicitly assuming $x$ and $y$ are non-negative from the way you've structured your equations, so you can fix a non-negative $y$ and try and figure out if there exists a positive solution $x$. That is, if we fix $y$, then
$$3x^2-75yx+28y^2=0$$
is a regular old quadratic equation in $x$. The solution will be some formula involving $y$, and you can check for which $y$ that formula gives a positive real number.