The ship left the harbor and is travelling $60$ kilometers downstream. Then it continues on the tributary river (upstream) $20$ kilometers. Travel took $7$ hours to finish. River speed is $1$ km/h. Calculate the speed of a ship.
$v$ is speed of the ship, then we write $v+1$ km/h when it's sailing downstream and $v-1$ km/h when it's sailing upstream? How to solve this as a quadratic equation?
Since you know that its speed going downstream is $v + 1$, and it traveled $60$ km downstream, can you express the time it took to travel downstream, in terms of $v$?
Next, you also know that it's speed going upstream is $v - 1$, and it traveled $20$ km upstream. Can you express the time it took to travel upstream, in terms of $v$?
Finally, the sum of those two times is given as 7 hours. That gives your equation to solve.
We begin by writing the equation, keeping in mind that $d = rt$ and the river can either add or take away from the boat's speed, depending on its direction. Is the boat is travelling downstream, the total rate is $r + 1,$ and if the boat is travelling upstream, its rate must be $r - 1.$ We have the equation, and we solve it as follows: $$\frac{60}{r + 1} + \frac{20}{r - 1} = 7$$ $$60(r - 1) + 20(r + 1) = 7(r^{2} - 1)$$ $$7r^{2} - 80r + 33 = 0.$$
We can solve this quadratic by factoring. We get that $(7r - 3)(r - 11) = 0.$ This yields $r = \frac{3}{7}$ and $r = 11.$ But notice that $r > 1$ (or else, the boat could not overcome the resistance of the water). Then we are left with the speed of the ship as $\boxed{11\frac{\text{km}}{\text{hr}}}.$
