Hello again, King_nic!

2) Five kilometers upstream from his starting point,

a rower passed a raft flowing with the current.

He rowed upstream for one more hour and then rowed back

and reached his starting point at the same time as the raft.

Find the speed of the current. Let $\displaystyle b$ = speed of the boat in still water (km/hr).

Let $\displaystyle c$ = speed of the current (km/hr). Code:

C b-c B 5 A
* - ← - ← - * - ← - ← - ← - ← - - *
* - → - → - * - → - → - → - → - - *
* - → - → - → - → - - *
B 5 A

Going upstream, the boat's speed is $\displaystyle (b-c)$ km/hr.

It started at $\displaystyle A$, went 5 km to $\displaystyle B$, where it met the floating log.

It continued upstream for an hour, traveling $\displaystyle (b-c)$ km to $\displaystyle C.$

Going downstream, the boat's speed is $\displaystyle (b+c)$ km/hr.

It went downstream $\displaystyle (b-c)$ km to $\displaystyle B.$

. . This took: .$\displaystyle \frac{b-c}{b+c}$ hours.

Then it went 5 km to $\displaystyle A.$

. . This took: .$\displaystyle \frac{5}{b+c}$ hours.

Since it met the log, the boat traveled for: .$\displaystyle 1 + \frac{b-c}{b+c} + \frac{5}{b+c}$ hours. .[1]

During this same time, the log went 5 km at $\displaystyle c$ km/hr.

. . This took: .$\displaystyle \frac{5}{c}$ hours. .[2]

Equate [1] and [2]: .$\displaystyle 1 + \frac{b-c}{b+c} + \frac{5}{b+c} \;=\;\frac{5}{c}$

Multiply by $\displaystyle c(b-c)(b+c)\!:\;\;c(b-c)(b+c) + c(b-c)^2 + 5c(b-c) \;=\;5(b-c)(b+c)$

. . which simplifies to: .$\displaystyle 2b^2c - 2bc^2 + 5bc - 5b^2 \:=\:0$

Since $\displaystyle b \neq 0$, divide by $\displaystyle b\!:\;\;2bc - 2c^2 - 5b + 5c\:=\:0$

Factor: .$\displaystyle 2c(b - c) -5(b-c) \:=\:0$

Factor: .$\displaystyle (b-c)(2c-5) \:=\:0$

If $\displaystyle b-c \:=\:0$, then: .$\displaystyle b = c$

The speed of the current equals the speed of the boat.

. . Then the boat could __not__ have gone upstream at all.

Therefore: .$\displaystyle 2c - 5 \:=\:0 \quad\Rightarrow\quad c \:=\:\frac{5}{2}$

The speed of the current is 2.5 km/hr.