Motorboat question

[Mi0]

I am struggling with this problem. It's from Gelfand's Algebra.

 

[MarkFL]

I would let $c$ be the speed of the current, and $v$ be the speed of the boat where there is no current and $d$ be the distance from $A$ to $B$...then we have:

[MATH]d=(v+c)a[/MATH]
[MATH]d=(v-c)b[/MATH]
Now, we are asked to find the time $t$ where:

[MATH]t=\frac{d}{v}[/MATH]
From the first two equations above, we find:

[MATH](v+c)a=(v-c)b[/MATH]
[MATH]v=\frac{c(a+b)}{b-a}[/MATH]
Hence:

[MATH]d=\left(\frac{c(a+b)}{b-a}-c\right)b[/MATH]
[MATH]d=\frac{2abc}{b-a}[/MATH]
Can you wrap it up now?

 

[MarkFL]

To follow up:

[MATH]t=\frac{\dfrac{2abc}{b-a}}{\dfrac{c(a+b)}{b-a}}=\frac{2ab}{a+b}[/MATH]