1. ## A mathematical adventure down the river...!

Q Tom was floating down the river on a raft, when 1 km lower down, Michael took to the water in a rowing boat. Michael rowed downstream at his fastest pace. Then he turned around and rowed back, arriving at his starting point just as Tom drifted by. If Michael's rowing speed in still water is ten times the speed of the current in the river, what distance had Michael covered before he turned his boat around?

Okay... First I drew a diagram, and defined some variables...

$V_{M} =$Michael's rowing velocity in still water

$V_{C} =$Velocity of the current in the river

$T_{1} =$Time taken for Michael to cover x km downstream

$T_{2} =$Time taken for Michael to cover x km upstream

$V_{T} =$Time taken for Tom to cover 1 km

We know $V_{M} = 10V_{C}$. We also know that $S = \frac{D}{T}$

Also, the velocity of the current will affect Michael's rowing velocity. So, Michael's speed downstream will be $V_{M} + V_{C} = 11V_{C}$, and $V_{M} - V_{C} = 9V_{C}$ when he turns around and moves upstream.

So, I got three equations, as follows:

$11V_{C} = \frac{x}{T_{1}}...[1]$

$9V_{C} = \frac{x}{T_{2}}...[2]$

For the third equation, I used the fact that $T_{1} + T_{2}$ equals the time taken for Tom to cover 1 km. So:

$T_{1} + T_{2} = \frac{1}{V_{T}}...[3]$

$\Rightarrow\frac{x}{11V_{C}} + \frac{x}{9V_{C}} = \frac{1}{V_{T}}$

$\Rightarrow\frac{9V_{C}x + 11V_{C}x}{99V_{C}^{2}} = \frac{1}{V_{T}}$

There are too many variables... How do I proceed? Or am I on the wrong track... and there's an easier way to get this done?

Help!

Thanks.

ILoveMaths07.

2. let $v$ = current speed
$10v$ = rowing speed
$t_1$ = time downstream for the rower
$t_2$ = time back upstream for the rower

for the "floater" ...

$v(t_1+t_2) = 1 \, km$

for the rower ...

$11v(t_1) - 9v(t_2) = 0 \, km$

from the rower's equation ...

$\frac{11}{9}t_1 = t_2$

sub into the floater's equation ...

$v\left(t_1 + \frac{11}{9}t_1\right) = 1$

$vt_1 = \frac{9}{20}$

rower's downstream distance = $11vt_1 = \frac{99}{20} \, km$